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Spatial patterns and maximum power in ecosystems

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Spatial patterns and maximum power in ecosystems
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SPATIAL PATTERNS AND MAXIMUM POWER IN ECOSYSTEMS


BY
JOHN R. RICHARDSON

















A DISSERTATION PRESENTED TO TAE GRADUATE SCHOOL OF THE
UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS OF THE DEGREE OF
DOCTOR OF PHILOSOPHY




UNIVERSITY OF FLORIDA

1988






























Copyright 1988

by

John R. Richardson









ACKNOWLEDGEMENTS


I 4ould like to express my sincere appreciation to Dr. Howard T. Odum, my committee chairman, for the insights and inspiration he gave during the completion of this work. Mis holistic views and open-mindedness provide an extremely fertile field to develop and pursue ideas in systems ecology. Other members of my committee (Drs. J.F. Alexander, G.R. Best, K.C. Ewel and C.L. Montague) provided useful feedback in class and with this project.

The support and patience of my wife Karen has

sustained me while my two children, Matthew and James, have provided joy and purpose for the completion of this dissertation.

Work was done in the Department of Environmental Engineering Sciences, University of Florida, and was supported by graduate research funding from the Graduate School of the University of Florida.


iii

















TABLE OF CONTENTS


ACKNOWLEDGEMENTS.........................................111

LIST OF FIGURES...... .....................................vi

ABSTRACT.................... .........-+.................. X

CHAPTER 1 INTRODUCTION..................................... 1

Historical Perspective................................... 2
Previous Models of Pulsing Patterns in Time and Space.2 Pattern Formation..... ................................3
Concepts of Pulsing, Patterns and Power.................13
Maximum Power in Systems.............................13
Design for Maximum Power.............................14
Pathway Configuration....... ......................... 15
Pulsing and Patterns in Ecosystems...................... 21
Succession and Disturbance...........................21
Edges....... ........... ......... .-- ....- ... 23
Hierarchies and Patches..............................24
Models............................... -... .................25
Gap Models and Patch Dynamics................ ..27
Spatial Systems and Models...........................28
Plan of Study......... ...... ............................32
Objectives...... .....................------. --- -.. 32
Data Site: Luquillo Rain Forest, Puerto Rico........33

CHAPTER 2 METHODS and MODELS..............................37

Simulation Procedures and Programs......................40
Simulation Models. ....................................... 41
Minimodel Tests......... .............................41
Pulse Model....... .............. ................ .....53
Pulse Model with Prey-Predator Sectors...............56
Spatial Model................. ............. .... ........59
Format for Spatial Graphs............................67
Measurement of Hierarchies at El Verde Site.............67

CHAPTER 3 RESULTS........ ................................. 71

Simulation of Three Path Model.......................... 71
Individual Pathway Tests.............................71
Frequency Studies........... .............. .............86
Simulation of Parallel Production-Consumption Model.....91
Single Run Simulations.......... .....................91
Multiple Run Simulations.............................. 100


iv









Initial Conditions and Total Energy Use ............. 112
Simulation of Pulse Model .............................. 117
Single Run Simulations..............................117
Multiple Run Simulations.............................123
Simulation of Pulse Model with Prey-Predator Sectors.. .133 Simulation of the Ring Model............................141
Simulation of Two Dimensional Surface Models...........151
Rain Forest Gaps and Hierarchies.......................164
Size Class Distributions.............................164
Gap Size Measurements...............................171
Comparison to Models................................171

CHAPTER 4 DISCUSSION.....................................178

Maximum Power Considerations...........................179
Power and Feedback With Paths of Higher Order.......179 Effect of Hierarchies on Performance................181
Power Used as a Function of Input Power.............132
Threshold for Stable Feedbacks and Pulsing..........182
Implications for Succession.............................184
Role of Individual Units.............................184
Succession and Pulsing.............................185
Spatial Pattern formation..............................186
Synchronous vs. Asynchronous Systems................136
Coupling of Spatial Units by Diffusion Processes... .187 Organization by Higher Level Consumers..............188
Power Use and Edge Effects...........................190
General Principles.....................................190

APPENDIX.................................................192

BIBLIOGRAPHY.............................................247

BIOGRAPHICAL SKETCH......................................254











LIST OF FIGURES
Figure Page

1 Spatial patterns based on chemical reaction
mechanisms........................................6

2 Hilborn (1979) model..............................8

3 The spatial development of cells based on
simple r-pentamino initial condition............12

4 Basic autocatalytic model with flow-limited
energy source...................................17

5 Basic multiple path model with three input
pathways representing differing feedback
regimes, linear, autocatalytic, and quadratic...20

6 Size class distribution of gaps formed in
tropical forest at Barro Colorado (Brokaw
1982)............................................26

7 Mite predator prey experiment (Huffaker
1958)............................................30

8 Size class distribution over time of plot of
trees in tropical forest at El Verde (Crow
1980)............................................35

9 Energy circuit language symbols (Odum 1983).....39

10 Three pathway model used to test effects of
various energy inputs on kinetic mechanisms.....43

11 Three pathway model with multiple drain pathways............................................47

12 Three pathway model with individual competing
units having single input pathways similar to
combined model..................................49

13 Parallel production-consumption model............52

14 Pulse model of tropical forest ecosystem
model...........................................55

15 Pulse model with additional prey-predator
sector...........................................58

16 Number of edge and center cells as a function
of total number of cells in a given square
area............................................62


4










17 Cell geometries possible for spatial models.....64

13 Format of spatial model display graphs..........69

19 Steady state power utilization of units in
the three path model (Figure 10) as a
function of input power (JO).....................73

20 Energy utilization of individual components
in the three path model in Figure 10............75

21 Steady state energy flows on various pathways
and combinations of pathways in the three
path model (Figure 10) as a function of input
power (JO).......................................77

22 Simulation of three path model in Figure 10.....80

23 Simulation of three path model with multiple
drain pathways in Figure 11. Percent power
used as a function of energy input (JO).........82

24 Simulation of three path competition model
with various pathways enabled (Figure 12).......85

25 Simulation of the three path model in Figure
10. Percent power used as a function of
frequency of the input power (JO=500)............83

26 Simulation of the three path model in Figure
10. Percent power used as a function of
frequency of the input power (JO=2000)...........90

27 Simulation of the three path model in Figure
10. Percent power used as a function of
frequency of the input power (JO=10000)..........93

28 Simulation of the parallel production-consumption
model in Figure 13. Model base run............96

29 Simulation of the parallel production-consumption
model in Figure 13..............................99

30 Simulation of the parallel production-consumption
model in Figure 13. Multiple simulations of
the model with available power increasing
from 50 to 300.................................102

31 Simulation of the parallel production-consumption
model in Figure 13. Multiple simulations of
the model with percent power used for entire
run vs input power..............................104


vii









32 Simulation of the parallel production-consumption
model in Figure 13. Run with available power
increasing from 50 to 300 and the initial
value of the consumer (Q4) equal to 50 (10x
base run in Figure 28)..........................107

33 Simulation of the parallel production-consumption
model in Figure 13. Multiple simulations of the model with available power held constant
(JO=100, base run value) and the initial
value of the consumer (Q4) varied from
1 to 6.........................................109

34 Simulation of the parallel production-consumption
model in Figure 13. Total percent power used
for entire run as a function of the initial
value of the consumer (Q4).....................111

35 Simulation of the parallel production-consumption
model in Figure 13. The initial value of weed
species (Q3) was varied from 0 to .5 and
input power was held constant (J0=100, base
run value)......................................114

36 Steady state values of percent power used as
a function of input energy and state variable
initial conditions for multiple simulation
runs of parallel production -consumption model
(Figure 13)......................................116

37 Simulation for pulse model (Figure 14) with
base run coefficients.........................119

38 Simulation of pulse model (Figure 14) without
a quadratic pathway (K7, K8, K9 = 0.0)..........122

39 Simulation of pulse model (Figure 14)
without a feedbacks into Q4 (K6, K8 = 0.0).....125

40 Multi-run simulation of the pulse model
(Figure 14) with variation in input energy.
(JO varied from 0 to 250).......................127

41 Multi-run simulation of pulse model (Figure
14) with variation in total carbon in model... .130

42 Multi-run simulation of pulse model (Figure
14) with variation is turnover time of
pulsing consumer. (K12 varied from
.01 to .5)......................................132


viii









43 Multi-run simulation of pulse model (Figure
14) with variation in quadratic pathway (K9
varied from 0.5E-6 to 0.53E-5 with K7 and K8
varied proportionately).........................135

44 Multi-run simulation of pulse model (Figure
14) with variation in linear pathway (Kl
varied from 0.0 to 0.12E-2 and K5 and KS
varied proportionately) with quadratic
pathway held at zero............................137

45 Simulation of pulse model with prey-predator
sectors (Figure 15).............................139

46 Simulation of pulse model (Figures 14 and 18)
with cells in a linear ring and diffusion
between consumers of each cell in ring
(DK=.1). Initial conditions of consumers
were set to near zero except for one "seed"
consumer at lower left corner of matrix which
was set to 100.................................143

47 Simulation of pulse model (Figures 14 and 18)
with cells in a linear ring but without diffusion. Initial conditions of producers and
consumers were set to random distribution
around ring....................................146

48 Simulation of pulse model (Figures 14 and 18)
with cells in a linear ring and a high level
of diffusion between consumers of each cell (DK=.1) and random distribution of producers
and consumers around ring......................148

49 Simulation of pulse model (Figures 14 and 18)
with cells in a linear ring and a low level of diffusion between consumers of each cell
(DK=.001) and random distribution of
producers and consumers around ring............150

50 Simulation of pulse model (Figure 14) with
cells arranged in two dimensions (Figure 18) without diffusion and with a constant energy
source.......................................... 153

51 Simulation of pulse model (Figure 14) with
cells arranged in two dimensions (Figure 18).
Energy source hierarchically is distributed
from center outward and no diffusion between
cells............................................155


ix








52 Simulation of pulse model (Figure 14) with
cells arranged in two dimensions (Figure 18).
Energy source is hierarchically distributed from center outward and diffusion is between
consumers of each cell (DK=.001)................158

53 Simulation of pulse model (Figure 14) with
cells arranged in two dimensions (Figure 18).
Energy source is randomly distributed and
diffusion is between consumers of each cell
(DK=.001).......................................160

54 Simulation of pulse model (Figure 14) with
cells arranged in two dimensions (Figure 18).
Energy source is randomly distributed and
diffusion between nutrient storages (Q4) of
each cell is set to high level (DK=.1)..........163

55 Simulation of the pulse model (Figure 14)
with cells arranged in two dimensions (Figure 18). Moving consumer model with search length set to one cell, no diffusion and hierarchical energy distribution........................166

56 Simulation of the pulse model (Figure 14)
with cells arranged in two dimensions (Figure 18). Moving consumer model with search length
set to five cells, no diffusion and
hierarchial energy distribution................168

57 Size class distribution of trees at El Verde
radiation site (November 1964)..................170

58 Size distribution of Cecropia gaps in tropical rainforest at El Verde.....................173

59 Size distribution of gaps in tropical rainforest pulsing model simulation (Figures 14
and 18) at time =760...........................177

60 Character set for displaying spatial graphs
on GIGI computer terminal for use with
screencopy to printer..........................196
















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

SPATIAL PATTERNS AND MAXIMUM POWER IN ECOSYSTEMS by

John R. Richardson

April, 1988

Chairman: Howard T. Odum
Major Department: Environmental Engineering Sciences

Studies of dynamic systems have shown that oscillations in time and space are related, both being generated by nonlinear, pulsing behavior that is derived from the mathematics of energy processing. Similar mathematics exist in chaos theory, bifurcation theory, and catastrophe theory.

Production-consumption models that simulate pulsing properties of ecological systems are of this class. This dissertation examines the spatial patterns and energetics of

autocatalytic and pulsing models as a paradigm for ecological and general systems. Configurations were tested with steady or varying resource availability for ability of the

model systems to maximize power as the criterion for utility and success. The spatial distribution of gaps generated by

simulations was compared to that observed in rain forests.

Models studied included (a) aggregated, singlecompartment autocatalytic designs; (b) parallel productionconsumption design; (c) production-consumption-recycle xi










designs; and (d) multiple cell spatial models each with a unit model but interconnected in different ways.

Models with autocatalytic feedbacks utilized more

power than the same models with only linear pathways. Percent power used increased with increasing available power.

Production-consumption models show multiple steady states with pulsing behavior as a transition between two steady states. Localized maxima of power use occur during pulsing but the overall power use is related to input power.

Spatial patterns of production and consumption in

spatial models were related to input energy patterns, the degree of connectivity between the individual cells in the model, and the hierarchical level of intercell connections. Large variations in patterns were accompanied with small changes in power utilized.

Edges of a spatial system can act as a source or sink for energy depending on the relationship between available energy inside and outside the boundaries and the degree of connectivity along the edges.

Basic autocatalytic production-consumption-recycle

models with different spatial conditions organize different spatial patterns while generating near total utilization of available power. The wide variety of spatial patterns results from dynamic adaptations for maximizing power for different spatial conditions. The simulation results resemble patterns in nature often attributed to random indeterminancy.


xii















CHAPTER 1

INTRODUCTION



Ecosystems develop patterns in time and space. Some of these patterns are generated by pulsing oscillatory processes. What sorts of interactions, organization and structure in an ecosystem lead to pulsing behavior, and how does this behavior affect the use of energy? What types of spatial patterns develop when ecosystems are influenced by pulsing in time and space? What are the energy implications of different pattern forming processes in ecosystems? What are the effects of pulsing on succession, competition, frequency response of producers and consumers, and coupling with external pulses?

This dissertation uses general systems models to analyze the effects of pulsing on pattern formation and overall power use as systems develop, build structure and organize in time and space. Simulation models using general systems principles and based on real ecosystems were used to test the role of pulsing behavior of consumers in organizing ecosystems over time and space. Data from a tropical ecosystem were used to calibrate pulsing and spatial models.


I






2


Historical Perspective


Previous Models of Pulsing Patterns in Time and Space

In many fields from chemistry, physics, and biology to astronomy, there are a variety of models, methods and techniques to describe and study systems that have discontinuities or other rapid fluctuations in their behavior. Some of these are catastrophe theory (Thom 1975), bifurcation theory, synergetics (Haken 1977a,1977b,1979), dynamical system theory (Rosen 1970), chaos and order (Prigogine 1980,1984, and Schaffer and Kot 1985), pulsing (Lotka 1920 and Odum 1982), pattern recognition, and morphogenesis (Meinhardt 1982). In all of these, processes being described are parts of nonlinear thermodynamically open systems. Energy constraints on these types of systems have not previously been well studied.

In the past, efforts to describe systems using classical thermodynamics centered on closed systems near equilibrium or open systems near steady state. In such systems, available energy is small. These approaches using equilibrium thermodynamics could not account for the behavior of many systems (Odum 1983, Prigogine 1984, Schaffer and Kot 1985).

Data with statistical anomalies are often difficult to analyze and methods are sometimes used to minimize fluctuations (Platt and Denman 1975). Systems that have aperiodic behavior, a great deal of noise, or time dependent changes in variance are not well suited to the normal










statistical methods. These 'unusual events' can be important in understanding how a system works (Weatherhead .1986).

Frequency analysis has been used for some time to study periodic behavior of systems (Platt and Denman 1975, and Emanuel, West and Shugart 1978). Fourier transformations decompose the output or behavior of a system into an additive series of sinusoidal processes. The variance is partitioned into a set of frequencies that when combined gives the output being measured. Aperiodic behavior or systems with known nonlinear components may also be studied with these techniques, but the results are often not useful. Some nonlinear systems with behavior described as 'chaotic' have frequency domain variance as noisy as the time domain variance (Abraham and Shaw 1984a, 1984b). Pattern Formation

Patterns in natural systems range from the smallest

molecular patterns of motion to the placement of the stars and galaxies in the universe. One of the most intriguing aspects of pattern formation is the similarity of patterns at differing time scales and sizes. From a systems point of view this would lead one to suspect that the processes are similar at each scale.

Chemically reacting systems give rise to various types of patterns (Bray 1921, Nicolis and Prigogine 1969, Winfree 1973, Haken 1977a, 1977b). The Belousov-Zhabotinski reaction, which makes fascinating patterns, is a simple









oxidation-reduction reaction involving malonic acid, bromate and a cerium catalyst (Winfree 1973). An example of the time and spatial development of this reaction is shown in Figure la.

Morphological development in biological systems has

been studied and modeled by Meinhardt (1982). Patterns form when autocatalytic growth in a system is combined with lateral inhibition (negative spatial feedbacks). Once autocatalytic activity starts, there must be a longer range negative feedback (spatial inhibition of the spread of this autocatalysis) or the whole system will pulse in a burst of autocatalytic consumption. This sets up spatial chemical gradients along which morphogenesis is thought to occur (Figure lb).

Hilborn (1979) experimented with predator-prey models based on an aquatic ecosystem. Hilborn's model had 100 spatial cells arranged in a linear chain with the ends connected to form a circle. Both predators and prey were allowed to diffuse across cell boundaries. The model was simulated with initial conditions set so that all cells had prey but only one cell had a predator. The model (Figure 2a) was allowed to iterate for 1000 time intervals, generating the pattern seen in Figure 2b. Further experiments showed that there was no tendency towards equilibrium in longer runs of the model.

The spatial development of insect eyes and insect legs has been modelled by Ransom (1981) using an autocatalytic
























Figure 1. Spatial patterns based on chemical reaction mechanisms.

(a) Spatial patterns generated by Belousev-Zhabotinski
chemical reaction (Prigogine 1980).


(b) Spatial patterns generated by simulation model
used to describe morphogenesis (Meinhard (1932).





0


a a a m m




- - - - - -


C04
























Figure 2. Hilborn's (1979) spatial model.

(a) Energy diagram of individual cell model

Equations for simulation model.

dX(i)=a*X(i) b*X(i)*X(i) (c*X(i)*Y(i)/(d+X(i)))
+h*X(i+l) +h*X(i-1) 2*h*X(i)

dY(i)=-e*Y(i) f*Y(i)*Y(i) + (g*X(i)*Y(i)/(d+X(i)))
+k*Y(i+l) +k*Y(i-l) 2*k*Y(i)

where i is the number of the subsystem in a linear loop.


(5) Simulation results of linear series of unit models
showing level of predator vs distance around loop.







S


0




x





















0



2












SPATIAL LOCATION









model. By allowing cells in the model to divide and migrate within given constraints, the model developed patterns similar to those in real insects. The model allowed simple random cell division with movement constrained to a hexagonal direction away from the center of the cell division.

Sergin (1978, 1979, 1980) studied the oscillatory behavior of long term climate variations using models that combine linear and nonlinear interactions of the heat capacities of the oceans and polar ice sheets. The period of the climatological events in these models is on the order of 10,000 to 100,000 years. The model of global temperatures varies in its behavior from steady state to oscillations based on small changes in areal coverage of continental ice sheets.

Pattern formation based on digital, rule based systems has been used to model biological systems. Examples such as cellular automata (Turing 1952 and Wolfram 1984) and a 'game of life' (Gardner 1970 and Poundstone 1985) generate complex spatial patterns from simple rules. The 'game of life' is generated on an N x N matrix where

1. Every active cell with two or three neighboring

cells survives to the next generation.

2. Each active cell with four or more neighbors

'dies' from overpopulation. Every active cell with one or no neighbors 'dies' from isolation.

3. Each empty cell adjacent to three 'live' neighbors

gives birth to a new cell.






10


Figure 3 is an example of the patterns generated from a simple five cell seed (R-pentomino) during 512 iterations. This pattern stabilizes (no more deaths and no more births) after 1103 iterations, although it is an oscillating steady state. Individual subsets of the final stable pattern oscillate.

The 'game of life' model has some of the features of autocatalysis (or cooperative behavior). Two or three live cells are required for survival or birth of new cells. It also has the feature of diffusive inhibition because individual cells that move out from a population center can become isolated and die. This rule-based system has no energy constraint that governs development and thus gives no energy basis for pattern formation.

The common theme that runs through these examples is one of combined interactions of autocatalytic growth with some form of inhibition, diffusion or other mechanism for preventing the autocatalytic growth from spreading too rapidly. A concept that is sometimes misunderstood or misinterpreted is that the terms fluctuation (Prigogine 1980, 1984) and bifurcation theory (Pacault 1977) refer to a change in the kinetics of reacting components of a system. This change in kinetics gives rise to the oscillations or pulses in the output.

The models in this dissertation also use combinations of autocatalytic and diffusion (linear) pathways to study

























Figure 3. The development of spatial patterns among cells based on simple r-pentamino initial condition (a) in 'a game of Life' simulation.

(a) Time = 9 (b) Time = 8
(c) Time = 16 (d) Time = 32 (e) Time = 64
(f) Time = 128 (g) Time = 256 (h) Time = 512








12


cl


I.


h






13


the possible mechanisms and energy consequences of pattern formation in ecosystems.



Concepts of Pulsing, Patterns and Power


Maximum Power in Systems

Although in the last century Podalinsky, Ostwald and Boltzman suggested energy use controlled system performance (Martinez-Alier 1987), Lotka (1922) made a more definitive statement. He stated that evolution proceeded in such a direction as to make the total energy flux through the system a maximum compatible with the constraints on the system. He related this to Ostwald's (1892) idea of all possible energy transformations, that one takes place which brings about the maximum transformation in a given time.

A theory of minimum entropy generation was put forth by Prigogine (Prigogine and Wiaume 1946) that a system evolved toward a stationary state characterized by the minimum entropy production compatible with the constraints on the system. He has since called this a failure and probably a special case of systems near equilibrium (Prigogine 1984). Prigogine (1978, 1980, 1982; Prigogine and Stengers 1984) now deals with systems far from equilibrium that have dynamic and oscillatory behavior. He has not postulated any definite theory about the energetic consequences of these types of systems.

Odum and Pinkerton (1955) proposed that natural systems tend to operate at that efficiency which produces a






14


maximum power output, a general restatement of Lotka's original idea of maximum energy flow but with an important distinction. Odum (1971, 1982, 1983a, and 1983b) further clarifies maximum power as useful power where 'use' is feedback of the product of energy use to amplify other pathways.

In describing cycles of life, death and regeneration, Calow (1978) has found that although Lotka's principle holds, there seem to be no a priori grounds for placing restrictions on how this use of energy should be achieved. He further stated that selection would have shifted in the course of time from one of maximizing speed to maximizing efficiency. This is a restatement of the strategy of ecosystem development utilizing r and K growth (Odum 1969).

Jantsch (1980) suggests than maximum engagement in

matter (i.e., energy storage) and maximum process intensity (i.e., entropy production) are criteria for ecosystem stability. Non-equilibrium structures thus come about by fluctuations in the mechanisms which result in modifications of the kinetic behavior of these structures. Design for Maximum Power

The important question here is how do systems build structure in order to maximize utilization of available power. Odum's theory (1971 and 1983) is that by feeding back energy (derived from structure that is being built) reinforcement occurs that increases efficiencies and energy






15


flow into the structure. Mechanisms must develop that build structure to capture the most energy possible. These feedback structures then have a prior energy use embodied in them (emergy, after embodied energy, of a structure has been defined as the total amount of energy used in developing these structures (Odum 1983 and 1986)). This dissertation looks at some of the possible kinetic pathways that feed back to process energy and the energetics of these pathways. Pathway Configuration

A simple model demonstrates several ways in which useful power can be increased (Figure 4, see description of symbols in Figure 9). This model is a single storage with autocatalytic production drawing on a flow-limited energy source (an energy source with constraints on the pathway, limiting the amount of energy that can be delivered).

The efficiency of a pathway can be increased if less energy is fed back to gain more energy. For a simple autocatalytic system (Figure 4a and 4b) this can be done by either using less energy to gain the same inflow (changing the value of K2 in the model) or by increasing the inflow for the same feedback (increasing K1 while concurrently decreasing K3). Because there are thermodynamic limits on any process, it may not be possible to improve designs to increase energy flows beyond thermodynamic limits.

The first law of thermodynamics requires the conservation of energy. This implies the following constraint on the production process of the model (Figure 4).
























Figure 4. Basic autocatalytic model with flow-limited energy source.

(a) Diagram with kinetic terms

dQ = Kl*JR*Q K2*JR*Q K4*Q
JR = JO / (l+KO*Q)

(b) Diagram with flow terms

RO = XO*JR*Q Rl = Kl*JR*Q R2 = K2*JR*Q R3 = K3*JR*Q
R4 = K4*Q





















Energy ~ source



























Energy so urce



JR


K2*JR*Q Storage



KO*JFWQ K Q

K4*0


K3*JR*Q



















R2 Storage



RO R1

//R4
R 3\\


17






18


K0*Jr*Q + K2*Jr*Q = Kl*Jr*Q + K3*Jr*Q (1)

Substitute R (flow) terms as abbreviations for terms in equation (1):

RO + R2 = R1 + R3 (2)

Inputs of energy of any process must equal the outputs. Efficiency is defined as:

Efficiency = (Output of useful power)/Inputs or in terms of our equation:

Efficiency = Rl/(RO+R2) (3)

where R3 is waste heat generated in the process (required by the second law of thermodynamics). Because R3 cannot be zero, there is a natural upper limit to the efficiency of any process.

Another method to increase energy flow from a flow

limited source is to have multiple pathways capture available energy, each effective at a different energy level (Figure 5). Multiple pathways (Jl,J2,J3) use stored energy to build structures to capture available energy. A linear, donor-controlled pathway (Jl) requires little structure and employs no feedback in order to capture energy, but has severe limitations (its efficiency cannot change) due to the dependency on the energy source. An autocatalytic pathway

(J2) feeds back embodied energy (structure built by the system) to draw in more energy. The quadratic pathway (J3) is a co-operative phenomenon in which the structure of the system is interacting with itself to feed back embodied energy to draw in more power. A system that develops such
























Figure 5. Basic multiple path model. Three input pathways represent different feedback regimes: linear (Jl), autocatalytic (J2), and quadratic (J3).
































Energy JO
source


20


x






K4*0






21


higher order feedback pathways may exhibit a greater rate of use of available energy.

This added quadratic pathway is available to utilize any energy left after the efficiency is raised to the upper limit for the autocatalytic pathway. This is a mechanism that can draw in energy that would normally be unavailable to the system. The quadratic pathway may have a high cost to develop and maintain this pathway but it enhances overall use of that extra energy by the whole system. This may give a competitive edge in some circumstances over systems without higher order pathways, particularly when available energy may be fluctuating. Available power will be increased by switching from one pathway to the other depending on the energy source. Some pathways are more efficient at low energy levels while others are more efficient at high energy levels, thus allowing such.systems to efficiently utilize fluctuating power sources.


Pulsing and Patterns in Ecosystems


Succession and Disturbance

Any climax state is eventually interrupted by disturbances that generate patches in which succession is reinitiated. The gaps in a forest may be generated by local outbreaks of consumers within the forest, tree mortality, or outside disturbances such as fires, hurricanes, volcanic activity, and landslides (Runkle 1985). The role of the landslide as a gap-forming mechanism has been described in






22


both temperate forests (Oliver 1981, and Veblin 1985) and tropical forests (Garwood 1979, and Leigh et al. 1982).

Disturbances (i.e., pulses) to an ecosystem can be generated from within or can come from outside the boundaries of an ecosystem and may vary in frequency and amplitude. The ability of an ecosystem to utilize available resources and adapt to these disturbances depends on the storages, structures and interactions within an ecosystem (Odum 1983). Hierarchical mechanisms may develop that capture and process energy at various levels and result in utilization of energy over a wider variety of input levels. Some mechanisms of interaction between parts of the ecosystem were studied in this dissertation to understand how systems may converge energy transformations and feedback controls to organize for higher productivity.

No unified theory of succession presented to date can be regarded as widely accepted (Anderson 1986). Horn (1976) wrote 'The sweeping generalization that can be safely made about succession is that it shows a bewildering variety of patterns.' Even the definitions of succession are widely varying. In this dissertation succession is regarded as a dynamic process in which the composition of an ecosystem changes through time, building structure and processing energy. This process eventually stabilizes in a climax from which there is a regression or loss of that structure due to disease, fire, treefall or other events. Seeding from another ecosystem or from storages in the soil from the






23


previous ecosystem regenerates a facsimile of the original ecosystem through a sequence of unidirectional stages that reaches a steady state system called a climax. This climax may be arrested at some point and in some cases succession may cycle between several stages. This definition is broader than most but is an attempt to describe the whole process instead of the more narrow 'growth-phase' definition.

Regression from a climax state may occur in several ways. In some cases it comes about as a pulse of consumption from within the ecosystem boundaries such as treefalls, landslides or disease outbreaks. It can also come about from disturbances from larger outside events such as hurricanes or drought. The frequency and amplitude of these disturbances tend to be inversely correlated: larger disturbances occur less frequently than smaller ones. This phenomenon is referred to as a hierarchy of disturbances (Bennett and Chorley 1978). The interaction of these disturbances along with the internal fluctuations may lead to the 'bewildering variety of patterns' to which Horn refers. Edges

Ecosystems can generally be broken up into subsystems that have uniform characteristics. These subsystems have boundaries where the composition changes from one particular type to another. The development of these edges may occur where differing types of energy interact with ecosystem components to generate patches and zones of transition. The






24


presence of many spatially distributed patches may be due to the production-consumption pulsing of components in the ecosystem.


Hierarchies and Patches

The frequency of disturbance based on internal cycles has been shown to be from 200-500 years in a variety of ecosystems (Emanuel, West and Shugart 1978, Runkle 1985). Distribution of disturbances over time varies from fairly constant low amplitude disturbances to long-period, high amplitude disturbances. The successional changes due to disturbances may be related to the size and scale of the disturbance (Peet and Christensen 1980, Peet 1981).

Brokaw (1982a, 1982b, and 1985a) found a hierarchical distribution in gap sizes in a tropical rain forest at Barro Colorado Island (Figure 6a). The area per size class is plotted vs. the size class (Figure 6b). This relationship may be important in determining patch dynamics. Brown (1980) suggested that size class distributions may be related to the emergy per size class (the emergy per size class is also related to the area per class). Brokaw calculated the turnover rate for the forest, based on the gap formation, to be from 85 to 128 years depending on the minimum size of the lowest class used.


Models

The simulation models used to study ecosystem behavior generally fall into two classes (Shugart 1984). One of these
























Figure 6. Size class distribution of gaps formed in tropical forest at Barro Colorado (Brokaw 1982).

(a) Distribution of gaps by diameter of gap.

(b) Distribution of gaps by area in gap.








26


Gap Distribution
Barro Colorado 21 20
19
1 17

15 1 4
1
12

10

E
z

6 5






4 -












3 0
30 50 70 90 110 130 150 170 190 210 230 250 270 290 310 330 350 Sl.e Close (--)







Gap Distribution
Sorro Colorado





700




500


400



2 300


o 200






30 50 70 90 110130 150170 190 210 230 250 270 290310 330 350 Sloe Close (mrr)






27


is based on the nonlinear 'Lotka-Volterra equations' and generally does not include outside influences. The other uses forced linear systems of differential equations and does have inputs from outside the system. Neither of these methods typically contains any spatial considerations and both deal with systems near equilibrium. Systems near equilibrium tend to move toward that equilibrium and are characterized by spatial uniformity (Prigogine 1984 and Field 1985).

In this study, open non-equilibrium models are developed that combine non-linear and -oscillatory interactions between production and consumption with outside forcing functions that provide resource controls. A pulsing, hierarchical model of production and consumption is used to generalize about succession and regression. Spatial interactions generated by this model are studied to understand the energetic and kinetic basis for pattern formation in ecosytems.


Gap Models and Patch Dynamics

Several previous studies based ecosystem models on disturbance gaps. The JABOWA forest simulator model by Botkin, Janak and Wallis (1972) keeps track of the birth, growth, and death of a group of trees from seedlings on to maturity within a certain gap size. Subroutines are used for crowding, shading, and response to individual nutrients and energy sources. The simulation then allows the gap to






29


develop a distribution of trees based on all of the input parameters. These gap models generally do not account for any outside disturbances that generate gaps.

Various gap models (Phipps 1979, Shugart and West 1980, Shugart, Mortlock, Hopkins, and Burgess 1980, Shugart and Noble 1981, Doyle 1982, Doyle, Shugart, and West 1982, Shugart 1984, and Pickett and White 1985) have been utilized to study forested ecosystems around the world. These models have various gap sizes ranging from 100m^2 to 833m^2.



Spatial Systems and Models

A spatial predator-prey insect microcosm was used by Huffaker (1958) to study two species of mites. The prey mite fed on oranges while the predator mite fed on the prey. In one set of experiments, the oranges were distributed in a 10x12 grid with partial barriers between the oranges and one prey placed on each of the 120 oranges. Five days later 27 predators were dispersed on the oranges. The resulting dynamics in populations both over time (8 months) and space are shown in Figure 7. In other experiments with oranges in different arrangements, the oscillatory behavior was not seen. Huffaker concluded that the predator-prey oscillation would only occur when there was migration from the outside or a sufficiently complex spatial arrangement of prey and barriers to allow localized growth of the prey followed by consumption by the predator.












Figure 7. Mite predator prey experiment (Huffaker 1958).

(a) Spatial distribution.
Prey concentration is shown by intensity of small
blocks (darker is higher density) and predator
locations are marked with small circles.

(b) Time series of total predators and prey in spatial
area. Letters on graph refer to the time series
for the spatial display next to the letter.



















Prey: 0-5 nil density (white), 6-25 low density (light stipple); 26-75 medium density (horizontal lines); 76 or over, high density (solid black). Predator: 1-8 (one white circle).

g 0 -- -5*- - >-J- - L - C ----. ~ ~~ CI .
-7
S- :: -u



U G



ME -. 50 -- D

200 e 40
A | .-- 0 -. -=!

I, 1 e 2 g '.





1500 5 2000 0





So- 30






S 1 S 25 25 35 5 1 5 20 25 30 5 10 I 20 25 30 0 0 35 20 25 30 5 10 15 20 25 30 5 10 5 20 25 30 5 10 20 25 30 5 10 15 20 25
JULY AUGUST SEPTEMBER OCTOBER NOVEMBER DECEMBER JaNUR Y fEBRUARY
SEWMACUL T A USE( PREY) ----- T OCCIDE T AtIS 5= PRfDA TOA )






31


In high altitude balsam fir forests in the northeastern United States, waves of tree loss and regeneration are thought to be formed by an interaction of the prevailing wind with the larger mature trees that are exposed along the gap-wave (Sprugel and Bormann 1981, and Sprugel 1984). The wind in this case acts to organize the disturbance cycle that occurs normally in this type of forest into a spatial wave pattern instead of randomly occurring patches.

The 'ohi'a dieback phenomenon in the rain forests of

Hawaii (Mueller-Dombois 1980) is a case of localized loss of trees in the forest not due to disease or insect pest. It was postulated that the effects were due to local soil moisture loss arising from some climate instability. Reproduction of the 'ohi'a was adequate enough to regenerate the forest after the dieback, thus providing a way for this shade intolerant species to become the primary canopy species without further succession. Climatic variability was thus used to an adaptive advantage.

Spatial modelling of ecosystems can be done in several different ways. By using a model based on the FORET simulation model (Shugart and West 1977) and spatially distributing the output of the model according to flooding conditions and hydroperiod, Pearlstine, McKellar and Kitchens (1985) suggested possible species changes due to changes in the hydroperiod caused by a river diversion in South Carolina. In this case the number of individual subcell models was kept small and the spatial distribution was based






32


on a combination of terrain relief, hydrology, and correlated output from the simulation model.

Another approach to spatial modelling is to divide the area into individual cells with a representative model in each cell with some interaction terms among the individual cells. This is the approach Costanza (1979) used in modelling the economic development of South Florida.

Simulations with individual models for each cell have certain advantages, because the interaction of neighboring cells influences the outcome. A serious disadvantage where the number of cells is large is the immense amount of computer time required for the simulations. By making the cell size larger this can be avoided, but loss of spatial detail occurs as the cell size increases. The sub-cell distribution modelling technique used by Pearlstine et al. (1985) has just the opposite advantages and disadvantages. The time requirements for simulation do not necessarily increase as the area of cells is increased, but individual intercell interactions are lost.


Plan of Study

Objectives

This study of energy use and pattern formation with production consumption models has several parts:

First, the energetics of different pathway configurations were tested using a series of minimodels. These models were manipulated to determine the energy use of






33


systems with different production and consumption kinetics and different combinations of components.

Second, a generalize production-consumption minimodel calibrated with tropical rainforest data was used to study the energetics of pulsing behavior.

Third, spatial pattern formation was investigated using the pulsing production-consumption model as subunits in a spatially distributed format. The spatial effects and energy implications of various patterns of energy inputs, edges, and lateral connectivity were determined.

These spatial simulations included several types of inter-block exchange. Hierarchical relationships are represented in these models when each consumer component interacts with more than one producer unit. The distribution of gaps developed by simulations was compared with gaps in the tropical rainforest in Puerto Rico.

Finally, insights and hypotheses were developed about behavior of ecological systems. Data site: Luquillo Rainforest, Puerto Rico

Data from the Lower Montane Rainforest in the Luquillo Mountains of Puerto Rico were used to compare some of the spatial simulations of pulsing and patches. Extensive studies on this forest were published previously (Odum and Pigeon, 1970).

Changes in structure and composition of a plot of tropical rain forest near El Verde in Puerto Rico over a period of 30 years were reported by Crow (1980). Data






34


included size class distributions taken in 1943, 1946, 1951 and 1976 (Figure 8).. It can be seen that there is a shift over time in the different size classes. The peak year for the 0-8 cm class is 1946 while the peak in the 8-12 cm class occurs in 1951 and the peak in the next three size classes occurs in 1976. The smallest number in the lower two classes also occurs in 1976. The last severe hurricane struck

Puerto Rico in 1932, and this movement through the size classes appears to be the growth and development of an age class of trees that grew back after the hurricane. The hurricane in this case acts as an organizing disturbance to reset succession of patches on a large scale.

The models simulated include the main integrative mechanisms observed in ecosystems for coupling production and consumption of spatially distributed units. Energy use of these configurations was obtained from the simulations to test the hypothesis that commonly observed organizational designs with a successional regime that alternates production and consumption, tend to maximize system power in the long run.
























Figure 8. Size class distribution over time of plot of trees in tropical forest at El Verde (Crow 1980).






36


Distribution of trees over time


Ii


12-16 16-20 20-24 24-28 28-32 >32
Size Clams (diameter In cm) 1946 = 1951 1976


700-


600500 400300 200100


E
Z


O-8 8-12

2 1943
















CHAPTER 2

METHODS AND MODELS


Ecosystem concepts, configurations, and models were

represented with energy circuit language from which simulation programs were derived. The energy circuit language, developed by H. T. Odum (Odum 1971, Odum and Odum 1981 and Odum, 1983), is a symbolic language for modelling ecosystems and their components. Elements of storages, flows, and interactions in this symbolic language keep track of the laws of energy conservation. The energy diagrams also show the correct kinetic interaction between parts of the system. The level of aggregation or disaggregation that is needed to

understand and model a system for a particular purpose can be achieved by drawing and revising diagrams using this energy circuit language. A diagram of most of the important symbols with a brief description of each is presented in

Figure 9.

One of the benefits of using the energy circuit

language is that it is possible to go from a conceptual model to the development of the differential equations

needed to simulate the model in a few steps. Each of the pathways on the diagram represents a flow that in turn can be represented by terms in the differential equations that


37

























Figure 9. Energy circuit language symbols (Odum 1983).







39


S



--c-e


Saiichong action more switchig actions.


A symbol that indicates one or


Produ=e- Unit that collects and transforms
low-qualty-energy under control interactions of high-quaty flows.


Self-limticng energy receer (Chapter 10) A unit that has a self-luniting output when mput dnves are high because there is a limiting constant quantity of maternal reacting on a circular pathway within.



Box Miscellaneous symbol to use for whatever unit
or function is labeled.



Contat-.ga-a amplifier A unit that delivers
an output in proportion to the input I but change by a constant factor as long as the energy source S is suaficient.



Trsnsoorin A unit that indicates a sale of goots
or seices (solid ine) is exchange (or payment of money (dashed). Price is shown as an external source.


Enery, ctrrtc A pathway whose fow is proptrtional to the quantity in the storage or source upstream. Source Outside source of energy delivenng forces
according to a program controlled from outside; a forcing functon.


Tank A compartment of energy storage within the
system storing a quantity as the balance of inflows and outflows; a otato variable.

Heat sink Dispersion of potential energy into heat
that accompanies all real transformnation processes and storage; loin of potential enerU from further use by the system.

Internctow Interactive intersection of two pathways coupled to produce an outflow in proportion to a function of both. control action of one flow on another; limiting factor action; work gate.


Consumer Umt that transforms energy
qualty, stores it, and feeds it back autscatalyttcally to improve inflow.






40


describe the changes in storage compartment (tank) values over time.


Simulation Procedures and Programs


The majority of the simulations in this dissertation were done in FORTRAN-4-PLUS on a Digital Equipment Corporation (DEC) PDP 11/34 with RSX-llM operating system. The graphical outputs of the simulations were displayed on a DEC VK-100 graphics terminal (General Image Generator and Interpreter or GIGI) connected to a Barco color monitor and DEC .LA-34 Decwriter. The GIGI terminal has a 760x240 pixel resolution and can display up to eight colors on a color monitor. In order to facilitate the graphics programming needed in my simulation models, I developed a set of FORTRAN subroutines with a more natural calling sequence to execute the ReGIS (Remote Graphics Instruction Set use by the GIGI

terminal) commands from the programs. This library of routines (GGLIB) is listed and documented in the Appendix.

Some of the goals of this dissertation were to examine the structure and function of systems in time and space and to determine how variation in coefficients may affect energy flows and storages of the systems. Graphical display programs were developed to project a simulated 3-D surface of the output of various state variables over time and over a range of input conditions. A special 3-D graphics display

program was written to display the output of these model simulations (program PLOTZ, Appendix).






41


The spatial models are broken down into cells that show the concentration of a given parameter in the individual

cell as a color block. For display on the color monitor this provides dramatic views of the model changes over time and space. In order to make hardcopy printouts a display character set was designed so the density of the dots in an individual cell was correlated to the color of the cell.

This provided a way of screen-dumping the images to paper and achieving patterns on paper that were similar to the ones on the video screen (See Appendix for a listing of the

character set).



Simulation Models

Minimodel Tests

First a group of minimodels were simulated to relate energy use to basic pathway designs. Then spatial models

with these configurations were studied for energy use and pattern formation.


Three path minimodel

In order to understand how a system processes variable

energy inputs, builds structure, and regulates or maximizes energy flows, a simple single tank model was simulated. The model is similar to the one described by Odum (1982) that has parallel pathways of different types competing for available energy (Figure 10). The model has a flow limited

source connected to a single storage (tank) by three different pathways; a linear pathway (Jl), an autocatalytic
























Figure 10. Three pathway model ised to test effects of various energy inputs on kinetic mechanisms.


Linear input: Autocatalytic input: Quadratic input:

dQ=Jl+J2+J3-K4*Q R=J0-Jl-KO*R*Q-K5*R*Q*Q


Jl=Kl*R J2=K2*Q*R J3=K3*Q*Q*R































ENERGY SOURCE


K 4mQ


43






44


pathway (J2), and a quadratic pathway (J3). The tank has a

linear drain.

The model represents a system that can change its use of three functional pathways to get energy. The linear pathway represents the energy flow that a system can receive without any feedback in this pathway, only pathway resistance to the flow. Because it is a donor controlled pathway, the system has no control on the flow. Diffusion pathways are an example of this type of energy flow. The linear pathway is very efficient because it takes almost nothing to receive the energy.

The autocatalytic pathway has a feedback from the system storage for interacting with an energy source to facilitate the capture of more energy. If energy is available to

support the storage this pathway may lead to a competitive advantage over the linear pathway. The efficiency of the autocatalytic pathway depends on the energy source, the storage and the pathway coefficient. A pathway of this type

has the capability of capturing more available energy.

The quadratic pathway has a self-stimulating feedback

(see equation on Figure 10) from the storage to capture available energy. Examples of cooperative feeding that may

fit this model are common in ecosystems such as pack hunting by some carnivores, cell and organ system interactions and the cooperative work by humans in developed nations.









This model was simulated in BASIC (program THREEPATH in Appendix) on a Digital Equipment Corporation (DEC) PDP 11/34 using a DEC VK-100 graphics terminal (GIGI). Measurements were made of the percent of the input power used while applying various levels of input power and varying the frequency of input power. Simulation runs were also made

with one or more of the three pathways set to zero to determine the impact of the various pathways on the overall system behavior and power utilization.

In conjunction with the three path model in Figure 10, a similar model with the same inputs but with additional

higher order drain pathways was simulated to determine the effects on total power usage (Figure 11). In any system that has crowding effects or high storage costs, these drain pathways may determine how the system processes energy. The model has a linear drain, an autocatalytic drain and a

quadratic drain.

The basic three path model was tested for the effects of size and turnover time on the percent power used for various power inputs by varying the drain coefficient (K4 on

Figure 10) in multiple simulation runs.

The percent power used when the three path model competes with individual storages with single pathways (Figure 12) was also simulated to see how the various pathways may help or hinder a system. The competitors are individual

tanks with single pathways corresponding to the three pathways in the three path model.

























Figure 11. Three pathway model with multiple drain pathways. Used to test effects of higher order drain pathways on threepath model.


Linear input: Autocatalytic input: Quadratic input: Linear drain: Autocatalytic drain: Quadratic drain:

dQ=Jl+J2+J3-J4-J5-J6 JR=JO-Jl-K2'*J2-K3'*J3


J 1=Kl*JR
J2=K2*Q*JR J3=K3*Q*Q*JR
J4=k4*Q 35=K5*Q*Q
J6=K6*Q*Q*Q








































Energy Jo source




JR


'17


X

X

X 3
X

J2 J
X J4 X

























Figure 12. Three pathway model with individual competing units having single input pathways similar to combined model. Coefficients in Appendix. Combination tank:

Linear input: Jl=Kl*JR
Atocatalytic input: J2=K2*Q*JR
Quadratic iput: J3=K3*Q*Q*JR

dQ=Jl+J2+J3-K4*Q

Single tanks:

Linear input: JX=Kl'*JR
dQl=JlX-K4*Ql

Autocatalytic input: J2X=K2'*Q2*JR
dQ2=J2X-K4*Q2

Quadratic input: J3X=K3'*Q3*Q3*JR
dQ3=J3X-K4*Q3

JR=JO-Jl-K2'*J2-K3'*J3-JlX-K2'*J2X-K3'*J3X






































Energy source J


X






J2










X J3X K4*03





02

J2X









K410






50


For any system to survive over the long term, it must fit into a regime of disturbances or catastrophic events from sources outside its own boundaries. The system must be

tuned to the frequencies of those systems that influence it in order to maximize power and survive. The three path model was simulated with various frequencies of power input to see how the various pathways process power at different frequencies and amplitudes.


Parallel production-consumption minimodel

A model with producers in parallel was used to study

the effects of competition among producers (Figure 13). The model had three producers, all having the same structure, with one aggregate consumer that was consuming all three and feeding back as a multiplier on the production function of each. It is a basic predator-prey model with competition among the different producers, along with feedback control and energy constraints in the form of a flow limited source. Instead of having combinations of pathways that can vary, this model had combinations of producers that could vary.

The producers had different turnover times and coefficients so that Ql, Q2, and Q3 represented climax, midsuccessional (shrub) and early successional (weed) species. The coefficient of consumption (the percent of each producer the consumer eats per unit time) for each producer was different. The weed species had a higher value than the shrub species, which was higher than the climax species

























Figure 13. Parallel production-consamption model.

Individual rate equations
R1 = Kl*Ql*JR*Q4 R2 = K2*Q2*JR*Q4 R3 = K3*Q3*JR*Q4
R4 = Dj*Ql RS = D2 *Q2 R6 = D3*Q3
R7 = K7*Ql*Q4 R3 = K8*Q2*Q4 R9 = K9*Q3*Q4
R10 = Fl*(Kl*Ql*JR*Q4 + K2*Q2*JR*Q4 + K3*Q3*JR*Q)
R11 = KO*(K7*QI*Q4 + K*Q2*Q4 + K9*Q3*Q4)
R12 = D4tQ4
JR = JO/(l + Ll*Ql*Q4 + L2*Q2*Q4 + L3*Q3*Q4)

Rate equations for state variables
dQl = R1 R4 R7 dQ2 = R2 R5 R8 dQ3 = R3 R6 R9
dQ4 = R11 R12 R10






52


al
R1 R7

R4 NR11 R10 R8 R12


02 2

R5 R9









R3 Q
x x










(Odum 1969). A list of coefficients is given in Appendix Table 3.

Several variations of this basic model were written in FORTRAN and BASIC computer languages and simulated on both a POP 11/34 and on a Heathkit H8. The source listing for the standard parallel production-consumption model (SUC10) is presented in the Appendix.

Pulse Model

A general pulsing ecosystem model (Figure 14) was designed to test various hypotheses about energy flows and pulsing, hierarchical organization, and spatial development of ecosystems. Some of the structure of the model was derived after the tests of the threepath model and the parallel production-consumption model. The model had many characteristics of ecosystems such as:

1. Flow limited resources (representing solar based
energy resources).
2. Nutrient storage within the boundaries of the model.
3. Units of production, consumption and storage.
4. Feedback of consumers on production through nutrient
recycle.
5. Consumption at low maintenance rates and at high
pulsing rates.
6. Production through a fast turnover storage into a
long turnover biomass storage.

The basic pulsing ecosystem model was tested for different flow rates, initial storages and energy inputs. From this, a baseline understanding of the dynamic behavior of the model and energy processing capabilities (as percent power used) was developed.

The pulse model (Figure 14) was similar to the one in Richardson and Odum (1981) with some changes in coefficients












Figure 14. Pulse model of tropical forest ecosystem model.


Individual rate equations:
Ri = Kl*Ql*Q4*JR
R2 = K2*Ql R3 = K3*Ql R4 = K4*Ql R = K5*Q2 R6 = K6*Q2
R7 = K7*Q2*Q3*Q3 RS = K8*Q2*Q3*Q3 R9 = K9*Q2*Q3*Q3
RIO = KIO*Ql*Q4*JR
RIu = Kll*Q2 R12 = K12*Q3
R13 = K13*QI*Q4*JR


Rate equations for state variables:
dQl = Rl R2
dQ2 = R3 R9 RI1 dQ3 = R7 + R5 R12
dQ4 = R12 + R8 + R6 R10 + R4
JR = JO/(1 + K13*QlkQ4)

















097


or










and flows to calibrate it to a tropical rain forest ecosystem. The original model was run on an Electronics Associates Incorporated model 2000 Analog/Hybrid computer. The models presented in this dissertation were simulated on a DEC POP 11/34. The multiple simulations of the pulse model were generated with a version of the program that would run 25 simulations while varying a coefficient or initial condition over those 25 runs and generate data files that were then displayed with the FORTRAN program PLOTZ (See Appendix). The source listing of the FORTRAN pulse program is in the Appendix.

The pulse model was calibrated with tropical forest ecosystem values for carbon flows and storages (Jordan and Drewry 1969, Odum and Pigeon 1970, and Brown, Lugo, Silander and Liegel 1983). The energy diagram of the model is given in Figure 14 and the equations, coefficients and initial conditions of the state variables are given in Appendix Table 4.


Pulse Model With Prey-Predator Sectors

An additional higher trophic level consumer was added to the pulsing consumer model (Figure 14) in order to test the relationship of turnover time and hierarchical matching of consumers (Figure 15). The extra consumer added to the model had the same structure as the lower level pulsing consumer (Q3), with both linear and quadratic pathways. This model was tested by varying the turnover time of the


S6












Figure 15. Pulse model with additional prey-predator
sector.


Individual rate equations:
Rl = Kl*Ql*Q4*JR
R2 = K2*Ql R3 = K3*Ql R4 = K4*Ql R5 = K5*Q2 R6 = K6*Q2
R7 = K7*Q2*Q3*Q3 RB = K8*Q2*Q3*Q3 R9 = K9*Q2*Q3*Q3
RIO = KlO*Ql*Q4*JR
RIl = Kll*Q2 R12 = K12*Q3
R13 = K13*Ql*04*JR R14 = K14*Q3*Q5*Q5 R15 = K15*Q3*Q5*Q5 R16 = K16*Q3*Q5*Q5
R17 = K17*Q5 R18 = KlB*Q3 R19 = Kl9*Q3 R20 = K20*Q3


Rate equations for state variables: dQl = Rl R2 dQ2 = R3 R9 Rll dQ3 = R7 + R5 R12 R1 R14 dQ4 = R4 + R6 + R12 + R12 RIO + R16 + R17 + R20 dQ5 = R15 R17 + R19 JR = JO/(l + K13*Ql*Q4)















S 00001 T
88 x 6M lID









highest level consumer (Q5) and measuring the percent power used and the level of the other storages in the system. Spatial Models

The models previously discussed were time domain models with no spatial effects. However, because ecosystems develop through time and space and spatial variations can be at least as important as variations in time, spatial models were developed and simulated to test hypotheses concerning spatial development of ecosystems such as energy processing and pattern formation and hierarchical control of pattern formation.

The basic spatial model was a collection of subunits, each one a pulsing consumer model (Figure 14). These subunits were organized in a spatial format. When this simple model was simulated in a spatial format, size effects, edge effects and the consumer range of influence can become important. Intercell interactions between individual producers, consumers, nutrients, and energy sources may be important in energy utilization and pattern formation.

Effects of edges in the spatial model were of interest in pattern formation and energy use. Special boundary conditions were defined for the model cells along the edge. These boundary cells were manipulated in the simulation model in order to study the effects of edges on energy use and pattern formation. The boundary cells were also manipulated to minimize the effect of edges in certain runs of the model.


59





60


Any ecosystem can be divided into edge and non-edge (center) parts. The amount of edge in an ecosystem is a function of the size and number of the individual patches within it. For a given area, as the number of subunits increases the percent of the subunits on the edge decreases (see Figure 16).

A 10x10 matrix was used in the spatial simulations, giving 36% of the total in edge cells and 64% in non-edge cells. This size model was chosen to reduce the edge and yet be small enough to simulate in a reasonable time. Computer runs for this model lasted approximately 3 hours on a PDP 11/34. A model with a center to edge ratio of 10:1 would need approximately 20 times as many cells. In order to test the effects of edges on the model, a single layer of cells was added around the outside edges of the 10x10 matrix, giving it a 12x12 total area (Figure 17). The outer layer was not acted as a buffer to approximate conditions of an edgeless system.


Arrangements of cells

In simulating a spatial model, many arrangements of

cells can be used. The simplest form used was a linear array with cells arranged in a linear ring. For two dimensional models the cell geometry chosen was a square. This was done for several reasons:























Figure 16. Number of edge and center cells as a function of total number of cells in a given square area.






62


























EDGE EFFECT
P~rmmt~r cnd Center 10





70 so50 AO 30

20

10



0 0.4 0.8 1.2 1.8 2 2.4
iThousands) NUMBER OF' CELLS a PERIMETER 4 CENTER
























Figure 17. Cell geometries considered for spatial models.

(a) Square matrix with each cell having 4 side
and 4 corner neighbors. Active l0xlO matrix
embedded in a 12x12 matrix. This one was
chosen for the spatial simulations.

(b) Hexagonal matrix with each cell having 6 side
neighbors.












000000000000

000000000000 0 0000000000 0 0000000000 0 0000000000
O0000000000

000000000 O0000000000


0000000000
0000000000
oooooooooo:, 0000000000 : :0000000000::. :. 0000000000 0000000000 0oooooooo0 0000000000 :0000000000






65


1. It simplified programming the model because two

dimensional arrays in FORTRAN are set up in rows and

columns.

2. It simplified writing the graphics routines to display the cells on a graphics terminal.

3. It reduced the edge effects of the model. Ring model

A modified version of the two dimensional spatial model was used to simulate a one dimensional case. The standard spatial pulse model was connected head to tail in a ring of 36 cells.


Two dimensional models

The simplest spatial implementation was the basic pulse model repeated over the 10xlO matrix with no interactions between individual cells. This model (program DSP1) was then simulated with three different energy forcing functions:

1. The energy source was hierarchically distributed

(highest energy input at the center of the matrix).

2. The energy source was evenly distributed.

3. The energy source was randomly distributed.


Energy inputs were scaled so the mean input over the whole matrix could be held constant for all energy types. Overall energy input could be varied to test pattern development and energy use with various energy levels.

Two different initial conditions were tested. A successional sequence was simulated with the initial values of





00


stored production (biomass, Q2 in Figure 14) set to a low

level. A steady state configuration was also used in which Q2 was set to a value just below the pulse threshold. The

nutrient tank (Q4) in each case was balanced to contain the remainder of carbon available in each cell.

This model tested different conditions and inputs.

1. Diffusion was allowed between nutrient tanks (Q4) of

each subunit. The base model (DSPl) allowed nutrients

to diffuse between cells at various diffusion rates.

The outer layer of non-reacting cells (see Figure 17)

had constant values for Q4 to allow tests of total

diffusion into and out of the cell matrix (diffusion

along the edges).

2. Diffusion was allowed between consumer tanks (Q3) of each subunit (program DSPlQ3). The outer non-reacting

cell layer was set to a constant value or was allowed to float (program DSPlQZ) at the average of the inner

l0xlO matrix to simulate a continuous sheet.


Simulations were run in which the consumer had a larger area or territory than the producer. A model variation (program DSPlC) was tested in which all of the consumer

tanks were clumped into one tank that aggregated consumption over the 10xlO matrix simultaneously. This version also had three different input energy patterns available, and allowed diffusion between nutrient (Q4) tanks.






67


The final variation was a model with production compartmentalized as before in individual cells but with free roaming consumers, not constrained by cell boundaries. One consumer was allowed to consume and move about the matrix according to a set of constraints. When the consumer grew above a preset size, it was split into two equal halves and each half was allowed to consume, move and split again. An upper limit of 100 was placed on the total number of consumers that could be generated during the run (the total in the 10x10 matrix of the previous model versions). This model also had three different energy inputs and diffusion of nutrients (Q4).


Format for Spatial Display Graphs

Data from the spatial pulsing model were displayed

using the format shown in Figure 18. The spatial distributions of the producers and consumers were shown at various times during the run (usually 50 years apart). The values of producers and consumers in individual cells were represented by the density of dots in the cell. The producer density increment was 2000 g/m^2 with a range of 0-16,000 g/m^2 while the consumer was represented by an increment of 50 g/m^2 and a range of 0-400 g/m^2.



Measurement of Hierarchies at El Verde Site


In order to compare hierarchical relationships that were generated in the model with those occurring in the











Figure 18. Format of spatial model display graphs.







8882888 8 888 8 8
~0
88 o 88 8 08 8 8


8ZEEcclcc]C
Qol88888o8 (=CIOCICICc8ca

rnrncc8c8c Q~c~cQCIcIQC cjlccccccc QcQ~~ccQ~ QCO II] ll~


x

xi~jiYH mnul 1jd


3 18
3Jwnswoo





70


tropical rain forest at El Verde, several measurements were made from data sets from the tropical rain forest study at

El Verde (1963-1967) in the Luquillo Mountains of Puerto Rico (Odum and Pigeon 1970).

A data set (2048 samples) characterizing the forest at the radiation site was generated by the U. S. Army Corps of Engineers (Rushing 1970). At the radiation site, every

plant 1.8 m. or taller was enumerated within a radius of 30 m. from the center of the site. Each plant was recorded

with the species name, height, diameter, crown diameter, exact location, and various other parameters.

Black and white negatives of aerial views of the radiation site (taken November 1963 before the radiation treatment) were printed as 8x10 inch photographs. Individual gaps characterized by the presence of Cecropia peltata (an

early successional species) were digitized from the photographs using a personal computer, Complot digitizer and digitizing program written especially for this purpose

(Measure3 in Appendix).















CHAPTER 3

RESULTS



Simulation of Three Path Model Individual Pathway Tests

The amount of energy flowing through each of the pathways in the three path model (Figure 10) depends on the total energy input to the model. As input power (JO) was

increased (Figure 19) steady state flows for each of the pathways changed. Each pathway predominates at certain times. The linear path had the largest power flow when input power was low, while the quadratic pathway had the highest flow at higher power inputs.

When input power was increased through time (Figure

20), there was no steady state, but, like Figure 19 when power increased, the energy flow shifted from the linear pathway to the autocatalytic and finally to the quadratic

path. The fraction of energy remaining (Jr/JO) also decreased over time. As input power increased, a greater

fraction of the input power was utilized.

The model was run with different pathway combinations (Figure 21) and with various power inputs. Each curve on the graph represents a steady state value for various combinations of pathways present in the simulation. Power used


71
























Figure 19. Steady state power utilization of units in the three path model (Figure 10) as a function of input power
(JO).














STEADY STATE PATH FLOW









-l e








in +r)



Hrn




.u

O E SI

+ M)






0:0:: ::: ,'..











Figure 20. Energy utilization of individual components in
the three path model in Figure 10. Input power is increasing through time.




















lihreepath. Model


_


J3


Jr/JO


JO


Time
























Figure 21. Steady state energy flows on various pathways and combinations of pathways in the three path model (Figure 10) as a function of input power (JO).


Linear pathway: Autocatalytic pathway: Quadratic pathway:


Jl=Kl*R J2=K2*Q*R
J3=K3*Q*Q*R











PERCENT POWER USED

Qi







-. +
-0 F N


01

E



-u

rr





73


at any given input was highest with all three pathways present. For any combination of pathways that contained the quadratic path (Jl+J2+J3 or J2+J3 or Jl+J3), power used increased with power input to reach the same asymptote (>95% power used). A slightly lower level was reached for pathways dominated by the autocatalytic pathway (J2 or J2+J1). This asymptote was approximately 90% power used with increasing power input. With only the linear pathway enabled, no change occurred in percent power used with increasing power.

A unique situation occurred when the quadratic pathway

(J3) existed alone. A low initial storage (Q) did not provide enough feedback on the J3 pathway to allow growth. Percent power used was never significant. The simulation with only J2 and J2+J3 showed zero percent power used at low input levels, then rose quickly at higher input power.

The size of the storage (Q) was varied to see the

effects on energy usage (Figure 22). This was achieved by varying the depreciation coefficient (K4) in multiple run while increasing power input in the three path model. At high values of K4 (fast turnover times), increases in percent power used at steady state with increasing power were small. With decreasing values of K4 (slower turnover

times), percent power used increased for the initial and final values of input power.

The addition of multiple drains with different structures (Figure 11) did not have as great an effect on the























Figure 22 Simulation of three path model in Figure 10. Percent power used as a function of energy input and size of drain coefficient (K4 varied from .02 to 2.0).











PERCENT POWER USED











-'
-i




0






-. 0
E
ma
























Figure 23. Simulation of three path model with multiple drain pathways in Figure 11. Percent power used as a function of energy input (JO).


Linear drain: Autocatalytic drain: Quadratic drain:


D4=k4*Q D5=K5*Q*Q D6=K6*Q*Q*Q











PERCENT POWER USED









00 0 0



M-u






OSi
*






83


model as multiple inflow pathways. The percent power used was lowest when all combinations of drain pathways were enabled (Figure 23). Percent power used increased with increasing input power. The highest value for percent power used was achieved when only the original linear drain was present. Any combination with the linear drain used less

power at low power inputs than the nonlinear pathways alone or in combination. The higher order drains enabled the

system to draw more power at low levels than when combined with linear pathways. This effect was opposite from that

with input pathways at very low power where the nonlinear pathways did not function well (see Figure 21).

The effects of adding competition pathways to the model (Figure 12) can be seen in Figure 24. In this case, each of

the competing pathways (single tanks Ql, Q2, and Q3 with individual pathways) were left on throughout the simulations. Here again the various pathways were disabled and simulations run with varying power inputs. The results were

similar in some ways to those in Figure 21 where at high power inputs the percent power used approached one of two asymptotes. The greatest percentage of power utilization occurred when all pathways were enabled and the lowest power

utilization occurred when only J2 or J3 were enabled. The addition of the extra competing storages increased the percent power used in each of the pathway combinations compared to Figure 21. These extra pathways were always there to use
























Figure 24. Simulation of three path competition model with various pathways enabled (Figure 12). Percent power used as a function of energy input (JO).


Linear pathway: Autocatalytic pathway: Quadratic pathway:


Jl=Kl*R J2=K2*Q*R J3=K3*Q*Q*R









PERCENT POWER USED










Z
w
C


0

;u









whatever power may be left over (particularly the linear path).


Frequency Studies

The basic three path model (Figure 10) was also used to

test the effects of different frequencies of input power on the model at three different power levels. At the lowest

power level (J0=500, Figure 25) the differences between pathways in percent power used was the greatest. The greatest frequency response occurred at low frequencies. The

frequency response was flat with only the linear path enabled. When all pathways were present, the percent power

used was highest with a peak at approximately 2 cycles. A peak of power utilization also occurred with the combinations of Jl+J2 and Jl+J3. The pathways that showed a minimum in the frequency response were composed of J2+J3 (the two nonlinear pathways combined) and J2. The quadratic pathway alone did nothing since no power was used (compare with Figure 21).

When the input power was increased to 2000 (Figure 26), the linear pathway showed no change in output with change in frequency and the quadratic pathway had no output. The

combination of Jl+J2 here again had a slight maximum at about 2 cycles while J2 alone had a maximum at zero cycles. The combination of all of the pathways (Jl+J2+J3) and Jl+J3 had a slight minimum of power utilization at about 8 cycles, while the combination of J2+J3 showed a slight minimum at about 2 cycles.
























Figure 25. Simulation of the three path model in Figure 10. Percent power used as a function of frequency of the input power (JO=500).


Linear pathway: Autocatalytic pathway: Quadratic pathway:


Jl=Kl*R J2=K2*Q*R J3=K3*Q*Q*R









PERCENT POWER USED


Of 'l'l 1 ^ -


z





10
c





m
-I


0 C
II


.LLLJ i, U L


ci

C (3


('4
Q


+z


a'


u


-




Full Text
171
Gap Size Measurements
The distribution of cecropia gaps at El Verde fall into
a hierarchical distribution (Figure 58). Figure 58a is the
size distribution of all four of the photographic plots
combined and Figure 58b shows the distributions of the
individual plots. The percentage of the total area that is
in the gap stage is 3.79% (Table 2). The values plotted in
Figure 58 are the actual areas measured in square inches on
the photograph. The figure shows a minimum size for the
gaps and a hierarchical distribution.
Comparison to Models
For each time slice that the spatial simulation model
printed a spatial pattern of producers and consumers, it
also printed a graph of the size distribution of the produ
cers and consumers (just to the right of the spatial pat
terns). The format of the distribution graph is not the
same as the size class distributions in Figure 57 but the
size distributions do represent the same class size phenom
enon. Depending on the energy input conditions and dif
fusion coefficients some of the size distributions had simi
lar relationships to the natural distribution (see Figures
51, 53 and 56) while others are quite different (see Figure
50). The pulsing in Figure 50 is totally synchronous while
the pulsing in Figures 51 and 53 are more spatially
asynchronous.


Figure 28. Simulation of the four sector succession model
in Figure 13. Model base run.
Legend:
PPU = Percent power used
PPD = Percent power drained
D = Diversity
B = Total Biomass
P = Productivity
Q1 = Climax species
Q2 = Mid successional species
Q3 = Early successional species
Q4 = Consumer


212
WRITE (3/100)ITILT,ISIZE,ITILT,(TEXT(I),I=1,N)
100 FORMAT('+ T(D',14,')(S',12,')(D',14,')',1H',A1,1H')
RETURN
END
C
C
C
SUBROUTINE LENGTH(TEXT,N)
BYTE TEXT(80)
IFL AG= 0
DO 20 1=80,1,-1
IF(TEXT(I).GT.32)IFLAG=1
N=I
IF (IFLAGEQ.0)G0T020
G0T099
20 CONTINUE
99 RETURN
END
C
C
C
SUBROUTINE GGFILL(IFLAG)
C SUBROUTINE TO TURN ON/OFF COLOR FILL CHARACTERISTIC
C IFLAG=0 NOFILL, IFLAG=1 FILL
C
WRITE(3,100)IFLAG
100 FORMATW(S',11,')')
RETURN
END


Time 2000
122


Figure 33. Simulation of the parallel production-
consumption model in Figure 13. Multiple simulations of the
model with available power held constant (J0=100, base run
value) and the initial value of the consumer (Q4) varied
from 1 to 6.
(a) Climax species (Ql)
(b) Intermediate producer species (Q2)
(c) Weed species (Q3)
(d) Consumer species (Q4)
(e) Percent power used (J0-Jr)/J0
(f) Total biomass


9


39
Energy circuit A pathway whose flow is propor
tional to the quantity in the storage or source upstream.
Source Outside source of energy delivering forces
according to a program controlled from outside; a forcing func
tion.
Tank A compartment of energy storage within the
system storing a quantity as the balance of inflows and outflows;
a state variable.
Heat sink Dispersion of potential energy into heat
that accompanies ail real transformation processes and storages;
Iocs of potential energy from further use by the system.
Interaction Interactive intersection of two path
ways coupled to produce an outflow in proportion to a function of
both; control action of one flow on another; limiting factor action:
work. gate.
Consumer Unit that transforms energy
quality, stores- it, and feeds it back autocatalyticaliy to improve
inflow.
Switching action A symbol that indicates one or
more switching actions.
-db
Producer Unit that collects and transforms
low-quality-energy under control interactions of high-quality
flows.
Self-limiting energy receiver (Chapter 10). A unit that has a
self-limiting output when input drives are high because there is a
limiting constant quantity of material reacting on a circular
pathway within.
Box Miscellaneous symbol to use for whatever unit
or function is labeled.
Constant-gain amplifier A unit that delivers
an output in proportion to the input I but changed by a constant
factor as long as the energy source S is sufficient.
Transaction A unit that indicates a sale of goods
or services (solid line) in exchange for payment of money
(dashed). Price is shown as an external source.


22
both temperate forests (Oliver 1981, and Veblin 1985) and
tropical forests (Garwood 1979, and Leigh et al. 1982).
Disturbances (i.e., pulses) to an ecosystem can be gen
erated from within or can come from outside the boundaries
of an ecosystem and may vary in frequency and amplitude.
The ability of an ecosystem to utilize available resources
and adapt to these disturbances depends on the storages,
structures and interactions within an ecosystem (Odum 1983).
Hierarchical mechanisms may develop that capture and process
energy at various levels and result in utilization of energy
over a wider variety of input levels. Some mechanisms of
interaction between parts of the ecosystem were studied in
this dissertation to understand how systems may converge
energy transformations and feedback controls to organize for
higher productivity.
No unified theory of succession presented to date can
be regarded as widely accepted (Anderson 1986). Horn (1976)
wrote 'The sweeping generalization that can be safely made
about succession is that it shows a bewildering variety of
patterns.' Even the definitions of succession are widely
varying. In this dissertation succession is regarded as a
dynamic process in which the composition of an ecosystem
changes through time, building structure and processing
energy. This process eventually stabilizes in a climax from
which there is a regression or loss of that structure due to
disease, fire, treefall or other events. Seeding from an
other ecosystem or from storages in the soil from the


001 mil
96


Figura 6. Size class distribution of gaps formed in
tropical forest at Barro Colorado (Brokaw 1982).
(a) Distribution of gaps by diameter of gap.
(b) Distribution of gaps by area in gap.


148


253
Shugart, H. H., D. C. West, and W. R. Emanuel. 1981.
Patterns and dynamics of forests: an application of
simulation models. In D. C. West, H. H. Shugart, and D.
B. Botkin (eds). Forest Succession Concepts and
Application. Springer-Verlag. New York. pp. 74-94.
Sprugel, D. G. 1984. Density, biomass, productivity, and
nutrient-cycling changes during stand development in
wave-generated balsam fir forests. Ecological
Monographs 54:165-186.
Sprugel, D. G., and F. H. Bormann. 1981. Natural
disturbance and the steady state in high-altitude
balsam fir forests. Science 211:390-393.
Thom, R. 1975. Structural Stability and Morphogenesis.
English translation by D.H. Fowler of French 1972
edition. Benjamin. New York.
Turing, A. M. 1952. The chemical basis of morphogenesis.
Phil. Trans. Royal Soc. London 273:37-72.
Urban, D. L., R. V. O'Neil, and H. H. Shugart. 1987.
Landscape ecology. Bioscience 37:119-127.
Veblen, T. T. 1985. Stand dynamics in Chilean Nothofagus
forests. In S. T. A. Pickett and P. S. White (eds).
The Ecology of Natural Disturbance and Patch Dynamics.
Academic Press. Orlando, pp. 35-51.
Weatherhead, P. J. 1986. How unusual are unusual events.
American Naturalist 128:150-154.
Winfree, A. T. 1973. Scroll shaped waves of chemical
activity in three dimensions. Science 181:937-938.
Wolfram, S. 1984. Cellular automata as models of
complexity. Nature 311:419-424.


760-- u 760
Time Time
2000 ^ r-40000


230
c
c
WRITE(4,114)
114 FORMAT(//1 Q1 Q2 Q3 Q4 TOTAL')
WRITE(4,1121)0110,2210,0310,04IC,Q4IC+Q3IC+Q2IC+Q1IC
WRITE(4,1122)M1,M2,M3,M4
WRITE(4,1123)Q1,Q2,Q3,Q4,Q4+Q3+Q2+Q1
1121 FORMAT(' INIT ',4(2X,G8.2),2X,G12.6)
1122 FORMAT(' MAX ',4(2X,G3.2))
1123 FORMAT(' FINAL',4(2X,G8.2),2X,G12.6)
119
120
WRITE (4,116) K1 ,K2,K3,K4,K5,K6,.K7 ,K8 ,K9 ,K10 ,K11 ,K12 ,K13 JO J
F0RMAT(/1X,'K1= ',G12.6,' K2= ',G12.6,' K3= ',G12.6/
+' K4= ',G12.6,' K5= ',G12.6,' K6= ',G12.6/
+ K7= ',G12.6,' K8= ',G12.6,' K9= ',G12.6/
+' K10=',G12.6,' K11=',G12.6,' K12=',G12.6/
+' K13=',G12.6,' J0= ',G12.6,' JR= ',G12.6)
EUSED=EUSED*DT
PAVAIL=PAVAIL*DT
PPU=100.*EUSED/PAVAIL
WRITE(4,119)VERS,DT,EUSED,PAVAIL,PPU
FORMAT(/' PULSE MODEL VERS',F6.3,' TIME STEP(DT) = ',F6.4/
+' TOTAL POWER USED = ',G15.6,' POWER AVAILABLE =',G15.6/
+' PERCENT POWER USED = ',G15.6)
WRITE(4,120)(FILE(I),1=1,3),E1,E2,E3,E4,E5,E6,E7,E8,E9,E10,
+E11,E12,E13,E14,E15
FORMAT(' DATA FILE NAME = ',3A4,1X,15A4)
IF(ICOPY.EQ.0)G0T0999
CALL GGON
WRITE(3,11001)
CALL GGOFF
999
END


235
IF (IETYP.EQ.3)GOTO220
E(I,J)=XMEAN*100.
220 CONTINUE
221 DO 250 1=2,11
DO 250 J=2,11
ETOT=ETOT+E(I,J)
250 CONTINUE
IF (IETYP.EQ.2)GOTO300 ¡CHANGED 3 TO 2 IN 3.0
SF=ETOT/(100.*100.)
ETOT=0.0
DO 270 1=2,11
DO 270 J=2,11
E(I,J)=E(I,J)*XMEAN/SF
ETOT=ETOT+E(I,J)
270 CONTINUE
C
C
C>>>> LOOP START <<<<<<
C
300 CONTINUE LOOP START
IT=T ¡GET INTEGER VALUE OF TIME FOR MOD FUNCTION
EUSED=0.0 ¡ENERGY USED PER TIME I.E. POWER
SPTEMP=0
PTEMP=0.
DO 400 1=2,11
DO 400 J=2,11
C.... RATE EQUATIONS
C
17=0
INUM=0
XEQ=00
C
IF (ICON(I,J).NE.0)XEQ=1.0
J0=E(I,J)
JR=J0/(1+K13*Q1(I,J)*Q4(I,J))
R1=DT*K1*Q1(I,J)*Q4(I,J)*JR
R2=DT*K2*Q1(I,J)
R3=DT*K3*Q1(I,J)
R4=DT*K4*Q1(I,J)
R5=DT*K5*Q2(I,J)
R6=DT*K6*Q2( I, J)
R10=DT*K10*Q1(I,J)*Q4( I,J)*JR
R11=DT*K11*Q2( I,J)
EUSED=EUSED+(J0-JR)*DT
C.... LEVEL EQUATIONS ....
C
PTEMP=PTEMP+R1 ¡PRIMARY PRODUCTION
Q1(I,J)=Q1(I,J)+R1R2
IF(Q1(I,J).LT.0.0)Q1(I,J)=0.0
Q2(I,J)=Q2(I,J)+R3-R11
Q4(I,J)=Q4(I,J)+R4+R6-R10
Q4(I,J)=Q4(I,J)+R5 *(1-XEQ)
C ADD LINEAR FLOW TO Q4 IF Q3 NOT THERE


47
1


66
stored production (biomass, Q2 in Figure 14) set to a low
level. A steady state configuration was also used in which
Q2 was set to a value just below the pulse threshold. The
nutrient tank (Q4) in each case was balanced to contain the
remainder of carbon available in each cell.
This model tested different conditions and inputs.
1. Diffusion was allowed between nutrient tanks (Q4) of
each subunit. The base model (DSP1) allowed nutrients
to diffuse between cells at various diffusion rates.
The outer layer of non-reacting cells (see Figure 17)
had constant values for Q4 to allow tests of total
diffusion into and out of the cell matrix (diffusion
along the edges) .
2. Diffusion was allowed between consumer tanks (Q3) of
each subunit (program DSP1Q3) The outer non-reacting
cell layer was set to a constant value or was allowed
to float (program DSP1QZ) at the average of the inner
10x10 matrix to simulate a continuous sheet.
Simulations were run in which the consumer had a larger
area or territory than the producer. A model variation
(program DSP1C) was tested in which all of the consumer
tanks were clumped into one tank that aggregated consumption
over the 10x10 matrix simultaneously. This version also had
three different input energy patterns available, and allowed
diffusion between nutrient (Q4) tanks.


CHAPTER 1
INTRODUCTION
Ecosystems develop patterns in time and space. Some of
these patterns are generated by pulsing oscillatory proces
ses. What sorts of interactions, organization and structure
in an ecosystem lead to pulsing behavior, and how does this
behavior affect the use of energy? What types of spatial
patterns develop when ecosystems are influenced by pulsing
in time and space? What are the energy implications of
different pattern forming processes in ecosystems? What are
the effects of pulsing on succession, competition, frequency
response of producers and consumers, and coupling with ex
ternal pulses?
This dissertation uses general systems models to ana
lyze the effects of pulsing on pattern formation and overall
power use as systems develop, build structure and organize
in time and space. Simulation models using general systems
principles and based on real ecosystems were used to test
the role of pulsing behavior of consumers in organizing eco
systems over time and space. Data from a tropical ecosystem
were used to calibrate pulsing and spatial models.
1


SPATIAL PATTERNS AND MAXIMUM POWER IN ECOSYSTEMS
3Y
JOHN R. RICHARDSON
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE
UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS OF THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA

Copyright 1988
by
John R. Richardson

ACKNOWLEDGEMENTS
I would like to express my sincere appreciation to Dr.
Howard T. Odum, my committee chairman, for the insights and
inspiration he gave during the completion of this work. His
holistic views and open-mindedness provide an extremely
fertile field to develop and pursue ideas in systems
ecology. Other members of my committee (Drs. J.F.
Alexander, G.R. Best, K.C. Ewel and C.L. Montague)
provided useful feedback in class and with this
project.
The support and patience of my wife Karen has
sustained me while my two children, Matthew and James,
have provided joy and purpose for the completion of
this dissertation.
Work was done in the Department of Environmental
Engineering Sciences, University of Florida, and was
supported by graduate research funding from the
Graduate School of the University of Florida.

TABLE OF CONTENTS
ACKNOWLEDGEMENTS iii
LIST OF FIGURES vi
ABSTRACT xi
CHAPTER 1 INTRODUCTION 1
Historical Perspective 2
Previous Models of Pulsing Patterns in Time and Space.2
Pattern Formation 3
Concepts of Pulsing, Patterns and Power 13
Maximum Power in Systems 13
Design for Maximum Power 14
Pathway Configuration 15
Pulsing and Patterns in Ecosystems 21
Succession and Disturbance 21
Edges 23
Hierarchies and Patches 24
Mode Is 25
Gap Models and Patch Dynamics 27
Spatial Systems and Models 28
Plan of Study 32
Objectives 32
Data Site: Luquillo Rain Forest, Puerto Rico 33
CHAPTER 2 METHODS and MODELS 37
Simulation Procedures and Programs 40
Simulation Models 41
Minimodel Tests 41
Pulse Model 53
Pulse Model with Prey-Predator Sectors 56
Spatial Model 59
Format for Spatial Graphs 67
Measurement of Hierarchies at El Verde Site 67
CHAPTER 3 RESULTS 71
Simulation of Three Path Model 71
Individual Pathway Tests 71
Frequency Studies 86
Simulation of Parallel Production-Consumption Model 91
Single Run Simulations 91
Multiple Run Simulations 100
i v

Initial Conditions and Total Energy Use 112
Simulation of Pulse Model 117
Single Run Simulations 117
Multiple Run Simulations 123
Simulation of Pulse Model with Prey-Predator Sectors... 133
Simulation of the Ring Model 141
Simulation of Two Dimensional Surface Models 151
Rain Forest Gaps and Hierarchies 164
Size Class Distributions 164
Gap Size Measurements 171
Comparison to Models 171
CHAPTER 4 DISCUSSION 173
Maximum Power Considerations 179
Power and Feedback With Paths of Higher Order 179
Effect of Hierarchies on Performance 181
Power Used as a Function of Input Power 132
Threshold for Stable Feedbacks and Pulsing 132
Implications for Succession 134
Role of Individual Units 184
Succession and Pulsing 185
Spatial Pattern formation 186
Synchronous vs. Asynchronous Systems 136
Coupling of Spatial Units by Diffusion Processes.... 187
Organization by Higher Level Consumers 183
Power Use and Edge Effects .190
General Principles .....190
APPENDIX 192
BIBLIOGRAPHY 247
BIOGRAPHICAL SKETCH 254
v

Figure
LIST OF FIGURES
Page
1 Spatial patterns based on chemical reaction
mechanisms 6
2 Hilborn (1979) model 8
3 The spatial development of cells based on
simple r-pentamino initial condition 12
4 Basic autocatalytic model with flow-limited
energy source 17
5 Basic multiple path model with three input
pathways representing differing feedback
regimes, linear, autocatalytic, and quadratic... 20
6 Size class distribution of gaps formed in
tropical forest at Barro Colorado (Brokaw
1982) 26
7 Mite predator prey experiment (Huffaker
1958) 30
8 Size class distribution over time of plot of
trees in tropical forest at El Verde (Crow
1980) 36
9 Energy circuit language symbols (Odum 1983) 39
10 Three pathway model used to test effects of
various energy inputs on kinetic mechanisms 43
11 Three pathway model with multiple drain path
ways 47
12 Three pathway model with individual competing
units having single input pathways similar to
combined model 49
13 Parallel production-consumption model 52
14 Pulse model of tropical forest ecosystem
model 55
15 Pulse model with additional prey-predator
sector 58
16 Number of edge and center cells as a function
of total number of cells in a given square
area 62
vi

17 Cell geometries possible for spatial models 64
13 Format of spatial model display graphs 69
19 Steady state power utilization of units in
the three path model (Figure 10) as a
function of input power (JO) 73
20 Energy utilization of individual components
in the three path model in Figure 10 75
21 Steady state energy flows on various pathways
and combinations of pathways in the three
path model (Figure 10) as a function of input
power (JO) 77
22 Simulation of three path model in Figure 10 80
23 Simulation of three path model with multiple
drain pathways in Figure 11. Percent power
used as a function of energy input (JO) 82
24 Simulation of three path competition model
with various pathways enabled (Figure 12) 35
25 Simulation of the three path model in Figure
10. Percent power used as a function of
frequency of the input power (J0=500) 88
26 Simulation of the three path model in Figure
10. Percent power used as a function of
frequency of the input power (J0=2000) 90
27 Simulation of the three path model in Figure
10. Percent power used as a function of
frequency of the input power (J0=10000) 93
28 Simulation of the parallel production-consumption
model in Figure 13. Model base run 96
29 Simulation of the parallel production-consumption
model in Figure 13 99
30 Simulation of the parallel production-consumption
model in Figure 13. Multiple simulations of
the model with available power increasing
from 50 to 300 102
31 Simulation of the parallel production-consumption
model in Figure 13. Multiple simulations of
the model with percent power used for entire
run vs input power 10 4
vi i

32
Simulation of tile parallel production-consumption
model in Figure 13. Run with available power
increasing from 50 to 300 and the initial
value of the consumer (Q4) equal to 50 (lOx
base run in Figure 28) 107
33 Simulation of the parallel production-consumption
model in Figure 13. Multiple simulations of
the model with available power held constant
(J0=100, base run value) and the initial
value of the consumer (Q4) varied from
1 to 6 109
34 Simulation of the parallel production-consumption
model in Figure 13. Total percent power used
for entire run as a function of the initial
value of the consumer (Q4) Ill
35 Simulation of the parallel production-consumption
model in Figure 13. The initial value of weed
species (Q3) was varied from 0 to .5 and
input power was held constant (J0=100, base
run va lue) 114
36 Steady state values of percent power used as
a function of input energy and state variable
initial conditions for multiple simulation
runs of parallel production -consumption model
(Figure 13) 116
37 Simulation for pulse model (Figure 14) with
base run coefficients 119
38 Simulation of pulse model (Figure 14) without
a quadratic pathway (K7, K8, K9 = 0.0) 122
39 Simulation of pulse model (Figure 14)
without a feedbacks into Q4 (K6, K8 = 0.0) 125
40 Multi-run simulation of the pulse model
(Figure 14) with variation in input energy.
(JO varied from 0 to 250) 127
41 Multi-run simulation of pulse model (Figure
14) with variation in total carbon in model....130
42 Multi-run simulation of pulse model (Figure
14) with variation is turnover time of
pulsing consumer. (K12 varied from
.01 to .5)
132

43
Multi-run simulation of pulse model (Figure
14) with variation in quadratic pathway (K9
varied from 0.5E-6 to 0.53E-5 with K7 and K3
varied proportionately) 135
44 Multi-run simulation of pulse model (Figure
14) with variation in linear pathway (Kll
varied from 0.0 to 0.12E-2 and K5 and KS
varied proportionately) with quadratic
pathway held at zero 137
45 Simulation of pulse model with prey-predator
sectors (Figure 15) 139
46 Simulation of pulse model (Figures 14 and 18)
with cells in a linear ring and diffusion
between consumers of each cell in ring
(DK=.l). Initial conditions of consumers
were set to near zero except for one "seed"
consumer at lower left corner of matrix which
was set to 100 143
47 Simulation of pulse model (Figures 14 and 18)
with cells in a linear ring but without dif
fusion. Initial conditions of producers and
consumers were set to random distribution
around ring 146
48 Simulation of pulse model (Figures 14 and 18)
with cells in a linear ring and a high level
of diffusion between consumers of each cell
(DK=.1) and random distribution of producers
and consumers around ring 148
49 Simulation of pulse model (Figures 14 and 18)
with cells in a linear ring and a low level
of diffusion between consumers of each cell
(DK=.001) and random distribution of
producers and consumers around ring 150
50 Simulation of pulse model (Figure 14) with
cells arranged in two dimensions (Figure 18)
without diffusion and with a constant energy
source 153
51 Simulation of pulse model (Figure 14) with
cells arranged in two dimensions (Figure 18).
Energy source hierarchically is distributed
from center outward and no diffusion between
ce 11s
155

52
Simulation of pulse model (Figure 14) with
cells arranged in two dimensions (Figure 18).
Energy source is hierarchically distributed
from center outward and diffusion is between
consumers of each cell (DK=.001) 158
53 Simulation of pulse model (Figure 14) with
cells arranged in two dimensions (Figure 18).
Energy source is randomly distributed and
diffusion is between consumers of each cell
(DK= .001) 160
54 Simulation of pulse model (Figure 14) with
cells arranged in two dimensions (Figure 18).
Energy source is randomly distributed and
diffusion between nutrient storages (Q4) of
each cell is set to high level (DK=.l) 163
55 Simulation of the pulse model (Figure 14)
with cells arranged in two dimensions (Figure
18). Moving consumer model with search length
set to one cell, no diffusion and hierarchi
cal energy distribution 166
56 Simulation of the pulse model (Figure 14)
with cells arranged in two dimensions (Figure
18). Moving consumer model with search length
set to five cells, no diffusion and
hierarchial energy distribution 163
57 Size class distribution of trees at El Verde
radiation site (November 1964) 170
58 Size distribution of Cecropia gaps in tropi
cal rainforest at El Verde 173
59 Size distribution of gaps in tropical rain
forest pulsing model simulation (Figures 14
and 18) at time =760 177
60 Character set for displaying spatial graphs
on GIGX computer terminal for use with
screencopy to printer 196
x

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
SPATIAL PATTERNS AND MAXIMUM POWER IN ECOSYSTEMS
by
John R. Richardson
April, 1988
Chairman: Howard T. Odum
Major Department: Environmental Engineering Sciences
Studies of dynamic systems have shown that oscillations
in time and space are related, both being generated by non
linear, pulsing behavior that is derived from the mathema
tics of energy processing. Similar mathematics exist in
chaos theory, bifurcation theory, and catastrophe theory.
Production-consumption models that simulate pulsing proper
ties of ecological systems are of this class. This dis
sertation examines the spatial patterns and energetics of
autocatalytic and pulsing models as a paradigm for ecolog
ical and general systems. Configurations were tested with
steady or varying resource availability for ability of the
model systems to maximize power as the criterion for utility
and success. The spatial distribution of gaps generated by
simulations was compared to that observed in rain forests.
Models studied included (a) aggregated, single
compartment autocatalytic designs; (b) parallel production-
consumption design; (c) production-consumption-recycle
xi

designs; and (d) multiple cell spatial models each with a
unit model but interconnected in different ways.
Models with autocatalytic feedbacks utilized more
power than the same models with only linear pathways. Per
cent power used increased with increasing available power.
Production-consumption models show multiple steady
states with pulsing behavior as a transition between two
steady states. Localized maxima of power use occur during
pulsing but the overall power use is related to input power.
Spatial patterns of production and consumption in
spatial models were related to input energy patterns, the
degree of connectivity between the individual cells in the
model, and the hierarchical level of intercell connections.
Large variations in patterns were accompanied with small
changes in power utilized.
Edges of a spatial system can act as a source or sink
for energy depending on the relationship between available
energy inside and outside the boundaries and the degree of
connectivity along the edges.
Basic autocatalytic production-consumption-recycle
models with different spatial conditions organize different
spatial patterns while generating near total utilization of
available power. The wide variety of spatial patterns
results from dynamic adaptations for maximizing power for
different spatial conditions. The simulation results
resemble patterns in nature often attributed to random
indeterminancy.

CHAPTER 1
INTRODUCTION
Ecosystems develop patterns in time and space. Some of
these patterns are generated by pulsing oscillatory proces
ses. What sorts of interactions, organization and structure
in an ecosystem lead to pulsing behavior, and how does this
behavior affect the use of energy? What types of spatial
patterns develop when ecosystems are influenced by pulsing
in time and space? What are the energy implications of
different pattern forming processes in ecosystems? What are
the effects of pulsing on succession, competition, frequency
response of producers and consumers, and coupling with ex
ternal pulses?
This dissertation uses general systems models to ana
lyze the effects of pulsing on pattern formation and overall
power use as systems develop, build structure and organize
in time and space. Simulation models using general systems
principles and based on real ecosystems were used to test
the role of pulsing behavior of consumers in organizing eco
systems over time and space. Data from a tropical ecosystem
were used to calibrate pulsing and spatial models.
1

2
Historical Perspective
Previous Models of Pulsing Patterns in Time and Space
In many fields from chemistry, physics, and biology to
astronomy, there are a variety of models, methods and tech
niques to describe and study systems that have discon
tinuities or other rapid fluctuations in their behavior.
Some of these are catastrophe theory (Thom 1975), bifurca
tion theory, synergetics (Haken 1977a,1977b,1979), dynamical
system theory (Rosen 1970), chaos and order (Prigogine
1980,1984, and Schaffer and Kot 1985), pulsing (Lotka 1920
and Odum 1982), pattern recognition, and morphogenesis
(Meinhardt 1982). In all of these, processes being de
scribed are parts of nonlinear thermodynamically open sys
tems. Energy constraints on these types of systems have not
previously been well studied.
In the past, efforts to describe systems using clas
sical thermodynamics centered on closed systems near equili
brium or open systems near steady state. In such systems,
available energy is small. These approaches using equili
brium thermodynamics could not account for the behavior of
many systems (Odum 1983, Prigogine 1984, Schaffer and Kot
1985).
Data with statistical anomalies are often difficult to
analyze and methods are sometimes used to minimize fluc
tuations (Platt and Denman 1975). Systems that have aperi
odic behavior, a great deal of noise, or time dependent
changes in variance are not well suited to the normal

3
statistical methods. These 'unusual events' can be impor
tant in understanding how a system works (Weatherhead 1986).
Frequency analysis has been used for some time to study
periodic behavior of systems (Platt and Denman 1975, and
Emanuel, West and Shugart 1978). Fourier transformations
decompose the output or behavior of a system into an addi
tive series of sinusoidal processes. The variance is parti
tioned into a set of frequencies that when combined gives
the output being measured. Aperiodic behavior or systems
with known nonlinear components may also be studied with
these techniques, but the results are often not useful.
Some nonlinear systems with behavior described as 'chaotic'
have frequency domain variance as noisy as the time domain
variance (Abraham and Shaw 1984a, 1984b).
Pattern Formation
Patterns in natural systems range from the smallest
molecular patterns of motion to the placement of the stars
and galaxies in the universe. One of the most intriguing
aspects of pattern formation is the similarity of patterns
at differing time scales and sizes. From a systems point of
view this would lead one to suspect that the processes are
similar at each scale.
Chemically reacting systems give rise to various types
of patterns (Bray 1921, Nicolis and Prigogine 1969, Winfree
1973, Haken 1977a, 1977b). The Belousov-Zhabotinski reac
tion, which makes fascinating patterns, is a simple

4
oxidation-reduction reaction involving raalonic acid, brmate
and a cerium catalyst (Winfree 1973). An example of the
time and spatial development of this reaction is shown in
Figure la.
Morphological development in biological systems has
been studied and modeled by Meinhardt (1982). Patterns form
when autocatalytic growth in a system is combined with
lateral inhibition (negative spatial feedbacks). Once auto
catalytic activity starts, there must be a longer range
negative feedback (spatial inhibition of the spread of this
autocatalysis) or the whole system will pulse in a burst of
autocatalytic consumption. This sets up spatial chemical
gradients along which morphogenesis is thought to occur
(Figure lb) .
Hilborn (1979) experimented with predator-prey models
based on an aquatic ecosystem. Hilborn1s model had 100
spatial cells arranged in a linear chain with the ends
connected to form a circle. Both predators and prey were
allowed to diffuse across cell boundaries. The model was
simulated with initial conditions set so that all cells had
prey but only one cell had a predator. The model (Figure
2a) was allowed to iterate for 1000 time intervals, gen
erating the pattern seen in Figure 2b. Further experiments
showed that there was no tendency towards equilibrium in
longer runs of the model.
The spatial development of insect eyes and insect legs
has been modelled by Ransom (1981) using an autocatalytic

Figure 1. Spatial patterns based on chemical reaction
mechanisms.
(a) Spatial patterns generated by 3elousev-Zhabotinski
chemical reaction (Prigogine 1980).
(b) Spatial patterns generated by simulation model
used to describe morphogenesis (Meinhard (1932).

9

Figure 2. Hilborn's (1979) spatial model.
(a) Energy diagram of individual cell model
Equations for simulation model.
dX(i) =a*X(i) b*X(i)*X(i) (c*X(i)*Y(i)/(d+X(i) ) )
+h*X( i +1) +h*X ( i 1) 2*h*X( i )
dY (i) =-e* Y (i) f Y (i) Y (i ) + (g*X (i) Y (i) / (d + X (i ) ) )
+k*Y(i+1) +k*Y(i 1) 2*k*Y(i)
where i is the number of the subsystem in a linear loop.
(b) Simulation results of linear series of unit models
showing level of predator vs distance around loop.

NUMBER OF PREDATORS
8
SPATIAL LOCATION

9
model. By allowing cells in the model to divide and migrate
within given constraints, the model developed patterns simi
lar to those in real insects. The model allowed simple
random cell division with movement constrained to a hex
agonal direction away from the center of the cell division.
Sergin (1978, 1979, 1980) studied the oscillatory be
havior of long term climate variations using models that
combine linear and nonlinear interactions of the heat cap
acities of the oceans and polar ice sheets. The period of
the climatological events in these models is on the order of
10,000 to 100,000 years. The model of global temperatures
varies in its behavior from steady state to oscillations
based on small changes in areal coverage of continental ice
sheets.
Pattern formation based on digital, rule based systems
has been used to model biological systems. Examples such as
cellular automata (Turing 1952 and Wolfram 1984) and a 'game
of life' (Gardner 1970 and Poundstone 1985) generate complex
spatial patterns from simple rules. The 'game of life' is
generated on an N x N matrix where
1. Every active cell with two or three neighboring
cells survives to the next generation.
2. Each active cell with four or more neighbors
'dies' from overpopulation. Every active cell
with one or no neighbors 'dies' from isolation.
3. Each empty cell adjacent to three 'live' neighbors
gives birth to a new cell.

10
Figure 3 is an example of the patterns generated from a
simple five cell seed (R-pentomino) during 512 iterations.
This pattern stabilizes (no more deaths and no more births)
after 1103 iterations, although it is an oscillating steady
state. Individual subsets of the final stable pattern
oscillate.
The 'game of life' model has some of the features of
autocatalysis (or cooperative behavior). Two or three live
cells are required for survival or birth of new cells. It
also has the feature of diffusive inhibition because indivi
dual cells that move out from a population center can become
isolated and die. This rule-based system has no energy
constraint that governs development and thus gives no energy
basis for pattern formation.
The common theme that runs through these examples is
one of combined interactions of autocatalytic growth with
some form of inhibition, diffusion or other mechanism for
preventing the autocatalytic growth from spreading too
rapidly. A concept that is sometimes misunderstood or mis
interpreted is that the terms fluctuation (Prigogine 1980,
1984) and bifurcation theory (Pacault 1977) refer to a
change in the kinetics of reacting components of a system.
This change in kinetics gives rise to the oscillations or
pulses in the output.
The models in this dissertation also use combinations
of autocatalytic and diffusion (linear) pathways to study

Figure 3. The development of spatial patterns among cells
based on simple r-pentamino initial condition (a) in 'a game
of Life'
simulation.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Time = 0
Time = 8
Time = 16
Time = 32
Time = 64
Time = 128
Time = 256
Time = 512

12

13
the possible mechanisms and energy consequences of pattern
formation in ecosystems.
Concepts of Pulsing, Patterns and Power
Maximum Power in Systems
Although in the last century Podalinsky, Ostwald and
Boltzman suggested energy use controlled system performance
(Martinez-Alier 1987), Lotka (1922) made a more definitive
statement. He stated that evolution proceeded in such a
direction as to make the total energy flux through the
system a maximum compatible with the constraints on the
system. He related this to Ostwald's (1892) idea of all
possible energy transformations, that one takes place which
brings about the maximum transformation in a given time.
A theory of minimum entropy generation was put forth by
Prigogine (Prigogine and Wiaume 1946) that a system evolved
toward a stationary state characterized by the minimum
entropy production compatible with the constraints on the
system. He has since called this a failure and probably a
special case of systems near equilibrium (Prigogine 1984).
Prigogine (1978, 1980, 1982; Prigogine and Stengers 1984)
now deals with systems far from equilibrium that have
dynamic and oscillatory behavior. He has not postulated any
definite theory about the energetic consequences of these
types of systems.
Odum and Pinkerton (1955) proposed that natural sys
tems tend to operate at that efficiency which produces a

14
maximum power output, a general restatement of Lotka's
original idea of maximum energy flow but with an important
distinction. Odum (1971, 1982, 1983a, and 1983b) further
clarifies maximum power as useful power where 'use' is
feedback of the product of energy use to amplify other
pathways.
In describing cycles of life, death and regeneration,
Calow (1978) has found that although Lotka's principle
holds, there seem to be no a priori grounds for placing
restrictions on how this use of energy should be achieved.
He further stated that selection would have shifted in the
course of time from one of maximizing speed to maximizing
efficiency. This is a restatement of the strategy of eco
system development utilizing r and K growth (Odum 1969) .
Jantsch (1980) suggests than maximum engagement in
matter (i.e., energy storage) and maximum process intensity
(i.e., entropy production) are criteria for ecosystem
stability. Non-equilibrium structures thus come about by
fluctuations in the mechanisms which result in modifications
of the kinetic behavior of these structures.
Design for Maximum Power
The important question here is how do systems build
structure in order to maximize utilization of available
power. Odum's theory (1971 and 1983) is that by feeding
back energy (derived from structure that is being built)
reinforcement occurs that increases efficiencies and energy

15
flow into the structure. Mechanisms must develop that build
structure to capture the most energy possible. These feed
back structures then have a prior energy use embodied in
them (emergy, after embodied energy, of a structure has been
defined as the total amount of energy used in developing
these structures (Odum 1983 and 1986)). This dissertation
looks at some of the possible kinetic pathways that feed
back to process energy and the energetics of these pathways.
Pathway Configuration
A simple model demonstrates several ways in which use
ful power can be increased (Figure 4, see description of
symbols in Figure 9). This model is a single storage with
autocatalytic production drawing on a flow-limited energy
source (an energy source with constraints on the pathway,
limiting the amount of energy that can be delivered).
The efficiency of a pathway can be increased if less
energy is fed back to gain more energy. For a simple auto
catalytic system (Figure 4a and 4b) this can be done by
either using less energy to gain the same inflow (changing
the value of K2 in the model) or by increasing the inflow
for the same feedback (increasing K1 while concurrently
decreasing K3). Because there are thermodynamic limits on
any process, it may not be possible to improve designs to
increase energy flows beyond thermodynamic limits.
The first law of thermodynamics requires the conserva
tion of energy. This implies the following constraint on
the production process of the model (Figure 4).

Figure 4. Basic autocatalytic model with flow-limited
energy source.
(a) Diagram with kinetic terms
dQ = Kl*JR*Q K2* JR*Q K4*Q
JR = JO / (1+K0*Q)
(b) Diagram with flow terms
RO = X0*JR*Q
R1 = Kl*JR* Q
R2 = K2*JR*Q
R3 = K3*JR*Q
R4 = K4*Q

i

18
K0*Jr*Q + K2*Jr*Q = Kl*Jr*Q + K3*Jr*Q (1)
Substitute R (flow) terras as abbreviations for terras
in equation (1):
R0 + R2 = R1 + R3 (2)
Inputs of energy of any process must equal the outputs.
Efficiency is defined as:
Efficiency = (Output'of useful power)/Inputs
or in terms of our equation:
Efficiency = R1/(R0+R2) (3)
where R3 is waste heat generated in the process (re
quired by the second law of thermodynamics). Because R3
cannot be zero, there is a natural upper limit to the effi
ciency of any process.
another method to increase energy flow from a flow
limited source is to have multiple pathways capture avail
able energy, each effective at a different energy level
(Figure 5). Multiple pathways (J1,J2,J3) use stored energy
to build structures to capture available energy. A linear,
donor-controlled pathway (J1) requires little structure and
employs no feedback in order to capture energy, but has
severe limitations (its efficiency cannot change) due to the
dependency on the energy source. An autocatalytic pathway
(J2) feeds back embodied energy (structure built by the
system) to draw in more energy. The quadratic pathway (J3)
is a co-operative phenomenon in which the structure of the
system is interacting with itself to feed back embodied
energy to draw in more power. A system that develops such

Figure 5. Basic multiple path model. Three input
pathways represent different feedback regimes:
linear (Jl), autocatalytic (J2), and quadratic (J3) .


21
higher order feedback pathways may exhibit a greater rate of
use of available energy.
This added quadratic pathway is available to utilize
any energy left after the efficiency is raised to the upper
limit for the autocatalytic pathway. This is a mechanism
that can draw in energy that would normally be unavailable
to the system. The quadratic pathway may have a high cost
to develop and maintain this pathway but it enhances overall
use of that extra energy by the whole system. This may give
a competitive edge in some circumstances over systems with
out higher order pathways, particularly when available
energy may be fluctuating. Available power will be in
creased by switching from one pathway to the other depending
on the energy source. Some pathways are more efficient at
low energy levels while others are more efficient at high
energy levels, thus allowing such systems to efficiently
utilize fluctuating power sources.
Pulsing and Patterns in Ecosystems
Succession and Disturbance
Any climax state is eventually interrupted by disturb
ances that generate patches in which succession is re
initiated. The gaps in a forest may be generated by local
outbreaks of consumers within the forest, tree mortality, or
outside disturbances such as fires, hurricanes, volcanic
activity, and landslides (Runkle 1985). The role of the
landslide as a gap-forming mechanism has been described in

22
both temperate forests (Oliver 1981, and Veblin 1985) and
tropical forests (Garwood 1979, and Leigh et al. 1982).
Disturbances (i.e., pulses) to an ecosystem can be gen
erated from within or can come from outside the boundaries
of an ecosystem and may vary in frequency and amplitude.
The ability of an ecosystem to utilize available resources
and adapt to these disturbances depends on the storages,
structures and interactions within an ecosystem (Odum 1983).
Hierarchical mechanisms may develop that capture and process
energy at various levels and result in utilization of energy
over a wider variety of input levels. Some mechanisms of
interaction between parts of the ecosystem were studied in
this dissertation to understand how systems may converge
energy transformations and feedback controls to organize for
higher productivity.
No unified theory of succession presented to date can
be regarded as widely accepted (Anderson 1986). Horn (1976)
wrote 'The sweeping generalization that can be safely made
about succession is that it shows a bewildering variety of
patterns.' Even the definitions of succession are widely
varying. In this dissertation succession is regarded as a
dynamic process in which the composition of an ecosystem
changes through time, building structure and processing
energy. This process eventually stabilizes in a climax from
which there is a regression or loss of that structure due to
disease, fire, treefall or other events. Seeding from an
other ecosystem or from storages in the soil from the

23
previous ecosystem regenerates a facsimile of the original
ecosystem through a sequence of unidirectional stages that
reaches a steady state system called a climax. This climax
may be arrested at some point and in some cases succession
may cycle between several stages. This definition is
broader than most but is an attempt to describe the whole
process instead of the more narrow 'growth-phase1'
definition.
Regression from a climax state may occur in several
ways. In some cases it comes about as a pulse of con
sumption from within the ecosystem boundaries such as tree-
falls, landslides or disease outbreaks. It can also come
about from disturbances from larger outside events such as
hurricanes or drought. The frequency and amplitude of these
disturbances tend to be inversely correlated: larger dis
turbances occur less frequently than smaller ones. This
phenomenon is referred to as a hierarchy of disturbances
(Bennett and Chorley 1978). The interaction of these dis
turbances along with the internal fluctuations may lead to
the 'bewildering variety of patterns' to which Horn refers.
Edges
Ecosystems can generally be broken up into subsystems
that have uniform characteristics. These subsystems have
boundaries where the composition changes from one particular
type to another. The development of these edges may occur
where differing types of energy interact with ecosystem
components to generate patches and zones of transition. The

24
presence of many spatially distributed patches may be due to
the production-consumption pulsing of components in the
ecosystem.
Hierarchies and Patches
The frequency of disturbance based on internal cycles
has been shown to be from 200-500 years in a variety of
ecosystems (Emanuel, West and Shugart 1978, Runkle 1985).
Distribution of disturbances over time varies from fairly
constant low amplitude disturbances to long-period, high
amplitude disturbances. The successional changes due to
disturbances may be related to the size and scale of the
disturbance (Peet and Christensen 1980, Peet 1981).
Brokaw (1982a, 1982b, and 1985a) found a hierarchical
distribution in gap sizes in a tropical rain forest at Barro
Colorado Island (Figure 6a). The area per size class is
plotted vs. the size class (Figure 6b). This relationship
may be important in determining patch dynamics. Brown
(1980) suggested that size class distributions may be
related to the emergy per size class (the emergy per size
class is also related to the area per class). Brokaw calcu
lated the turnover rate for the forest, based on the gap
formation, to be from 85 to 128 years depending on the
minimum size of the lowest class used.
Models
The simulation models used to study ecosystem behavior
generally fall into two classes (Shugart 1984). One of these

Figura 6. Size class distribution of gaps formed in
tropical forest at Barro Colorado (Brokaw 1982).
(a) Distribution of gaps by diameter of gap.
(b) Distribution of gaps by area in gap.

Total Area per Class (m*m) Number of Gaps
26
Gap Distribution
Barro Colorado
Size Class (m^m)
Gap Distribution
Size Class (m^m)

27
is based on the nonlinear 'Lotka-Volterra equations' and
generally does not include outside influences. The other
uses forced linear systems of differential equations and
does have inputs from outside the system. Neither of these
methods typically contains any spatial considerations and
both deal with systems near equilibrium. Systems near equi
librium tend to move toward that equilibrium and are char
acterized by spatial uniformity (Prigogine 1984 and Field
1985).
In this study, open non-equilibrium models are de
veloped that combine non-linear and -oscillatory interactions
between production and consumption with outside forcing
functions that provide resource controls. A pulsing, hier
archical model of production and consumption is used to
generalize about
succession and
reg
ression.
Spatial inter
actions generated
by this
model
are
studied
to understand
the energetic and
kinetic
basis
for
pattern
formation in
ecosyterns.
Gap Models and Patch Dynamics
Several previous studies based ecosystem models on
disturbance gaps. The JABOWA forest simulator model by
Botkin, Janak and Wallis (1972) keeps track of the birth,
growth, and death of a group of trees from seedlings on to
maturity within a certain gap size. Subroutines are used
for crowding, shading, and response to individual nutrients
and energy sources. The simulation then allows the gap to

28
develop a distribution of trees based on all of the input
parameters. These gap models generally do not account for
any outside disturbances that generate gaps.
Various gap models (Phipps 1979, Shugart and West 1980,
Shugart, Mortlock, Hopkins, and Burgess 1980, Shugart and
Noble 1981, Doyle 1982, Doyle, Shugart, and West 1982,
Shugart 1984, and Pickett and White 1985) have been utilized
to study forested ecosystems around the world. These models
have various gap sizes ranging from 100m'"2 to 833m~2.
Spatial Systems and Models
A spatial predator-prey insect microcosm was used by
Huffaker (1958) to study two species of mites. The prey
mite fed on oranges while the predator mite fed on the prey.
In one set of experiments, the oranges were distributed in a
10x12 grid with partial barriers between the oranges and one
prey placed on each of the 120 oranges. Five days later 27
predators were dispersed on the oranges. The resulting dy
namics in populations both over time (8 months) and space
are shown in Figure 7. In other experiments with oranges in
different arrangements, the oscillatory behavior was not
seen. Huffaker concluded that the predator-prey oscillation
would only occur when there was migration from the outside
or a sufficiently complex spatial arrangement of prey and
barriers to allow localized growth of the prey followed by
consumption by the predator.

Figure 7. Mite predator prey experiment (HufEaker 1958)
(a) Spatial distribution.
Prey concentration is shown by intensity of small
blocks (darker is higher density) and predator
locations are marked with small circles.
(b) Time series of total predators and prey in spatial
area. Letters on graph refer to the time series
for the spatial display next to the letter.

Prey: 0-5 nil density (white); 6-25 low density (light stipple); 26-75 medium density (horizontal lines); 76 or
over, high density (solid black). Predator; 1-8 (one white circle).
OCCIDEN TALI S

31
In high altitude balsam fir forests in the northeastern
United States, waves of tree loss and regeneration are
thought to be formed by an interaction of the prevailing
wind with the larger mature trees that are exposed along the
gap-wave (Sprugel and Bormann 1981, and Sprugel 1984). The
wind in this case acts to organize the disturbance cycle
that occurs normally in this type of forest into a spatial
wave pattern instead of randomly occurring patches.
The 'ohi'a dieback phenomenon in the rain forests of
Hawaii (Mueller-Dombois 1980) is a case of localized loss of
trees in the forest not due to disease or insect pest. It
was postulated that the effects were due to local soil
moisture loss arising from some climate instability. Repro
duction of the 'ohi'a was adequate enough to regenerate the
forest after the dieback, thus providing a way for this
shade intolerant species to become the primary canopy
species without further succession. Climatic variability
was thus used to an adaptive advantage.
Spatial modelling of ecosystems can be done in several
different ways. By using a model based on the FORET simula
tion model (Shugart and West 1977) and spatially distribu
ting the output of the model according to flooding condi
tions and hydroperiod, Pearlstine, McKellar and Kitchens
(1985) suggested possible species changes due to changes in
the hydroperiod caused by a river diversion in South
Carolina. In this case the number of individual subcell
models was kept small and the spatial distribution was based

32
on a combination of terrain relief, hydrology, and cor
related output from the simulation model.
Another approach to spatial modelling is to divide the
area into individual cells with a representative model in
each cell with some interaction terms among the individual
cells. This is the approach Costanza (1979) used in model
ling the economic development of South Florida.
Simulations with individual models for each cell have
certain advantages, because the interaction of neighboring
cells influences the outcome. A serious disadvantage where
the number of cells is large is the immense amount of
computer time required for the simulations. By making the
cell size larger this can be avoided, but loss of spatial
detail occurs as the cell size increases. The sub-cell
distribution modelling technique used by Pearlstine et al.
(1985) has just the opposite advantages and disadvantages.
The time requirements for simulation do not necessarily
increase as the area of cells is increased, but individual
intercell interactions are lost.
Plan of Study
Objectives
This study of energy use and pattern formation with
production consumption models has several parts:
First, the energetics of different pathway config
urations were tested using a series of minimodels. These
models were manipulated to determine the energy use of

33
systems with different production and consumption kinetics
and different combinations of components.
Second, a generalize production-consumption minimodel
calibrated with tropical rainforest data was used to study
the energetics of pulsing behavior.
Third, spatial pattern formation was investigated using
the pulsing production-consumption model as subunits in a
spatially distributed format. The spatial effects and
energy implications of various patterns of energy inputs,
edges, and lateral connectivity were determined.
These spatial simulations included several types of
inter-block exchange. Hierarchical relationships are rep
resented in these models when each consumer component inter
acts with more than one producer unit. The distribution of
gaps developed by simulations was compared with gaps in the
tropical rainforest in Puerto Rico.
Finally, insights and hypotheses were developed about
behavior of ecological systems.
Data site: Luquillo Rainforest, Puerto Rico
Data from the Lower Montane Rainforest in the Luquillo
Mountains of Puerto Rico were used to compare some of the
spatial simulations of pulsing and patches. Extensive
studies on this forest were published previously (Odum and
Pigeon, 1970).
Changes in structure and composition of a plot of
tropical rain forest near El Verde in Puerto Rico over a
period of 30 years were reported by Crow (1980). Data

34
included size class distributions taken in 1943, 1946, 1951
and 1976 (Figure 8)-. It can be seen that there is a shift
over time in the different size classes. The peak year for
the 0-8 cm class is 1946 while the peak in the 3-12 cm class
occurs in 1951 and the peak in the next three size classes
occurs in 1976. The smallest number in the lower two clas
ses also occurs in 1976. The last severe hurricane struck
Puerto Rico in 1932, and this movement through the size
classes appears to be the growth and development of an age
class of trees that grew back after the hurricane. The
hurricane in this case acts as an organizing disturbance to
reset succession of patches on a large scale.
The models simulated include the main integrative mech
anisms observed in ecosystems for coupling production and
consumption of spatially distributed units. Energy use of
these configurations was obtained from the simulations to
test the hypothesis that commonly observed organizational
designs with a successional regime that alternates pro
duction and consumption, tend to maximize system power in
the long run.

Figure 8. Size class distribution over time of plot of
trees in tropical forest at El Verde (Crow 1980).

36
Distribution
of trees over time

CHAPTER 2
METHODS AND MODELS
Ecosystem concepts, configurations, and models were
represented with energy circuit language from which simula
tion programs were derived. The energy circuit language,
developed by H. T. Odum (Odum 1971, Odum and Odum 1981 and
Odum, 1983), is a symbolic language for modelling ecosystems
and their components. Elements of storages, flows, and
interactions in this symbolic language keep track of the
laws of energy conservation. The energy diagrams also show
the correct kinetic interaction between parts of the system.
The level of aggregation or disaggregation that is needed to
understand and model a system for a particular purpose can
be achieved by drawing and revising diagrams using this
energy circuit language. A diagram of most of the important
symbols with a brief description of each is presented in
Figure 9.
One of the benefits of using the energy circuit
language is that it is possible to go from a conceptual
model to the development of the differential equations
needed to simulate the model in a few steps. Each of the
pathways on the diagram represents a flow that in turn can
be represented by terms in the differential equations that
37

Figure 9. Energy circuit language symbols (Odum 1983).

39
Energy circuit A pathway whose flow is propor
tional to the quantity in the storage or source upstream.
Source Outside source of energy delivering forces
according to a program controlled from outside; a forcing func
tion.
Tank A compartment of energy storage within the
system storing a quantity as the balance of inflows and outflows;
a state variable.
Heat sink Dispersion of potential energy into heat
that accompanies ail real transformation processes and storages;
Iocs of potential energy from further use by the system.
Interaction Interactive intersection of two path
ways coupled to produce an outflow in proportion to a function of
both; control action of one flow on another; limiting factor action:
work. gate.
Consumer Unit that transforms energy
quality, stores- it, and feeds it back autocatalyticaliy to improve
inflow.
Switching action A symbol that indicates one or
more switching actions.
-db
Producer Unit that collects and transforms
low-quality-energy under control interactions of high-quality
flows.
Self-limiting energy receiver (Chapter 10). A unit that has a
self-limiting output when input drives are high because there is a
limiting constant quantity of material reacting on a circular
pathway within.
Box Miscellaneous symbol to use for whatever unit
or function is labeled.
Constant-gain amplifier A unit that delivers
an output in proportion to the input I but changed by a constant
factor as long as the energy source S is sufficient.
Transaction A unit that indicates a sale of goods
or services (solid line) in exchange for payment of money
(dashed). Price is shown as an external source.

40
describe the changes in storage compartment (tank) values
over time.
Simulation Procedures and Programs
The majority of the simulations in this dissertation
were done in FORTRAN-4-PLUS on a Digital Equipment Corpora
tion (DEC) PDP 11/34 with RSX-11M operating system. The
graphical outputs of the simulations were displayed on a DEC
VK-100 graphics terminal (General Image Generator and Inter
preter or GIGI) connected to a Barco color monitor and DEC
.LA-34 Decwriter. The GIGI terminal has a 760x240 pixel
resolution and can display up to eight colors on a color
monitor. In order to facilitate the graphics programming
needed in my simulation models, I developed a set of FORTRAN
subroutines with a more natural calling sequence to execute
the ReGIS (Remote Graphics Instruction Set use by the GIGI
terminal) commands from the programs. This library of rou
tines (GGLIB) is listed and documented in the Appendix.
Some of the goals of this dissertation were to examine
the structure and function of systems in time and space and
to determine how variation in coefficients may affect energy
flows and storages of the systems. Graphical display pro
grams were developed to project a simulated 3-D surface of
the output of various state variables over time and over a
range of input conditions. A special 3-D graphics display
program was written to display the output of these model
simulations (program PLOTZ, Appendix).

41
The spatial models are broken down into cells that show
the concentration of a given parameter in the individual
cell as a color block. For display on the color monitor this
provides dramatic views of the model changes over time and
space. In order to make hardcopy printouts a display char
acter set was designed so the density of the dots in an
individual cell was correlated to the color of the cell.
This provided a way of screen-dumping the images to paper
and achieving patterns on paper that were similar to the
ones on the video screen (See Appendix for a listing of the
character set) .
Simulation Models
Minimodel Tests
First a group of minimodels were simulated to relate
energy use to basic pathway designs. Then spatial models
with these configurations were studied for energy use and
pattern formation.
Three path minimodel
In order to understand how a system processes variable
energy inputs, builds structure, and regulates or maximizes
energy flows, a simple single tank model was simulated. The
model is similar to the one described by Odum (1982) that
has parallel pathways of different types competing for
available energy (Figure 10). The model has a flow limited
source connected to a single storage (tank) by three dif
ferent pathways; a linear pathway (Jl), an autocatalytic

Figure 10. Three pathway model used to test effects of
various energy inputs on kinetic mechanisms.
Linear input: J1=K1*R
Autocatalytic input: J2=K2*Q*R
Quadratic input: J3=K3*Q*Q*R
dQ=J1+J2+J3-K4*Q
R=J0-J1-K0*R*Q-K5*R*Q*Q


44
pathway (J2), and a quadratic pathway (J3). The tank has a
linear drain.
The model represents a system that can change its use
of three functional pathways to get energy. The linear
pathway represents the energy flow that a system can re
ceive without any feedback in this pathway, only pathway
resistance to the flow. Because it is a donor controlled
pathway, the system has no control on the flow. Diffusion
pathways are an example of this type of energy flow. The
linear pathway is very efficient because it takes almost
nothing to receive the energy.
The autocatalytic pathway has a feedback from the sys
tem storage for interacting with an energy source to facili
tate the capture of more energy. If energy is available to
support the storage this pathway may lead to a competitive
advantage over the linear pathway. The efficiency of the
autocatalytic pathway depends on the energy source, the
storage and the pathway coefficient. A pathway of this type
has the capability of capturing more available energy.
The quadratic pathway has a self-stimulating feedback
(see equation on Figure 10) from the storage to capture
available energy. Examples of cooperative feeding that may
fit this model are common in ecosystems such as pack hunting
by some carnivores, cell and organ system interactions and
the cooperative work by humans in developed nations.

45
This model was simulated in BASIC (program THREEPATH in
Appendix) on a Digital Equipment Corporation (DEC) PDP 11/34
using a DEC VK-100 graphics terminal (GIGI). Measurements
were made of the percent of the input power used while
applying various levels of input power and varying the
frequency of input power. Simulation runs were also made
with one or more of the three pathways set to zero to deter
mine the impact of the various pathways on the overall
system behavior and power utilization.
In conjunction with the three path model in Figure 10,
a similar model with the same inputs but with additional
higher order drain pathways was simulated to determine the
effects on total power usage (Figure 11). In any system
that has crowding effects or high storage costs, these
drain pathways may determine how the system processes energy.
The model has a linear drain, an autocatalytic drain and a
quadratic drain.
The basic three path model was tested for the effects
of size and turnover time on the percent power used for
various power inputs by varying the drain coefficient (K4 on
Figure 10) in multiple simulation runs.
The percent power used when the three path model com
petes with individual storages with single pathways (Figure
12) was also simulated to see how the various pathways may
help or hinder a system. The competitors are individual
tanks with single pathways corresponding to the three path
ways in the three path model.

Figure 11. Three pathway
ways. Used to test effects
threepath model.
Linear input:
Autocatalytic input:
Quadratic input:
Linear drain:
Autocata lytic drain:
Quadratic drain:
multiple drain path-
order drain pathways on
J1=K1*JR
J2=K2*Q*JR
J3=K3*Q*Q*JR
J4=k4*Q
J5=K5*Q*Q
J6=K5*Q*Q*Q
model with
of higher
dQ=J1+J2+J3-J4-J5-J6
JR=J0-J1-K2'*J2-K3'*J3

47
1

Figura 12. Three pathway model with individual competing
units having single input pathways similar to combined
model. Coefficients in Appendix.
Combination tank:
Linear input:
Autocatalytic input:
Quadratic input:
dQ=Jl+J2+J3-K4*Q
Single tanks:
Linear input:
dQl=JlX-K4*Ql
Autocatalytic input:
dQ2=J2X-K4*Q2
J1=K1*JR
J2=K2*Q*JR
J3=K3*Q*Q*JR
J1X=K1'*JR
J2X=K21*Q2*JR
Quadratic input: J3X=K3'*Q3*Q3*JR
dQ3=J3X-K4*Q3
JR=J0-J1-K2'*J2-K3'*J3-J1X-K2'*J2X-K3'*J3X

49

50
For any system to survive over the long term, it must
fit into a regime of disturbances or catastrophic events
from sources outside its own boundaries. The system must be
tuned to the frequencies of those systems that influence it
in order to maximize power and survive. The three path
model was simulated with various frequencies of power input
to see how the various pathways process power at different
frequencies and amplitudes.
Parallel production-consumption minimodel
A model with producers in parallel was used to study
the effects of competition among producers (Figure 13). The
model had three producers, all having the same structure,
with one aggregate consumer that was consuming all three and
feeding back as a multiplier on the production function of
each. It is a basic predator-prey model with competition
among the different producers, along with feedback control
and energy constraints in the form of a flow limited source.
Instead of having combinations of pathways that can vary,
this model had combinations of producers that could vary.
The producers had different turnover times and coef
ficients so that Ql, Q2, and Q3 represented climax, mid-
successional (shrub) and early successional (weed) species.
The coefficient of consumption (the percent of each producer
the consumer eats per unit time) for each producer was
different. The weed species had a higher value than the
shrub species, which was higher than the climax species

Figure 13. Parallel production-consumption model.
Individual rate equations
R1 = K1*Q1* JR*Q4
R2 = K2*Q2*JR*Q4
R3 = K3*Q3*JR*Q4
R4 = D1*Q1
R3 = D2*Q2
R6 = D3*Q3
R7 = K7*Q1*Q4
R3 = K8*Q2*Q4
R9 = K9*Q3*Q4
RIO = FI* (K1*Q1*JR*Q4 + K2*Q2*JR*Q4 + K3*Q3*JR*Q)
Rll = KO* (K7*Q1*Q4 + K8*Q2*Q4 + K9*Q3*Q4)
R12 = D4*Q4
JR = J0/(1 + L1*Q1*Q4 + L2*Q2*Q4 + L3*Q3*Q4)
equations
for state va
dQl =
R1 -
R4 -
R7
dQ2 =
R2 -
R5 -
R8
dQ3 =
R3 -
R6 -
R9
dQ4 =
Rll
- R12
- RIO

52

53
(Odum 1969) A list of coefficients is given in Appendix
Table 3.
Several variations of this basic model were written in
FORTRAN and BASIC computer languages and simulated on both a
PDP 11/34 and on a Heathkit H8. The source listing for the
standard parallel production-consumption model (SUC10) is
presented in the Appendix.
Pulse Model
A general pulsing ecosystem model (Figure 14) was de
signed to test various hypotheses about energy flows and
pulsing, hierarchical organization, and spatial development
of ecosystems. Some of the structure of the model was
derived after the tests of the thredpath model and the
parallel production-consumption model. The model had many
characteristics of ecosystems such as:
1. Flow limited resources (representing solar based
energy resources).
2. Nutrient storage within the boundaries of the model.
3. Units of production, consumption and storage.
4. Feedback of consumers on production through nutrient
recycle.
5. Consumption at low maintenance rates and at high
pulsing rates.
6. Production through a fast turnover storage into a
long turnover biomass storage.
The basic pulsing ecosystem model was tested for dif
ferent flow rates, initial storages and energy inputs. From
this, a baseline understanding of the dynamic behavior of
the model and energy processing capabilities (as percent
power used) was developed.
The pulse model (Figure 14) was similar to the one in
Richardson and Odum (1981) with some changes in coefficients

r
Figure 14.
Pulse model of
tropical forest
ecosystem model.
Individua'
L rate
equations:
Rate eqiiati
ons
for state
variables:
R1 =
K1*Q1*
Q4* JR
dQl =
R1
- R2
R2 =
K2*Q1
dQ2 =
R3
- R9 Rll
R3 =
K3*Q1
dQ3 =
R7
+ R5 R12
R4 =
K4*Q1
dQ4 =
R12
+ R8 + RS
- RIO + R4
R5 =
K5*Q2
JR =
JO/
(1 + K13*Q1
*Q4)
R6 = K6*Q2
R7 = K7*Q2*Q3*Q3
R8 = K8*Q2*Q3*Q3
R9 = K9*Q2*Q3*Q3
RIO = K10*Q1*Q4*JR
Rll = K11*Q2
R12 = K12*Q3
R13 = K13*Q1*Q4*JR


56
and flows to calibrate it to a tropical rain forest ecosys
tem. The original model was run on an Electronics
Associates Incorporated model 2000 Analog/Hybrid computer.
The models presented in this dissertation were simulated on
a DEC PDP 11/34. The multiple simulations of the pulse
model were generated with a version of the program that
would run 25 simulations while varying a coefficient or
initial condition over those 25 runs and generate data files
that were then displayed with the FORTRAN program PLOTZ (See
Appendix). The source listing of the FORTRAN pulse program
is in the Appendix.
The pulse model was calibrated with tropical forest
ecosystem values for carbon flows and storages (Jordan and
Drewry 1969, Odum and Pigeon 1970, and Brown, Lugo, Silander
and Liegel 1983). The energy diagram of the model is given
in Figure 14 and the equations, coefficients and initial
conditions of the state variables are given in Appendix
Table 4.
Pulse Model With Prey-Predator Sectors
An additional higher trophic level consumer was added
to the pulsing consumer model (Figure 14) in order to test
the relationship of turnover time and hierarchical matching
of consumers (Figure 15). The extra consumer added to the
model had the same structure as the lower level pulsing
consumer (Q3), with both linear and quadratic pathways.
This model was tested by varying the turnover time of the

Figure 15. Pulse model with additional prey-predator
sector.
Individual rate equations:
Rate equations for state variables
R1 =
K1*Q1*Q4*JR
dQl = R1 R2
R2 =
K2*Q1
dQ2 = R3 R9 Rll
R3 =
K3*Q1
dQ3 = R7 + R5 R12 R18 -
R14
R4 =
K4*Q1
dQ4 = R4 + R6 + R12 + R12 -
RIO
R5 =
K5*Q2
+ R16 + R17 + R20
R6 =
K6*Q2
dQ5 = R15 R17 + R19
R7 =
K7*Q2*Q3*Q3
JR = J0/(1 + K13*Q1*Q4)
R8 =
K8*Q2*Q3*Q3
R9 =
K9*Q2*Q3*Q3
RIO
= K10*Q1*Q4*JR
Rll
= K11*Q2
R12
= K12*Q3
R13
= K13*Q1*Q4*JR
R14
= K14*Q3*Q5*Q5
R15
= K15*Q3*Q5*Q5
R16
= K16*Q3*Q5*Q5
R17
= K17*Q5
R18
= K18*Q3
R19
= K19*Q3
R20
= K20*Q3

CO
un

59
highest level consumer (Q5) and measuring the percent power
used and the level of the other storages in the system.
Spatial Models
The models previously discussed were time domain models
with no spatial effects. However, because ecosystems de
velop through time and space and spatial variations can be
at least as important as variations in time, spatial models
were developed and simulated to test hypotheses concerning
spatial development of ecosystems such as energy processing
and pattern formation and hierarchical control of pattern
formation.
The basic spatial model was a collection of subunits,
each one a pulsing consumer model (Figure 14). These sub
units were organized in a spatial format. When this simple
model was simulated in a spatial format, size effects, edge
effects and the consumer range of influence can become
important. Intercell interactions between individual produ
cers, consumers, nutrients, and energy sources may be im
portant in energy utilization and pattern formation.
Effects of edges in the spatial model were of interest
in pattern formation and energy use. Special boundary con
ditions were defined for the model cells along the edge.
These boundary cells were manipulated in the simulation
model in order to study the effects of edges on energy use
and pattern formation. The boundary cells were also manip
ulated to minimize the effect of edges in certain runs of
the model.

60
Any ecosystem can be divided into edge and non-edge
(center) parts. The amount of edge in an ecosystem is
a function of the size and number of the individual patches
within it. For a given area, as the number of subunits
increases the percent of the subunits on the edge decreases
(see Figure 16) .
A 10x10 matrix was used in the spatial simulations,
giving 36% of the total in edge cells and 64% in non-edge
cells. This size model was chosen to reduce the edge and
yet be small enough to simulate in a reasonable time. Com
puter runs for this model lasted approximately 3 hours on a
PDP 11/34. A model with a center to edge ratio of 10:1
would need approximately 20 times as many cells. In order
to test the effects of edges on the model, a single layer of
cells was added around the outside edges of the 10x10
matrix, giving it a 12x12 total area (Figure 17). The outer
layer was not acted as a buffer to approximate conditions of
an edgeless system.
Arrangements of cells
In simulating a spatial model, many arrangements of
cells can be used. The simplest form used was a linear array
with cells arranged in a linear ring. For two dimensional
models the cell geometry chosen was a square. This was done
for several reasons:

Figure 16. Number of edge and center cells as a function of
total number of cells in a given square area.

PERCE NI
62
EDGE EFFECT
Prim*tr and Cntr
PERIMETER -* CENTER


Figura 17. Cell geometries considered for spatial models.
(a) Square matrix with each cell having 4 side
and 4 corner neighbors. Active 10x10 matrix
embedded in a 12x12 matrix. This one was
chosen for the spatial simulations.
(b) Hexagonal matrix with each cell having 6 side
neighbors.

,0000000000
.oooooooooo-
oooooooooo;
::oooooooooo
,0000000000
.0000000000
.'.'OOOOOOOOOO-.
..oooooooooo;.
.'.-OOOOOOOOOO
.'..oooooooooo:.
! 000000000000!
1000000000000 i
OOOOOOOOOOOO!
iOOOOOOOOOOOO
OOOOOOOOOOOO
OOOOOOOOOOOO i
OOOOOOOOOOOO1
OOOOOOOOOOOO
000000000000:
OOOOOOOOOOOO
OOOOOOOOOOOO 1
OOOOOOOOOOOO j
1 1

1. It simplified programming the model because two
dimensional arrays in FORTRAN are set up in rows and
columns.
2. It simplified writing the graphics routines to dis
play the cells on a graphics terminal.
3. It reduced the edge effects of the model.
Ring model
A modified version of the two dimensional spatial model
was used to simulate a one dimensional case. The standard
spatial pulse model was connected head to tail in a ring of
36 cells.
Two dimensional models
The simplest spatial implementation was the basic pulse
model repeated over the 10x10 matrix with no interactions
between individual cells. This model (program DSP1) was
then simulated with three different energy forcing functions
1. The energy source was hierarchically distributed
(highest energy input at the center of the matrix).
2. The energy source was evenly distributed.
3. The energy source was randomly distributed.
Energy inputs were sea
whole matrix could be held
Overall energy input could
opment and energy use with
Two different initial
led so the mean input over the
constant for all energy types,
be varied to test pattern devel-
various energy levels,
conditions were tested. A suc
cessions! sequence was simulated with the initial values of

66
stored production (biomass, Q2 in Figure 14) set to a low
level. A steady state configuration was also used in which
Q2 was set to a value just below the pulse threshold. The
nutrient tank (Q4) in each case was balanced to contain the
remainder of carbon available in each cell.
This model tested different conditions and inputs.
1. Diffusion was allowed between nutrient tanks (Q4) of
each subunit. The base model (DSP1) allowed nutrients
to diffuse between cells at various diffusion rates.
The outer layer of non-reacting cells (see Figure 17)
had constant values for Q4 to allow tests of total
diffusion into and out of the cell matrix (diffusion
along the edges) .
2. Diffusion was allowed between consumer tanks (Q3) of
each subunit (program DSP1Q3) The outer non-reacting
cell layer was set to a constant value or was allowed
to float (program DSP1QZ) at the average of the inner
10x10 matrix to simulate a continuous sheet.
Simulations were run in which the consumer had a larger
area or territory than the producer. A model variation
(program DSP1C) was tested in which all of the consumer
tanks were clumped into one tank that aggregated consumption
over the 10x10 matrix simultaneously. This version also had
three different input energy patterns available, and allowed
diffusion between nutrient (Q4) tanks.

67
The final variation was a model with production com
partmentalized as before in individual cells but with free
roaming consumers, not constrained by cell boundaries. One
consumer was allowed to consume and move about the matrix
according to a set of constraints. When the consumer grew
above a preset size, it was split into two equal halves and
each half was allowed to consume, move and split again. An
upper limit of 100 was placed on the total number of con
sumers that could be generated during the run (the total in
the 10x10 matrix of the previous model versions). This
model also had three different energy inputs and diffusion
of nutrients (Q4).
Format for Spatial Display Graphs
Data from the spatial pulsing model were displayed
using the format shown in Figure 18. The spatial distribu
tions of the producers and consumers were shown at various
times during the run (usually 50 years apart). The values
of producers and consumers in individual cells were repre
sented by the density of dots in the cell. The producer
density increment was 2000 g/m~2 with a range of 0-16,000
g/m-'2 while the consumer was represented by an increment of
50 g/m"2 and a range of 0-400 g/m~2.
Measurement of Hierarchies at El Verde Site
In order to compare hierarchical relationships that
were generated in the model with those occurring in the

Figure 18. Format of spatial model display graphs.

69
X
Ux
OX
X)l
z: 2:
CJ
(X
ux
CJ^
IDU
1=11
<£
X
OOOOOOOOOO
OOOOOOOOOO
OOOOOOOOOO
OOOOOOOOOO
OOOOOOOOOO
OOOOOOOOOO
OOOOOOOOOO
OOOOOOOOOO
OOOOOOOOOO
OOOOOOOOOO
GGGGGGQGG
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QQGQQQQQQQ
QQQQQQQQQQ
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70
tropical rain forest at El Verde, several measurements were
made from data sets from the tropical rain forest study at
El Verde (1963-1967) in the Luquillo Mountains of Puerto
Rico (Odum and Pigeon 1970).
A data set (2048 samples) characterizing the forest at
the radiation site was generated by the U. S. Army Corps of
Engineers (Rushing 1970). At the radiation site, every
plant 1.8 m. or taller was enumerated within a radius of 30
m. from the center of the site. Each plant was recorded
with the species name, height, diameter, crown diameter,
exact location, and various other parameters.
Black and white negatives of aerial views of the radia
tion site (taken November 1963 before the radiation treat
ment) were printed as 8x10 inch photographs. Individual
gaps characterized by the presence of Cecropia peltata (an
early successional species) were digitized from the photo
graphs using a personal computer, Complot digitizer and
digitizing program written especially for this purpose
(Measure3 in Appendix).

CHAPTER 3
RESULTS
Simulation of Three Path Model
Individual Pathway Tests
The amount of energy flowing through each of the path
ways in the three path model (Figure 10) depends on the
total energy input to the model. As input power (JO) was
increased (Figure 19) steady state flows for each of the
pathways changed. Each pathway predominates at certain
times. The linear path had the largest power flow when
input power was low, while the quadratic pathway had the
highest flow at higher power inputs.
When input power was increased through time (Figure
20), there was no steady state, but, like Figure 19 when
power increased, the energy flow shifted from the linear
pathway to the autocatalytic and finally to the quadratic
path. The fraction of energy remaining (Jr/JO) also de
creased over time. As input power increased, a greater
fraction of the input power was utilized.
The model was run with different pathway combinations
(Figure 21) and with various power inputs. Each curve on
the graph represents a steady state value for various com
binations of pathways present in the simulation. Power used
71

Figura 19. Steady state power utilization oE units in the
three path model (Figure 10) as a function of input power
(JO) .

STEADY STATE PATH FLOUI
S+3

Figure 20. Energy utilization of individual components in
the three path model in Figure 10. Input power is increasing
through time.

in
r-

Figure 21. Steady state energy flows on various pathways
and combinations of pathways in the three path model (Figur
10) as a function of input power (JO).
Linear pathway:
Autocatalytic pathway:
Quadratic pathway:
J1=K1*R
J2=K2*Q*R
J3=K3*Q*Q*R

PERCENT POWER USED
< : ^ i n ^ n i

78
at any given input was highest with all three pathways pres
ent. For any combination of pathways that contained the
quadratic path (J1+J2+J3 or J2+J3 or J1+J3), power used
increased with power input to reach the same asymptote (>95%
power used). A slightly lower level was reached for path
ways dominated by the autocatalytic pathway (J2 or J2+J1).
This asymptote was approximately 90% power used with in
creasing power input. With only the linear pathway enabled,
no change occurred in percent power used with increasing
power.
A unique situation occurred when the quadratic pathway
(J3) existed alone. A low initial storage (Q) did not pro
vide enough feedback on the J3 pathway to allow growth.
Percent power used was never significant. The simulation
with only J2 and J2+J3 showed zero percent power used at low
input levels, then rose quickly at higher input power.
The size of the storage (Q) was varied to see the
effects on energy usage (Figure 22). This was achieved by
varying the depreciation coefficient (K4) in multiple run
while increasing power input in the three path model. At
high values of K4 (fast turnover times), increases in per
cent power used at steady state with increasing power were
small. With decreasing values of K4 (slower turnover
times), percent power used increased for the initial and
final values of input power.
The addition of multiple drains with different struc
tures (Figure 11) did not have as great an effect on the

Figure 22 Simulation of three path model in Figure 10.
Percent power used as a function of energy input and size of
drain coefficient (K4 varied from .02 to 2.0).

INPUT POWER
PERCENT POWER USED
iuu>uimsjco(e
'm

Figure 23. Simulation of three path model with multiple
drain pathways in Figure 11. Percent power used as a
function of energy input (JO).
Linear drain:
Autocatalytic drain:
Quadratic drain:
04=k4*Q
D5=K5*Q*Q
D6=K6*Q*Q*Q

PERCENT POWER USED
CO
O

83
model as multiple inflow pathways. The percent power used
was lowest when all combinations of drain pathways were
enabled (Figure 23). Percent power used increased with
increasing input power. The highest value for percent power
used was achieved when only the original linear drain was
present. Any combination with the linear drain used less
power at low power inputs than the nonlinear pathways alone
or in combination. The higher order drains enabled the
system to draw more power at low levels than when combined
with linear pathways. This effect was opposite from that
with input pathways at very low power where the nonlinear
pathways did not function well (see Figure 21).
The effects of adding competition pathways to the model
(Figure 12) can be seen in Figure 24. In this case, each of
the competing pathways (single tanks Ql, Q2, and Q3 with
individual pathways) were left on throughout the simula
tions. Here again the various pathways were disabled and
simulations run with varying power inputs. The results were
similar in some ways to those in Figure 21 where at high
power inputs the percent power used approached one of two
asymptotes. The greatest percentage of power utilization
occurred when all pathways were enabled and the lowest power
utilization occurred when only J2 or J3 were enabled. The
addition of the extra competing storages increased the per
cent power used in each of the pathway combinations compared
to Figure 21. These extra pathways were always there to use

Figure 24. Simulation of three path competition model with
various pathways enabled (Figure 12). Percent power used as
a function of energy input (JO).
Linear pathway:
Autocatalytic pathway:
Quadratic pathway:
J1=K1* R
J2=K2*Q*R
J3=K3*Q*Q*R

PERCENT POWER USED
CO
U1

96
whatever power may be left over (particularly the linear
path) .
Frequency Studies
The basic three path model (Figure 10) was also used to
test the effects of different frequencies of input power on
the model at three different power levels. At the lowest
power level (J0=500, Figure 25) the differences between
pathways in percent power used was the greatest. The great
est frequency response occurred at low frequencies. The
frequency response was flat with only the linear path en
abled. When all pathways were present, the percent power
used was highest with a peak at approximately 2 cycles. A
peak of power utilization also occurred with the combin
ations of J1+J2 and J1+J3. The pathways that showed a min
imum in the frequency response were composed of J2+J3 (the
two nonlinear pathways combined) and J2. The quadratic
pathway alone did nothing since no power was used (compare
with Figure 21).
When the input power was increased to 2000 (Figure 26),
the linear pathway showed no change in output with change in
frequency and the quadratic pathway had no output. The
combination of J1+J2 here again had a slight maximum at
about 2 cycles while J2 alone had a maximum at zero cycles.
The combination of all of the pathways (J1+J2+J3) and J1+J3
had a slight minimum of power utilization at about 8 cycles,
while the combination of J2+J3 showed a slight minimum at
about 2 cycles.

Figure 25. Simulation of the three path model in Figure
10. Percent power used as a function of frequency of the
input power (J0=500).
Linear pathway:
Autocatalytic pathway:
Quadratic pathway:
J1=K1*R
J2=K2*Q*R
J3=K3*Q*Q*R

PERCENT
88
a
L
0)
D
£C
Ul
3
o
0.
90.
. : 1
1 : 1 = 1 : i 1 ; l 1
ALL
! 1 : .
8$.

J1+J3
70.
J1 + J2
J2+J3
-
6.
^

50.
J2
40.
"
30.
J1

20.

10.

J3

0.
1
' j 1 I ; I
= 1 ~
0. 5. 10. 15 20 25. 30 35. 40 45 50.
INPUT FREQUENCY
Figure 26. Simulation of the three path model in Figure
10. Percent power used as a function of frequency of the
input power (J0=2000).
Linear pathway:
Autocatalytic pathway:
Quadratic pathway:
J1=K1*R
J2=K2*Q*R
J3=K3*Q*Q*R

PERCENT POWER USED
V£>
O

91
When the input power was raised to 10000 (Figure 27)
the percent power used went up for all combinations of
pathways except the linear path. The quadratic pathway was
operational at this high power level but with a significant
minimum at 2 cycles per run. Other combinations had small
minima and maxima that are hard to see at the scale of this
graph.
The response of the model to various frequencies and
power input is shown in Table 1. Simulation runs with
pathway J1+J2 had a maximum in percent power used at all
three power inputs while the combination of J2+J3 had a
minimum in percent power used at all three power inputs.
The combination of all pathways (J1+J2+J3) has a peak of
maximum percent power utilization at low power and low
frequency input. At higher power levels percent power util
ization (with all three pathways enabled) was lower with
some shifting in the frequency at which this occurs.
Simulation of Parallel Production-Consumption Model
Single Run Simulations
The parallel production model showed a successional
pattern with the initial dominant species (Q3, with the
fastest turnover) growing up, then declining as Q2 became
the dominant species and finally Q1 (with the slowest turn
over) reached a maximum and then dropped back to a slightly
lower steady state (Figure 28). The consumer (Q4, with the

Figure 27. Simulation of the three path model in Figure
10. Percent power used as a function of frequency of the
input power (J0=10000).
Linear pathway:
Autocatalytic pathway:
Quadratic pathway:
J1=K1*R
J2=K2*Q*R
J3=K3*Q*Q*R

PE RCENT POWER USED
cs>

94
Table 1. Frequency response (mnimums and mximums) of
three path model (Figure 10) with varying input power.
Pathway
combination
INPUT POWER
J0=500
J0=2000
J0=10000
J1+J2+J3 (All)
MAX (2)
MIN (8)
MIN (3)
J2+J3
MIN (2)
MIN (2)
MIN (2)
J1+J3
MAX (2)
MIN (8)
MIN (3)
J1+J2
MAX (2)
MAX (2)
MAX (2)
J1
N/R
N/R
N/R
J2
MIN (2)
MAX (0)
MAX (0)
J3
N/O
N/O
MIN (2)
Numbers in parenthesis are the frequencies at which the
maximum or minimum occurs.
N/R signifies there was no frequency response for this set
of pathways.
N/O signifies there was no power uses at these inputs

Figure 28. Simulation of the four sector succession model
in Figure 13. Model base run.
Legend:
PPU = Percent power used
PPD = Percent power drained
D = Diversity
B = Total Biomass
P = Productivity
Q1 = Climax species
Q2 = Mid successional species
Q3 = Early successional species
Q4 = Consumer

001 mil
96

97
longest turnover time per unit) also rose to a steady state
value. During this time, the productivity climbed to a
local maxima, then dropped slightly, finally climbing to a
slightly higher steady state. The percent power used for
the whole run was 95.5%.
In order to test the role of each of the producers
early in the simulation, a series of runs were made with the
initial condition of one of the producer species set to zero
(Figure 29 a,b,c). With no initial climax species (Ql)
present (Figure 29a) the shrub species (Q2) became dominant
in the final steady state. The percent power used for the
run was 94.7%, slightly less than the base run configura
tion. This configuration did not support as high a level of
consumer (Q4) compared to the base model run (75.2 vs.
90.8) .
When the shrub species (Q2) was absent (Figure 29b),
the percent power used for the run and steady state values
for the consumers were similar to the base run. Without the
shrub species present to compete during the middle period,
the final climax species (Ql) peaked earlier and higher than
in the base run.
When the weed species (Q3) was initially absent (Figure
29c), the system was not self sustaining. The primary
reason was that during the early part of the simulation, the
consumer (Q4) was dependent on the weed species (Q3). With
no Q3 present, the consumer Crashed very quickly. The whole
system then crashed because the consumer feeds back in

Figure 29 Simulation of the parallel production-consumption
model in Figure 13. See Figure 28 for legend and ordinate
scale.
(a) Simulation run with initial value of climax
species (Ql) set equal to zero.
(b) Simulation run with initial value of interme
diate species (Q2) set equal to zero.
(c) Simulation run with initial value of weed
species (Q3) set equal to zero.

99
100

100
the production function of all of the producers in the
system.
When the model was simulated with no initial consumer
(Q4) it crashed even faster (not shown) than in Figure 29c
because of the feedbacks in the model from the consumer to
the producers.
Multiple Run Simulations
The behavior of the parallel production model with
varying input power is seen in Figure 30a-f. In this set of
runs the base model was run for 100 time units. For each
successive run, the input power (JO) was increased, varying
from 50 to 300. As the energy input increased, the peaks of
the producers were higher (Q1-Q3), with Q3 (the weed
species) showing the most change in amplitude (Figure 30c).
The climax species (Ql) peaked sooner as the input power in
creased .
The simulation of succession to a climax was thus
speeded up by increasing the energy input at lower levels,
but at higher levels the increase in energy had little
effect on the transition to dominance of the climax species.
The effect of increasing energy input was also seen in
the level of the consumer (Q4 in Figure 30d) With in
creasing power, the consumer was maintained at a propor
tionately higher steady state.
For this set of simulations, as the input power in
creased, the percent power used increased asymptotically
(Figure 31). There was a diminishing return on the input

Figure 30. Simulation of the parallel production-
consumption model in Figure 13. Multiple simulations of the
model with available power increasing from 50 to 300.
(a) Climax species (Ql)
(b) Intermediate producer species (Q2)
(c) Weed species (Q3)
(d) Consumer species (Q4)
(e) Percent power used (J0-Jr)/J0
(f) Total biomass

102
Q4 1
Tme
Time
100.

Figure 31. Simulation of the parallel production-
consumption model in Figure 13. Multiple simulations of
the model with percent power used for entire run vs input
power. See Figure RS3a-f.

PERCENT POWER USED PERCENT POWER USED
104
M,
9.
39.
79.
69.
59.
49.
39.
29.
19.
9.
9 39 69 99. 129. 159. 189 219 249 279 399.
INPUT POWER (J9)
-1-1. I
[ ; I : I
-1
I ; I 1
INPUT POWER

105
power as the effect was greater at low power than it was at
higher levels of power.
When the input power was varied as in the previous
example (50 to 300) but the initial condition of the con
sumer was started at a higher level (Q4INIT=50, lOx base run
value) the results were similar to the previous run but
damped (Figure 32a-f). The shift in time of the peak of the
climax species (Ql) was less than before and the amplitudes
of the initial peaks of Q2 and Q3 were less. Percent power
used per time increment also was higher in the earlier
stages of this run compared to the previous run (compare
Figure 32e with 30e). With higher initial levels of the
consumer, the model generated more power earlier through the
feedback of the consumer on the producers.
When the input power was held constant (J0=100) and the
initial condition of the consumer (Q4) varied, the model
displayed two different behaviors (Figure 33a-f). With few
consumers initially, the system crashed, unable to proceed
through the normal growth sequence. When the initial quant
ity of the consumers (Q4) was above a critical level, the
system grew and went through a normal growth sequence. A
sharp transition occurred in the percent power used as
Q4INIT was increased (Figure 34).
Because the consumer (Q4) was feeding back as a multi
plier to the producers, some minimum critical value must
exist for the consumer population to stabilize this model.

Figure 32. Simulation of the parallel production-
consumption model in Figure 13. Run with available power
increasing from 50 to 300 and the initial value of the
consumer (Q4) equal to 50 (lOx base run in Figure 28).
(a) Climax species (Ql)
(b) Intermediate producer species (Q2)
(c) Weed species (Q3)
(d) Consumer species (Q4)
(e) Percent power used (J0-Jr)/J0
(f) Total biomass

107
50 o
Tim
Tim
100

Figure 33. Simulation of the parallel production-
consumption model in Figure 13. Multiple simulations of the
model with available power held constant (J0=100, base run
value) and the initial value of the consumer (Q4) varied
from 1 to 6.
(a) Climax species (Ql)
(b) Intermediate producer species (Q2)
(c) Weed species (Q3)
(d) Consumer species (Q4)
(e) Percent power used (J0-Jr)/J0
(f) Total biomass

109
Tim
Time
100

Figure 34. Simulation of
consumption model in Figu
for entire run as a funct
consumer (Q4). This repr
33e.
the parallel
re 13. Total
ion of the in
esents a cros
production-
percent power
itial value of
s section of F
used
the
igure

PERCENT POWER USED
111
1. 1.5 2. 2.5 3. 3.5 4. 4.5 5. 5.5 S.
INITIAL VALUES, Q4

112
Either immigration or a temporary auxiliary support system
is necessary to start a system of this class.
Similarly, when the input power was held constant
(J0=100) and the initial condition of the weed species (Q3)
was varied (Figure 35a-f) the system crashed at low levels
of Q3, but at higher levels it was stable (see Figure 29c
for a single run with Q3=0).
The system response was different with changes in the
initial conditions of Q1 and Q2 (refer to Figures 29a and
29b) because the consumer was not as dependent upon them for
its survival early in the simulation.
Initial Conditions and Total Energy Use
The behavior of the parallel production-consumption
model with different initial conditions for the state var
iables (Ql, Q2, Q3, and Q4) and input power was tested. In
this set of simulations, the total percent power used was
measured for each simulation run while varying the input
power and the initial condition of the state variables one
at a time (Figure 36).
In all four cases when JO was low, the model was unable
to utilize the energy available to it. When the input power
was above a certain point then the model was able to utilize
the input energy with two exceptions. When Q3 (weed
species) was very low, the percent power used rose to a
plateau then fell when the input energy went above a certain
level. The model was unstable under these conditions.

Figure 35. Simulation of the parallel production-
consumption model in Figure 13. The initial value of weed
species (Q3) was varied from 0 to .5 and input power was
held constant (J0=100, base run value).
(a) Climax species (Ql)
(b) Intermediate producer species (Q2)
(c) Weed species (Q3)
(d) Consumer species (Q4)
(e) Percent power used (J0-Jr)/J0
(f) Total biomass

114
Time
Time
100

Figure 36. Steady state values of percent power used as a
function of input energy and state variable initial
conditions for multiple simulation runs of parallel
production-consumption model (Figure 13).
(a) Vary input energy and
(b) Vary input energy and
(c) Vary input energy and
(d) Vary input energy and
Q1 (Climax species)
Q2 (Intermediate producer)
Q3 (Weed Species)
Q4 (Consumer)

Percent
Power Used
100
300
Percent
Power Used
116

117
When Q4 was below a certain threshold the system could
not be sustained regardless of the input energy. After an
initial threshold level of consumers was reached, the system
was stable, similar to that described above for Q3. Since
Q4 has a direct feedback on Ql, Q2, and Q3, the interaction
of these in the production term can determine whether or not
the system was stable. If the value of Q4 was too low then
there was little production and the system crashed.
Simulation of the Pulse model
Single Run Simulations
A simulation of the base run pulse model (Figure 14) is
shown in Figure 37a. As Q2 increased, the available carbon
or nutrient carbon tank (Q4) decreased proportionately. As
the stored biomass increased there was a threshold level at
which the consumer (Q3) began to grow rapidly and pulsed.
This pulse consumed Q2 and released the carbon back into the
available carbon pool (Q4). The threshold of pulsing was
dependent on the level of both Q2 and Q3. The level of Q3
before the pulse was, however, directly related to the level
of Q2 and the input diffusion pathway. After the pulse, the
consumer (Q3) decayed back to a low level.
The cycle repeats itself at a frequency of approximate
ly 325 years. The power used varied during the simulation
with the highest rate occurring shortly after the pulse,
when the nutrients have been concentrated in Q4 as available
carbon.

Figure 37. Simulation for pulse model (Figure 14) with
base run coefficients (See Appendix).
(a) Base run of model.
(b) Input energy one-half of base run.
(c) Input energy two times the base run.
Legend:
PPU = Percent power used
Q1 = Production unit
Q2 = Stored biomass
Q3 = Pulse consumer
Q4 = Nutrient storage

I J
O Q2/Q4 48000
o ppu mo
i
Q2/Q4 48000
i l
0 PPU 100
I 1
o PPU 100
119

120
If the input energy was less (J0=50, half of the base
run) then the pulse came at a later time (Figure 37b) and
the frequency of pulsing had a longer period. The production
was lower and the stored biomass (Q2) took longer to reach
the level that would trigger the pulse in the consumer (Q3).
When the input energy was raised to twice the level of
the base run (J0 = 200), the consumer pulsed only one time
(Figure 37c) and then remained at a low level instead of
decaying away entirely as in the base run. With a low level
of consumer, the stored biomass was not able to build up and
remained at a lower steady-state level.
The total power used for each of these runs was related
to the input power. As the input power went up, the percent
power used also went up from 93.3 at 50%, to 96.5 at 100%
and 98.3 at 200%.
The quadratic pathway between the stored biomass (Q2)
and the consumer (Q3) was responsible for the pulsing much
as the autocatalytic pathway of a Lotka-Volterra model is
responsible for its oscillating limit-cycle behavior. With
only the linear path between Q2 and Q3, the behavior was not
pulsing or oscillatory (Figure 38). The stored biomass grew
while the nutrients were used up. In this time frame (760
years), the values did not reach a steady state and 93% of
the available power was used. When simulated for 2000 time
units (Figure 38b) the percent power used dropped off to a
low steady state value. The system became nutrient limited

Figure 38. Simulation of pulse model (Figure 14) without
a quadratic pathway (K7, K8r K9 = 0.0).
(a) Simulation for 760 years.
(b) Simulation for 2000 years.
Legend:
PPU = Percent power used
Q1 = Production unit
Q2 = Stored biomass
Q3 = Pulse consumer
Q4 = Nutrient storage

Time 2000
122

123
because most of the nutrients were tied up in the stored
biomass.
When the pulse model was run without feedbacks into Q4
(pathways R6 and R8 cut off) the model continued to pulse
but began to decline (Figure 39). The percent power used
dropped as the level of Q4 dropped until one final pulse and
then everything decayed to a low steady state condition.
Multiple-run Simulations
When the input power was increased, the result was most
noticeable on the stored biomass (Q2) and the consumer (Q3,
Figure 40). At low values of JO there was no pulsing within
the time frame of the simulation (760 years). As JO was
increased, the pulsing began as a result of the stored
biomass (Q2) increasing to a threshold level at which Q3
pulsed and consumed the stored biomass (Q2). As JO was
further increased, the pulsing frequency increased. At high
levels of JO the first pulse decayed and the system switched
to a steady state with Q2 being maintained at a low level
(see Figure 37c for example). The total power used (Figure
40e) increased linearly as JO increased with small fluc
tuations over time due to the pulsing of Q3. The percent
power used (Figure 4 0 f) was less than 80% for low values of
JO then rose rapidly through the pulsing and leveled off as
JO approached 250. The percent power used was reduced by
the initial consumption but returned to a maximum after the
pulse. The percent power used increased as the available

Figure 39. Simulation of pulse model (Figure 14) without
feedbacks into Q4 (K6, K8 = 0.0)
U)
Simulation
for
760 years.
(b)
Simulation
for
2000 years
Legend:
PPU = Percent power used
Q1 = Production unit
Q2 = Stored biomass
Q3 = Pulse consumer
Q4 = Nutrient storage

Time 2000
125

Figure 40. Multi-run simulation of the pulse model (Figu
14) with variation in input energy. (JO varied from 0 to
250) .
(a) Production unit (Ql).
(b) Stored biomass (Q2).
(c) Pulse consumer (Q3).
(d) Nutrient storage (Q4).
(e) Power used (JO-Jr)
(f) Percent power used 100*(J0-Jr)/(JO)

Time Time
Percent Power
Used
oooo, aa. aaipoo.

128
power was increased (similar to three path models seen
earlier) with local maxima immediately after the pulse.
The total amount of nutrients in the system also had an
important effect on the behavior of the model (Figure 41a-
f). At higher initial levels of Q4 there was little effect
on the model. At these higher ranges, the model was no
longer nutrient limited but was energy limited. At low
values for the initial concentration of Q4 the pulsing
greatly affected the labile production (Ql), the stored
biomass (Q2) and the pulsing consumer (Q3). At the lowest
level of Q4, there was no pulsing, Q2 remained at a low
steady state value, and Q3 also remained at a low steady
state value. There was a small shift in the pulsing fre
quency at the lowest initial levels of Q4 but no frequency
shift at the higher levels. The power used was greatly
affected at low initial levels of Q4 but rose only slightly
at higher values of Q4. For the same amount of change in
Q4, the variability of the power used was greater when Q4
was small than when Q4 was high. However, the percentage
change was greater in the beginning than at the end.
The turnover time of the pulsing consumer affected the
behavior of the system and use of power (Figure 42a-f). The
pulse model was simulated with the value of the drain coef
ficient (K12) of the consumer (Q3) varied with each run. As
the turnover time increased, the frequency of pulsing shift
ed to a shorter period with the amplitude decreasing until
there is no pulse at all but a continually rising consumer.

Figure 41. Multi-run simulation of pulse model (Figure
14) with variation in total carbon in model. (Q4 varied
from 2000 gC/m2 to 100,000 gC/m2.
(a) Production unit (Ql) .
(b) Stored biomass (Q2).
(c) Pulse consumer (Q3).
(d) Nutrient storage (Q4).
(e) Power used (JO-Jr)
(f) Percent power used 100*(J0-Jr)/(JO)

zzzz
o rr

Figure 42. Multi-run simulation of pulse model (Figure
14) with variation is turnover time of pulsing consumer.
(K12 varied from .01 to .5).
(a) Production unit (Ql).
(b) Stored biomass (Q2).
(c) Pulse consumer (Q3).
(d) Nutrient storage (Q4).
(e) Power used (JO-Jr)
(f) Percent power used 100*(JO-Jr)/(JO)

760-- u 760
Time Time
2000 ^ r-40000

133
This implies there is a 'window' of size for the consumer to
pulse.
Changing the rate constant (K9) of the quadratic path
way caused the pulsing consumer to change frequency, in
creasing the frequency of pulsing with an increasing coef
ficient value (Figure 43). There was a point in this set of
simulations where the pulsing ceases but in this case the
size of the consumer remains small. When the quadratic
pathway became dominant at low consumer levels, the system
did not pulse but completely consumed the stored biomass
storage (Q2).
When simulated without the quadratic pathway and chang
ing the coefficient of the linear pathway (Kll), the model
did not pulse, the consumer (Q3) remained at a low level and
the stored biomass (Q2) built up (Figure 44, compare to
single run figure 38). As the linear pathway increased,
there was a slight increase in the consumer (Q3) with less
of a build-up in the stored biomass (Q2). In all cases,
through time the power use and percent power used dropped
off.
Simulation of Pulse Model with Prey-Predator Sectors
Simulation of the pulse model with an additional prey-
predator sector (Figure 15) investigated how turnover time
is related to hierarchical consumers (Figure 45). With a
drain coefficient on Q5 the same as or larger than that of
the normal pulsing consumer (K17=0.05 or 0.5), the effect

Figure 43. Multi-run simulation of pulse model (Figure
14) with variation in quadratic pathway (K9 varied from
0.5E-6 to 0.53E-5 with K7 and K8 varied proportionately).
(a) Production unit (Ql) .
(b) Stored biomass (Q2).
(c) Pulse consumer (Q3) .
(d) Nutrient storage (Q4).
(a) Power used (JO-Jr)
(f) Percent power used 100*(JO-Jr)/(JO)

o
5E-5
135

Figure 44. Multi-run simulation of pulse model (Figur
14) with variation in linear pathway (Kll varied from
to 0.12E-2 and K5 and K6 varied proportionately) with quad
ratic pathway held at zero.
(a) Production unit (Ql).
(b) Stored biomass (Q2).
(c) Pulse consumer {Q3).
(d) Nutrient storage (Q4).
(e) Power used (JO-Jr)
(f) Percent power used 100*(J0-Jr)/(JO)
O ID

2T f. 40000

Figure 45. Simulation of pulse model with prey-predator
sectors (Figure 15).
(a)Simulation with turn-over time of higher level
pulsing consumer (Q5) set equal to lower level
pulsing consumer (Q3).
(b)Simulation with turn-over time of higher level
pulsing consumer (Q5) set to ten times longer than
the turn-over time of lower level pulsing consumer
(Q3) .
(c)Simulation with turn-over time of higher level
pulsing consumer (Q5) set to one hundred times
longer than the turn-over time of the lower level
puling consumer (Q3).

0009
UJ!i
0
0009

140
was hardly detectable in the simulation result (Figure 45a).
The frequency of pulsing was not changed and the power
utilized was only negligibly changed. The higher level
consumer (Q5) was near zero for the entire simulation.
When the model was run with a turnover time (K17=0.005)
of the top consumer (Q5) longer than the normal pulsing
consumer (Figure 45b), pulsing occurred at the normal fre
quency but the higher level consumer grew over time until it
began pulsing. The period of pulsing became longer and the
pulse amplitude of the stored producer and nutrient storages
became greater. The normal pulsing consumer (Q3) remained
at a low level, acting as a feeder to the higher level
consumer (Q5). The power utilized dropped slightly to 95.3.
When the turnover time of the higher level consumer
(Q5) was raised by another order of magnitude (K17= 0.0005)
the outcome was quite different (Figure 45c). The higher
level pulsing consumer (Q5) climbed toward an asymptote
while the stored production (Q2) also climbed to a steady
state value. the normal pulsing consumer (Q3) again re
mained at a low level. In this case the nutrients (Q4)
became tied up in the stored biomass (Q2) the power used
dropped to 36.1 at steady state. The percent power used for
the entire simulation was 48.2.

141
Simulation of the Ring Model
The linear array ring model was simulated with high
diffusion (DK=.l) between consumers in adjacent cells
(Figure 46). Initially the concentration of producers and
consumers around the ring was constant except for a single
consumer at a high level (Q3 (2,2) =100.; lower left hand
corner of consumer matrix). At T=50 years (Figure 46A), the
consumers had pulsed in both directions around the ring and
completely encircled the ring by T=100 (46B). At T=150
(46C), the production was beginning to spread around the
ring from the lower left corner and continued through T=
200, 250, 300, 350 (46D-H). The consumers again began to
grow (T= 350, 46H) and spread around the ring again. This
was followed by another wave of production and consumption
(T= 500-750, 46J-0).
In runs with lower diffusion (0.01) between consumers,
the pulse wave traveled slow enough that the wave only moved
part way around the entire ring before the internal pulse
frequency allowed the remainder of the consumers to pulse,
thus stopping the wave. With an even lower diffusion coef
ficient of 0.001, the wave moved 3 cells before stopping.
With a diffusion coefficient of 0.01, the wave moved 10
cells before being stopped by the natural internal pulse
frequency.
A different pattern developed when the producers and
consumers in the model were distributed in a random pattern
around the ring (the individual cell concentration of pro-

Figure 46. Simulation of pulse model (Figures 14 and 18)
with cells in a linear ring and diffusion between consumers
of each cell in ring (DK=.l). For each time unit (e.g. A=0)
density of producer and consumer in the matrix is shown
along with size class distribution. The time series from A
to 0 summarizes the temporal pattern of totals in matrix.
Initial conditions of consumers were set to near zero except
for one "seed" consumer at lower left corner of matrix which
was set to 100.
Spatial pallara
X- \ ..
CONSUMER
CELLS
>14000
M 12000
13 ,ooo
jSsj sooo
W. S000
$8 4000
**? 2000
g/a.2
I >380
300
2S0
200
ISO
:>*' too
jff 80
Consumar
alta olaaaas
s/-a

143
m
1 11 J
1 II J
1 II J

Ul 1
Ip
|garro
J :
: S:: :
WBMMW333J
u
r j
< a

¡
L
Li? mam a
Q
D
m
i
X

144
ducers and consumers was constant and the same as the homo
geneous initial conditions) and diffusion set to zero
(Figure 47). The output was based entirely on the random
field from the initial conditions. Each individual cell
model was producing and consuming at the same rate but there
was no spatial synchronization of the cells. The pattern
repeated itself over time (compare T=50, 47A with T=700,
47N) .
When diffusion was set at a high level (0.1) between
the consumers, with the same random initial distribution of
producers and consumers, the resulting pattern was quite
different (Figure 48). The pulsing consumers moved in a
wave around the ring followed by a wave of production (T=50,
100, 150, 200, 250, 300; Figure 48A-F) followed by another
wave of consumption beginning just prior to T=350 (48G).
This was similar to the simulation in Figure 46 that began
with a homogeneous initial distribution of producers and
consumers and had waves of consumption and production around
the ring.
When the model was run with random distribution of
producers and consumers (Figure 49) and a low value of
diffusion (DK=0.001), the spatial pattern that developed had
some properties of both of the two previous runs. Because
speed of movement was less with a lower value of diffusion,
a number of focal points for pulse waves were generated
which then run into each other and stop. The production
follows the pattern of consumption with multiple foci.

Figure 47. Simulation of pulse
with cells in a linear ring but
time unit (e.g. A=0) density of
matrix is shown along with size
time series from A to 0 summari
totals in matrix.Initial condit
sumers were set to random distr
model (Figures 14 and 18)
without diffusion. For each
producer and consumer in the
class distribution. The
zes the temporal pattern of
ions of producers and con-
ibution around ring.
I
>14000

M
12000
Jfl
1
10000
6000 g'"2
I
6000
98
4000
2000
. a
0

>960
900
260
200 9/a
160
100
lx el**

146
a Mi STM a* a
a -
a : L '

a m ass a* a
a i
l j
SS 9 'SHSi.'.i "SS
D

m -S3
8 1 m

t, j
a* a .. *
>£- >?,
- i S-:
I I
I 1
aa~ 9:::..:\v a
r
asa
Lj
s Jii

i i
j
"N
IM 1
i
!
1
i
-> i
3 33 dHMl ril
Ijg na
1 L 1
^ i
I
^ !1

Figure 43. Simulation of pulse model (Figures 14 and 18)
with cells in a linear ring and a high level of diffusion
between consumers of each cell (DK=.l) and random distribu
tion of producers and consumers around ring. For each time
unit (e.g. A=0) density of producer and consumer in the
matrix is shown along with size class distribution. The
time series from A to 0 summarizes the temporal pattern of
totals in matrix.
>14000
12000
10000
000 /2
000
-4000
2000
Consumer
Iso oloeso*

148

Figure 49. Simulation of pulse model (Figures 14 and 18)
with cells in a linear ring and a ,low level of diffusion
between consumers of each cell (DK=.001) and random distri
bution of producers and consumers around ring. For each
time unit (e.g. A=0) density of producer and consumer in the
matrix is shown along with size class distribution. The
time series from A to 0 summarizes the temporal pattern of
totals in matrix.
1
>14000
12000
10000
000 S
I
000
4000
2000
-JO
Produoer
lx* olaatet
v
Contunar
tlx* !

150

151
Simulation of Two Dimensional Surface Models
The simplest simulation of the two dimensional pulsing
model, with no diffusion and an evenly distributed energy
source (Figure 50), had a time series output identical to
the basic pulse model (Figure 37a). Even though the model
was disaggregated into 100 cells, each of the cells was
identical. In this run, each of the cells was synchronized
(by the initial conditions) and the pulsing was based only
on the internal frequency of the model (T=250, 50E and
T=600, 50L). There was little change in the size distribu
tion of the producers and the consumers during the
simulation.
The influence of an energy source that is hierarchical
ly distributed from the center of the matrix outward gen
erates a different pattern (Figure 51). The production was
higher in the center of the matrix than at the outer edges.
In this simulation without diffusion there was no edge
effect. The first pulse came at the center of the matrix
(highest input energy) and then moved outward to the edge in
a series of pulses. The production and consumption then
continued to oscillate. The frequency of pulsing in each
individual cell depended on the intensity of the energy
input to that cell (see also Figure 40a-f). The center
cells pulsed at a higher frequency than the outer cells due
to differences in input energy. The time series of the

Figure 50. Simulation of pulse model (Figure 14) with
cells arranged in two dimensions (Figure 18) without
diffusion and with a constant energy source. For each time
unit (e.g. A=0) density of producer and consumer in the
matrix is shown along with size class distribution. The
time series from A to 0 summarizes the temporal pattern of
totals in matrix.
Spatial paitara
CONSUMES
CELLS
Oaaalty
cala
O'"*2
,>14000
13000
j§ 10000
fix sooo
S# 000
W 4000
fff 2000
9/m2
%lm2

153

Figure 51. Simulation of pulse model (Figure 14) with
cells arranged in two dimensions (Figure 18). Energy
source hierarchically is distributed from center outward and
no diffusion between cells. For each time unit (e.g. A=0)
density of producer and consumer in the matrix is shown
along with size class distribution. The time series from A
to 0 summarizes the temporal pattern of totals in matrix.
000
4000
2000
>960
900
260
200 */*
160
100

155

156
simulation had sharp peaks due to the different frequencies
of pulsing of the independent cells. The size distributions
of the producers and consumers were based on the input
energy and are grouped accordingly. Without diffusion, the
pattern formed was entirely dependent on the hierarchical
pattern of the input energy.
The addition of diffusion between the consumers of each
cell for the previous model smoothed out the time series for
the consumers and producers (Figure 52). A low level of
diffusion (DK=0.001) enabled the first pulsing cells (lo
cated at the center of the matrix) to affect the neighboring
cells, thus spreading the pulse wave out over the matrix.
In this simulation the size distribution of the producers
and consumers tended to smooth out over time. The edge
effects were minimized in this simulation by allowing the
outer non-reactive ring of consumer cells to float at a
value that was the average of the total consumers in the
matrix.
Diffusion between the consumers at a low level had a
much greater effect in this two dimensional version of the
model than in the one dimensional ring version of the model.
When the two dimensional version was run with a random
energy source and a low diffusion coefficient (Figure 53,
DK = 0.001) the effect was similar to that seen in Figure 52.
In this case, local foci of high productivity (caused by
locally high values of input energy) led to pulses that
spread over the entire matrix. This simulation was dif-

Figure 52. Simulation of pulse model (Figure 14) with
cells arranged in two dimensions (Figure 18). Energy
source is hierarchically distributed from center outward and
diffusion is between consumers of each cell (DK=.001). For
each time unit (e.g. A=0) density of producer and consumer
in the matrix is shown along with size class distribution.
The time series from A to 0 summarizes the temporal pattern
of totals in matrix.
Sn.tlal si,, ,1a.
distribution
1000 1
fcL
l
10 CELLS
10 CELL
40.0001
I
S\ duo.,.
10
10
I >14000
tsooo
10000
000
000
§5 4000
*** 2000
_Jo
Con.ua>.,
1*0 olo.so*

158

Figure 53. Simulation of pulse model (Figure 14) with
cells arranged in two dimensions (Figure 18). Energy
source is randomly distributed and diffusion is between
consumers of each cell (DK=.001). For each time unit (e.g.
&=0) density of producer and consumer in the matrix is shown
along with size class distribution. The time series from A
to 0 summarizes the temporal pattern of totals in matrix.
I >14000
uooo
10000
000
000
$*§ 4000
aooo
/*
__vjo L_Jo
Produaar Conauar
ilia olaaaaa alza olaaaaa

lt
:1; '::i'.:';;
: : i : si :
o
¡a Irjjji-'ii
* IsTlik
a
* ::i'3s;"
m
t 1:
fjbtfe
LUI
ill

!:
ii#
ULUCI
. : : -
m
.vnW Ttcjis
Ejs^to)-jsh w¡
s s _
.
k
/ 1"

LIU
yj
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nn
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c
uu

161
ferent from the random ring simulations (Figures 48 and 49)
in that the input energy was randomly distributed while in
the case of the ring model the initial producer-consumer
pairs were randomly distributed. Little synchronization of
the matrix occurred because the random energy distribution
caused locally high concentrations of producers every time
there was a pulse. In the simulation of the ring with
randomly distributed producers and consumers, at a high
level of diffusion the pulse wave moved fast enough to
reset all of the producers and consumers to similar values.
With a low diffusion value, the wave traveled so slowly that
it did not get around the ring, and multiple foci of pulsing
developed.
Simulation with diffusion between the nutrient com
partments of each cell (Q4) of the model instead of to the
consumers (Q3) can be seen in Figure 54. With a random
distribution of energy and a high level of diffusion
(DK=0.1) the pulsing was almost totally uncoupled. By the
end of the run (T=750, Figure 540) there was constant pul
sing in one cell or another, and the overall level of pro
ducers as seen in the time series graph was fairly constant.
The spatial configuration of the model was also tested
with a moving consumer. This is similar to the diffusion
runs of the model but represents an active process with
discontinuous (non-uniform) movement of consumers from cell
to cell. The consumer was allowed to search for the largest
producer to consume before moving. The model was tested

Figure 54. Simulation of pulse model (Figure 14) with
cells arranged in two dimensions (Figure 18). Energy
source is randomly distributed and diffusion between
nutrient storages (Q4) of each cell is set to high level
(DK=.l). For each time unit (e.g. A=0) density of producer
and consumer in the matrix is shown along with size class
distribution. The time series from A to 0 summarizes the
temporal pattern of totals in matrix.


a
>14000
a
>380
M
12000
M
300
4
10000
280
ooo g/2
200
000
180
V.w
4000
100
2000
;;;
80
oduO
0
m Co
0

163

164
with a hierarchical energy input and a consumer search
length of 1 cell (Figure 55) and a search length set to 5
cells (Figure 56). The simulations are quite different in
both the spatial patterns generated and in the time series
graph of the simulation.
When limited to a search length of 1 cell, the con
sumption pattern moved like a wave from left to right across
the producers after starting in the center. With a longer
search length (Figure 56), the consumption began in the
center and spread out in a circular pattern over the pro
ducers. There are two of these waves of consumption during
the time of the simulation for the search length of 5. The
run with a search length of 1 cell has slower consumption
and only moves across the field once.
Rain Forest Gaps and Hierarchies
Size Class Distributions
Three different size class distributions (Figure 57)
were generated from the data set from the radiation site at
El Verde to characterize the hierarchical patterns in the
vegetation. Figure 57a represents the distribution of
plants by diameter. This can be compared to the data from
Crow (1980) in Figure 8. The distribution of plants by
crown diameter (Figure 57b) and by height (Figure 57c) was
hierarchical. The sampling technique affected the results
in the lowest size classes.

Figure 55. Simulation of the pulse model (Figure 14) with
cells arranged in two dimensions (Figure 18). Moving
consumer model with search length set to one cell, no
diffusion and hierarchical energy distribution. For each
time unit (e.g. A=0) density of producer and consumer in the
matrix is shown along with size class distribution. The
time series from A to 0 summarizes the temporal pattern of
totals in matrix.
4000
M 12000
m loooo
000
££ ooo
4000
2000
9/m2
i/-2

166
- 1
IIll
/

L
kfrM
m u
m
|| fiis
WW
1:p fl
*a
II :: : :
r

Figure 56. Simulation of the pulse model (Figure 14) with
cells arranged in two dimensions (Figure 18) Moving
consumer model with search length set to five cells, no
diffusion and hierarchial energy distribution. For each
time unit (e.g. A=0) density of producer and consumer in the
matrix is shown along with size class distribution. The
time series from A to 0 summarizes the temporal pattern of
totals in matrix.
1X4000
? 2000
10000
000
000
4000
2000
'2
/a

168

Figure 57. Size class distribution of trees at El Verde
radiation site (November 1964)
(a) Size class distribution by diameter
(b) Size class distribution by crown diameter
(c) Size class distribution by height

170
Diameter Distribution
Height Distribution
Radiation Site
Wmmm
LZn Y7\ I
10 12 14 16 18 20 22 24 26 26

171
Gap Size Measurements
The distribution of cecropia gaps at El Verde fall into
a hierarchical distribution (Figure 58). Figure 58a is the
size distribution of all four of the photographic plots
combined and Figure 58b shows the distributions of the
individual plots. The percentage of the total area that is
in the gap stage is 3.79% (Table 2). The values plotted in
Figure 58 are the actual areas measured in square inches on
the photograph. The figure shows a minimum size for the
gaps and a hierarchical distribution.
Comparison to Models
For each time slice that the spatial simulation model
printed a spatial pattern of producers and consumers, it
also printed a graph of the size distribution of the produ
cers and consumers (just to the right of the spatial pat
terns). The format of the distribution graph is not the
same as the size class distributions in Figure 57 but the
size distributions do represent the same class size phenom
enon. Depending on the energy input conditions and dif
fusion coefficients some of the size distributions had simi
lar relationships to the natural distribution (see Figures
51, 53 and 56) while others are quite different (see Figure
50). The pulsing in Figure 50 is totally synchronous while
the pulsing in Figures 51 and 53 are more spatially
asynchronous.

Figure 58. Size distribution of Cecropia gaps in tropical
rainforest at El Verde.
(a) Distribution of gaps in all five photographs.
(b) Distribution of gaps in each individual photograph.

Number per Class Number per class
173
a Size Distribution of Cecropia Gaps
0.02.02.040-4 .0606 .0808 .1 01 0.1 21 2.141 4.1 61 6.1 8 >.18
Size Class (sq. inches)
/I Combined data
b Size Distribution of Cecropia Gaps
from Aerial Photographs
Size Class (sq. inches)
pic2 yz7A pics
PIC8
CT71 PIC1

174
Table 2. Area o£ gaps digitized from photographs of Luquillo
tropical rain forest.
Picture number
1
2
5
8
Total
Number of gaps
53
43
35
74
205
Mean
0.0761
0.0517
0.0455
0.0473
0.0554
Std. Error
0.0212
0.0096
0.0043
0.0083
0.0066
Minimum
0.0093
0.0125
0.0094
0.0067
0.0067
Maximum
1.121
0.360
0.100
0.5525
1.121
Area % of total
5.38
2.97
2.12
4.66
3.785
* Means are not significantly different p=.005

175
The gap size distribution was measured for a set of
spatial simulations with input energy distributed hierarchi
cally, evenly and randomly. Figure 59a represents a com
bined gap size distribution measured from a set of three
different simulation runs using a hierarchical input energy
source. The gap size distribution is skewed to one set of
large gaps and a few smaller patches. With a random energy
source the results (Figure 59b) resemble the size class
distribution of the natural system (Figure 58) with more
small patches and fewer large ones. With an evenly dis
tributed input energy source and no diffusion, the system
pulses in a synchronous manner that generates a gap the size
of the simulation (100%) with each pulse. With diffusion
present, the patch size is dependent on the edge effect. If
the edge effect is canceled the result is the same; however
with a diffusive loss or gain along the edge, the patch size
is reduced from 100% due to the uncoupling of the synch
ronous pulsing at the edges.

Figure 59. Size distribution of gaps in tropical rainforest
pulsing model simulation (Figures 14 and 18) at time =760.
(a)
Size
class
distribution
from
three separate
runs
with
hierarchical
energy
distribution.
(b)
Size
class
distribution
f rom
three separate
run
with r
andom energy
distri
bution.
model
model

Number Number
177
Size Class Distribution
Size Class Distribution
Random Energy Input
0.0.2.02.0404.0606.0808.1 01 O. 1 21 21 41 4. 1 61 6.1 8 >.18
Class Size (Percent of simulation area)

CHAPTER 4
DISCUSSION
Many of the characteristics of ecosystem function were
generated by the simulations in this dissertation. Energy
increased with growth. Net production alternated with pul
sing net consumption. Hierarchical patterns in space re
sulted from oscillations in time. Edge effects developed.
There were similarities with succession observed in nature.
Many characteristics of ecosystems were generated by mini
models that had autocatalysis, recycling, parallel pathways
of different order, spatial intercell exchanges and hier
archical distribution of time constants. In other words,
simple models emulated many features of more complex
ecosystems.
The spatial model in this dissertation differed from
many previous spatial ecosystem models that used individual
species growing and interacting together (Botkin, Janak and
Wallis 1972, Phipps 1979, Doyle 1982). This model was a
unit ecosystem model that combined all of the species into
compartmentalized production, consumption and nutrient
storages. This simplified the model but kept many of the
ecosystem characteristics.
178

179
Maximum Power Considerations
The class of models studied here duplicate real systems
by reinforcing pathways that process more power. The feed
backs simulate useful power processing. These models link
kinetics and energetics in ways observed in nature.
Power and Feedback With Paths of Higher Order
Systems that generate higher order pathways to cap
ture varying energy flows may offer a competitive advantage.
The maximum power implication is that as systems develop
feedbacks (higher order pathways) they can extract more
energy from the source. Lotka (1922) stated that as long as
there was untapped available energy, systems were capable of
growth when rates of flow increased through the system.
Odum (1982 and 1983) added that as systems mature they feed
back energy which amplifies other pathways and maximizes
power. The multiple pathway configuration shown in the
three path model provides a possible mechanism for this to
occur. In the three path model simulations (Figures 19 -
27) the linear pathway had a fixed efficiency while the
autocatalytic and quadratic pathways had variable effi
ciencies (see Figure 20 and 21) depending on the input
power.
The development of multiple pathways in a system is
incurred at some energy cost to the system. The energy
costs associated with developing and maintaining the non
linear pathways must be competitive to survive. For systems

180
with small storages (i.e. fast turnover times), the quad
ratic pathway can be non-functional (Figure 21, pathway J3).
Because non-linear pathway flows are a function of both the
energy source and the storage, there are conditions when the
pathway has a threshold for operation (Figure 21, pathway J2
and J2+J3 and Figure 27 pathway J3). Low energy systems may
not have enough energy available to allow development of
these higher order pathways.
Human systems may be a good example of how these path
ways may operate. Nomadic, subsistence societies can be
considered as basically linear systems that utilize avail
able resources with few or no feedbacks. By developing
autocatalytic feedbacks, primitive societies move up to
developing societies building structures to process more
energy (farming, mining, transportation and manufacturing).
As growth continues, systems develop within society that
have higher order quadratic feedbacks to facilitate proces
sing energy (communications, banking and finance, and infor
mation systems). Because the higher order pathways are
dependent on storages and energy flows, the structures may
not be stable with reduced energy.
For a system pathway to utilize fluctuating energy
flows, it must have enough structure to sustain the system
when the non-linear pathways are not functioning (at lower
energy levels). While the nonlinear pathways were dependent
on the frequency and amplitude of input energy (Figures 25,
26 and 27) the linear pathway had no frequency dependency

181
and thus provided energy to the system under all input
regimes. A system with a combination of pathways then shows
greater stability under fluctuating regimes and maximizes
power with increasing energy inputs.
Multiple pathway models have been used to describe a
variety of systems. A disaster model using multiple path
ways (linear and autocatalytic) has been used to describe
earthquakes and floods (Alexander 1978). Models of chemical
reacting systems have often used multiple pathway models to
describe the kinetics of the reactions ("Brusselator",
Nicolis and Prigogine 1977 and "Oregonator", Field and Noyes
1974). "Chaotic systems" are often modeled with multiple
non-linear pathways (Abraham and Shaw 1984b).
Effect of Hierarchies on Performance
Hierarchical subunits of a system generally have in
creasing turn-over times with increasing trophic levels
(Allen and Starr 1982, Urban, O'Neill and Shugart 1987).
The addition of an extra consumer (adding a level to the
hierarchy) of the pulse model (Figure 15) must have the
appropriate turnover time to survive. If the turn-over time
was too short, not enough energy was available to that level
of the hierarchy to sustain it and the added level did not
survive (Figure 45a). If the turnover time was too long,
the rate of power use dropped and the whole system collapsed
(Figure 45c). The appropriate size consumer modified the
output behavior of the model (pulsing with a longer period),
but the system was stable and utilized slightly more power.

182
The highest level of the hierarchy in this model determined
the frequency and scale of pulsing. Therefore there are
optimum turnover times for maximum performance.
Conversely, as input power increases, a higher level of
consumers may be supported. This was seen in the parallel
production-consumption model (Figure 32d) and the pulse
model (Figure 40c).
Power Used as a Function of Input Power
The general trend for all of the models tested here was
that as the input power increased, the percent of input
power that is utilized increased. This occurred in the
three path model (Figures 21,- 22, 23), the parallel
production-consumption model (Figures 30e, 32e, and 36), the
pulse model (Figure 40f) and the spatial models. This
appears to be a function of the non-linear pathways that
feed energy back to increase the efficiency with increasing
available energy. Individual simulations of these models
with only linear pathways did not show this behavior.
Threshold for Stable Feedbacks and Pulsing
The pulse model exhibited a double threshold phenom
enon. At low power inputs the model did not pulse and at
high power inputs the model did not pulse (Figures 37 and
40). In the middle power range, the model pulsed and the
pulse frequency was a function of the input power. Local
ized maxima of power utilization may occur in the pulsing
range due to synchronization of inputs with natural internal

183
frequencies (Richardson and Odum, 1981). This double
threshold behavior has also been shown in a wide variety of
prey-predator model configurations (Kuno 1987). Oscillating
chemical reactions exhibit this multiple output state be
havior (Field 1985).
At low power levels, the pulsing model supported a
constant low amount of consumers (dependent on the linear
pathway) while at high power levels the consumer was at a
constant higher level (sustained by both the linear and
quadratic pathways) with the producer at a low level. This
was also the case in the chemical reactions and prey-
predator models described above. Models with this behavior
may describe a variety of ecosystems that show various
levels of producers and consumers. A grassland ecosystem
such as the Serengeti (McNaughton 1985) may be an example of
low levels of producers supporting high levels of consumers.
A similar dependence of the highest trophic level on
the input energy was also exhibited with the parallel
production-consumption model (Figure 32) although this model
did not pulse. It should be noted with this model that the
consumer level increased and the 'climax' producer did not.
The pulsing model did not pulse when the consumer quad
ratic pathway (Figure 38) was removed, the consumer built up
to a steady state, and the percent power used declined.
There was a lot of structure in.the higher level of the
hierarchy but the system was not effective at using the
extra power that was available. Competitively, a system

184
with this structure may be at a disadvantage and could be
eliminated through consumption by a higher level of the
hierarchy or competition by other systems at the same level
of the hierarchy.
If a system was not materials conservative (feedbacks
from the consumer to the nutrient storage cut off or di
verted, Figure 39) then the system ceased pulsing and ran
down. The system had no feedback pathways and so did not
capture all of the available energy.
Implications for Succession
Role of Individual Units
Early successional producers can be thought of as pre
paring the way for succession to occur. Although early
successional species may have other roles, in the parallel
production-consumption model (Figure 13) they can be seen as
providing an energy source to the consumer level of the
model as the rest of the system builds up. When the early
successional species was at a low level, the consumer level
(Q4) remained low (Figure 35). This low consumer level did
not feed back enough to the producers to stabilize the
system and the system crashed. As the early successional
species (Q3) reached a threshold initial condition, suf
ficient structure was built and the system progressed to a
steady state.

185
If the consumer level in a successional system is too
low then the system may not be stable. In the parallel
production-consumption model, the consumer provided a feed
back on the producers through the input production multi
plier and through consumption on the producers. When the
consumer was at a level that was too low, succession as
depicted by the model (Figure 33 and 36) did not begin. At
some initial threshold level of consumers, the model pro
ceeded through a successional sequence.
In developing management plans for revegetating sites
disturbed by mining, intensive agriculture or natural dis
turbances, it is imperative that careful attention be paid
to the whole structure of the ecosystem that is being re
built. Without the proper mix of early, middle, and late
successional producers along with a set of consumers that
match the producers, the restablishment of a natural suc
cessional sequence may be retarded or destroyed.
Succession and Pulsing
The role of pulsing in succession may be that in some
systems it is necessary to have the pulsed recycle to main
tain energy flows near maximum levels. Several cases of the
pulsing model (Figures 38 and 39) showed that when recycling
was disturbed power use dropped. Certain types of sucession
may need an alternation of production and consumption at a
frequency that allows the maximum use of available energy.
Systems in which available nutrients become bound in the

biomass may benefit by the fast release from a pulse of
consumption and recycle.
186
Spatial Pattern Formation
Synchronous vs. Asynchronous Systems
When a spatially organized system is totally synchron
ized (all subunits behaving as one) the system may be like
a monoculture with little pattern formation other than that
of the local source inputs. In this state, pattern diver
sity is low. Where cells are not all synchronized with each
other, patterns can develop that are dependent on the
asynchronous nature of the individual subunits as well as
the local energy sources.
When the spatial model was simulated with all of the
individual cells uncoupled (not linked through intercell
diffusion processes) and totally synchronized (all cells
begun with the same initial conditions and an even energy
distribution), no pattern was generated (Figure 50). The
level of producers and consumers was the same in each cell
at every point in time.
Any variation in the energy input over the matrix area
lead to individual cells pulsing at frequencies depending on
the energy level local to that area (Figure 51). Although
the pattern was quite different from the synchronized one,
the energy use is the same (Table 5 in Appendix).

187
Coupling of Spatial Units by Diffusion Processes
In any ecosystem, spatially distributed subunits are
connected to each other through a variety of processes.
Nutrients and seeds can be carried spatially by transport
from wind, water and animal activity. Predation by con
sumers tends to reorganize the vegetation community struc
ture. The degree to which subunits are connected to one
another is strongly reflected in the patterns that may
develop.
Connectivity between subunits tends to decrease the
asynchronous behavior caused by local energy differences.
With a low level of diffusion (Dk=.001) the pulsing behavior
was propagated across cell boundaries (compare Figure 52
with Figure 51). At higher levels of diffusion (not shown),
the effect was to increase the synchronous nature of the
pulsing across the matrix. Energy use with various levels
of diffusion did not change appreciably (Table 5 in
Appendix).
In a single dimension system (ring model) the effect of
diffusion was similar. At high levels of diffusion (Figure
48) pulses were propagated around the entire ring, while at
a lower level of diffusion the propagation was confined to
local areas (Figure 49). The asynchronous pulsing (Figure
47) was thus organized into a more synchronized spatial
pattern depending on the degree of connection between the
individual cells.

188
The level of the hierarchy in which inter-cell coupling
takes place plays an important part in the development of
spatial patterns. When this coupling took place at the
level of nutrient exchange from cell to cell (Figure 54),
the effect was hardly detectable, even at high diffusion
levels.
Spatial patterns generated are not totally dependent on
the natural energy inputs but organize using those natural
energy regimes. Spatial diversity thus depends on the the
landscape energy pattern, the interactions between the sub
units, the hierarchy level of the interaction and the
existing pattern of vegetation.
Most of the models in this dissertation used only
diffusive coupling between spatial subunits of the model.
Many systems have more complex interactions between subunits
than this simple linear coupling. The active transport
systems of biological systems are good examples of the more
complex coupling that can occur in living systems. The
moving consumer model represents a more complex coupling
between individual cell units.
Organization by Higher Level Consumers
The role of the consumer in these models was very im
portant in organizing pattern formation. When the spatial
model was simulated with one consumer spread evenly over the
matrix, the result was exactly the same as when the model
was simulated with all cells uncoupled and a single consumer
in each cell of the matrix (Figure 50). In this case, the

189
synchronous organization of individual consumers (100 total)
over the entire matrix of cells mimiced the effects of a
large consumer with the same territory. The percent power
used for each of these simulations was the same (Table 5 and
6 in Appendix).
Coupling of the consumers from cell to cell by dif
fusion organized the consumer action over the whole matrix
depending on the strength of that coupling. Low levels of
diffusive coupling generated local areas of organization by
the consumers (Figure 53) while strong coupling organized
the disturbance over the entire matrix, (not shown but
similar to Figure 50).
Active coupling between subunits by consumers was simu
lated using a moving consumer model (Figures 55 and 56). In
this case, very different patterns were formed with a smal
ler number of consumers. The action of organizing the
entire landscape (10x10 matrix) was achieved with fewer
consumers. The energy use was not significantly different
from the other spatial simulations (Table 7 in Appendix).
The efficiency of active coupling may be higher than passive
(diffusion) coupling.
Organization at a higher level tends to have a larger
effect in generating patterns. Some of this may due to a
type of 'memory' generated in the landscape by the
disturbance-succession sequence generated by these pulsing
production consumption models. As the system pulses, small
differences between individual cells generate further dis-

190
continuities. These small differences act as information
storage for future pattern development.
Power Use and Edge Effects
No system exists in an infinite plane without edges.
Edges were manipulated in the spatial models to understand
their role in pattern formation. Some of the simulations
allowed consumers to diffuse into or out of the spatial
matrix at high and low levels of diffusion.
When the consumer level on the outside ring was kept at
a low value (0.0), the percent power used decreased (Table 8
in Appendix) with increasing rates of diffusion. If the
outside buffer had a high value for the consumer (Q3 equal
100) then just the reverse was seen. With increasing rates
of diffusion there was an increase in the percent power
used. This implies that consumer exchange can act as an
energy source or a drain in a system depending on the
relationship of the system to its surrounding area through
its edges.
General Principles
The following are some general principles suggested by
model studies, which may be useful hypotheses in future
experimental studies.
1) Multiple pathways increase efficiencies and enable
better use of fluctuating energy sources. Multiple steady
states can result from one basic configuration. The

191
kinetics of these pathway configurations are similar to
others studied by chaos theory, bifurcation theory and
catatastrophe theory.
2) Hierarchical structure is expressed in kinetics as
increasing turnover times with increasing territory. Path
ways of control of production-consumption systems must match
the turnover time of the appropriate hierarchical level in
order to cause reinforcement.
3) In early successional systems there may be critical
minimum stocks of producers and consumers for a system to
grow.
4) Similar maximum power processing may be achieved by
a wide variety of spatial patterns.
5) Connectivity in systems has a greater role in
pattern formation at higher levels of the hierarchy.
Control of patterns and patchiness through consumer control
is highly dependent on the spatial connectivity of the
consumers.
6) Patch size m
the consumer and the
7) Some of the
simplified for human
be grouped according
ay be related to the turnover time of
spatial connectivity of the consumers,
great complexity of ecosystems may be
comprehension if varied mechanisms can
to the basic kinetics, energetics and
hierarchical roles they perform.

APPENDIX
192

193
Table 3.
Coefficient values for parallel production-consumption
model in Figure 13.
K1
.003
Production coefficient for
Q1
K2
.005
Production coefficient for
Q2
K3
.007
Production coefficient for
Q3
D1
.1
Drain coefficient for Q1
D2
.2
Drain coefficient for Q2
D3
.3
Drain coefficient for Q3
K7
.006
Consumption coefficient for
Q1
K8
.015
Consumption coefficient for
Q2
K9
.040
Consumption coefficient for
Q3
D4
.08
Drain coefficient for Q4
K0
.1
Intake coefficient for Q4
FI
.01
Feedback loss coefficient for Q4

194
Table 4
Steady state values, coefficients and flows for pulse model.
J0
100.
Sunlight normalized to 100
JR
4.0817993
Available sunlight at ground level
Q1
1000.
Labile storage (Primary producer)
Q2
10000.
Stored Biomass
Q3
50.
Pulse consumer
Q4
30000.
Nutrients (Available carbon)
K1
.00000417
R1
510.63309
K2
.5
R2
500.
K3
.05
R3
50.
K4
.45
R4
450.
K5
.00005
R5
.5
K6
.00045
R6
4.5
K7
.0000002
R7
5.
K8
.0000018
R8
45.
K9
.000002
R9
50.
K10
.00000417
R10
510.63309
Kll
.0005
Rll
5.
K12
.05
R12
2.5
K13
7.833E-7
(Jordan and Drewry 1969, Odum and Pigeon 1970, and Brown
Lugo, Silander and Liegel 1983)

Figure 60. Character set for displaying spatial graphs on
GIGI computer terminal suitable for use with screen copy
onto printer. Each dot pattern is represented by the hexa
decimal code on the left edge of each plot.
(a) 80 dots
(b) 40 dots
(c) 27 dots
(d) 20 dots
(e) 16 dots
(f) 12 dots
(g) 7 dots
(h) 3 dots

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197
Table 5. Percent power used as a function of input energy
sources and diffusion in different ecosystem levels.
Successional
Steady state
(a) No Diffusion
Energy Distribution
Percent
Power Used
Hierarchical
96.5
96.6
Even
96.5
96.6
Random
96.6
96.6
(b) Diffusion
between nutrient (Q4) tanks
(Successional initial conditions)
Diffusion rate
.001 .01
.1
Energy Distribution
Percent Power
Used
Hierarchical
96.5 96.5
95.5
Even
96.5 96.5
96.5
Random
96.6 96.6
95.6
(c) Diffusion
between consumer (Q3) tanks
(Successional initial conditions)
Diffusion rate
.001 .01
.1
Energy Distribution
Percent Power
Used
Hierarchical
96.6 96.6
96.6
Even
96.5 96.5
96.5
Random
95.6 96.5
96.5
(a) Model DSP1. See Figure 50 for example run.
(b) Model DSP1. See Figure 54 for example run.
(c) Model DSP1QZ See Figure 53 for example run.

198
Table 6. Percent power used for various runs of DSP1C
spatial model having only one consumer equally distributed
across the entire production matrix.
Successional state Steady state
High initial condition for consumer Q3 (5000).
Hierarchical 96.5 96.5
Even 96.5 96.6
Random 96.4 96.4
Low initial condition for consumer Q3 (50).
Hierarchical 96.5 96.4
Even 96.5 96.4
Random 96.4 96.3

199
Table 7. Percent power used for DSP100 model as a function
of search length and input energy type.
Search length
(cells)
Successional
state
Steady
state
Hierarchical distribution
1
96.4
(17) (a)
96.5
(31)
2
96.6
(28)
95.6
(39)
3
96.6
(36)
96.5
(39)
4
95.6
(36)
96.6
(43)
5
96.5
(33) (b)
96.5
(45)
Even distribution
1
95.7
(15)
crash
(13)
2
96.4
(30)
95.5
(30)
3
96.4
(38)
96.5
(34)
4
96.4
(38)
96.5
(42)
5
96.5
(32)
96.5
(40)
Random distribution
1
96.4
(20)
95.7
(32)
2
96.6
(34)
96.6
(38).
3
96.5
(40)
96.5
(44)
4
96.6
(42)
96.5
(45)
5
96.5
(41)
96.6
(45)
(n) indicates number of consumers at end of simulation.
(a) see Figure 55
(b) see Figure 56

200
Table 8. Percent power used as a function of different
energy input sources and diffusion rates. Edge effect model
with different levels of consumers (Q3) on outside (buffer)
edge of spatial matrix.
Diffusion rate
.001
.01
.1
Value of Q3
Energy
Percent Power Used
on outside
Distribution
edge
Hierarchical
96.6
96.5
96.1
O
o
Even
96.5
96.5
95.9
Random
96.6
96.5
95.9
Hierarchical
96.6
96.5
96.7
50.
Even
96.6
96.7
96.6
Random
96.6
96.7
96.6
Hierarchical
96.6
96.7
96.9
100.
Even
96.6
96.8
96.8
Random
96.6
96.9
96.9

201
PROGRAM SUCIO
C SUCGGX
C VERS 1.1
C FEBRUARY 5, 1984
C
BYTE FILE(16),ESC,DES(40)
REAL M1,M2,M3,M4,M9
REAL K1, K2 K3 K4, K5 ,K6,K7,K8,K9,K0,L1,L2,L3,J,J0
DIMENSION FILNAM(6),IY(50,200)
DATA FILE/16*0/
DATA DES/40*0/
FT1(A,B)=ABS(AINT(A/B)-A/B)
D(X,Y)=(X/Y)*ALOG(X/Y)
C
C
WRITE(5,100)
100 FORMAT(1X,' SUCGGM GENERATES 6 DATAFILES',
&/' BE SURE THAT THEY DONT ALREADY EXIST',
&/' WHAT IS THE DATA FILE FOR THIS MODEL RUN ?')
READ(5,101)(FILE(I),1=1,16)
101 FORMAT(16A1)
C
C
C WRITE(5,1011)
C1011 FORMAT(' WHICH Q TO SAVE (1,2,3,4,5=% POW USED,6=BIOMSS)'$)
C READ(5,1012)IQSAV
C1012 FORMAT(13)
WRITE( 5,1013)
1013 FORMAT(' WHAT IS THE INCREMENT IN JO? [R] '$)
READ(5,1014)XINC
1014 FORMAT(G15.5)
C
C
c WRITE(5,99)
C 99 FORMAT(' HOW LONG TO RUN? ')
C READ(5,98)TIME
C 98 FORMAT(F6.0)
TIME=100. ¡X.1
C WRITE(5,981)
C981 FORMAT(' DO YOU WANT A HARDCOPY? (1-YES,0-NO)'$)
C READ (5,982)ICOPY
C982 FORMAT(12)
OPEN(UNIT=1,NAME=FILE,TYPE='OLD',FORM='UNFORMATTED')
READ(1)E1,E2,E3,E4,E5,E6,E7,E8,E9,E10,E11,E12,E13,E14,E15,
C COEFFICIENTS **********************************************
+NUM,K0,K1,K2,K3,K4,K5,K6,K7,K8,K9,D1,D2,D3,D4,L1,L2,L3,F1
+,JO,Q1INIT,Q2INIT,Q3INIT,Q4INIT
CLOSE(UNIT=1)
C INITIAL CONDITIONS********************************************
XJ0INI=J0
NSLICE=25
NCNTS=150
1 CONTINUE
C

202
C BIG OUTER LOOP
C
DO 1062 IQSAV-1,6
C
C
NRUN=0
2 CONTINUE
J0=XJ0INI+NRUN*XINC
NRUN=NRUN+1
T=0
PERCNT=0
PAVAIL=0
PUSED=0
DUSED=0
Q9=0
BIOMSS=0
M1=0
M2=0
M3=0
M4=0
M9=0
P=0
BMAX=0
Q1=Q1INIT
Q2=Q2INIT
Q3=Q3INIT
Q4=Q4INIT
Q1SIZE=30.
Q2SIZE=5.
Q3SIZE=1.
Q4SIZE=20.
C SET OUTPUT VECTOR AND FLAG **********************************
DT=. 1
C WRITE(5,108)DT
C 108 FORMAT(' TIME INTERVAL DT= ',F5.3)
C ISTEP=1/DT !Dr'S PER T
C IPLOT=0 !PLOTTING INTERVAL IN DT'S
C ITCNT= 0 ¡ITERATION COUNTER
C WRITE(5,1081)
C1081 FORMAT(' WHAT IS PLOTTING INTERVAL PER TIME UNIT [I] '$)
C READ(5/1082)IPLOT
C1082 FORMAT(12)
C WRITE(5,1083)
C1083 FORMAT(1X,' INPUT VALUES FOR SIZES Q1-Q4 [R] '$)
C READ (5,1084)Q1SIZE,Q2SIZE,Q3SIZE,Q4SIZE
C1084 FORMAT(4F8.3)
C WRITE GIGI STARTUP INFORMATION
C CALL GGON
C CALL GGINIT
C CALL GGERA
C CALL GGAXIS(0,0,767,479)
C CALL GGBOX(7,0,0,767,479)
C CALL GGBOX7,0,0,767,350)
C

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io *
_ U) Kj _
O \
II
C-I -*
3 3 3 3 3 + + Jd-'WWM>iQlp^
vOifcCJto-*XZ)3at*>JdCy>lniH*iUi
lOi + Cjl I IH *
_fpo ;0Q O O XO
d ii) to -4 Z
JO I
I VO o
* -*
fO
o
OJ
ITCNT=ITCNT+1
RATE EQUATIONS'

20000
21000
22000
23000
24000
25000
26000
C
C
C
C
C
C
C
C
C
C
C
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
1101
11011
25001
GOTO(21000,22000,23000,24000,25000,26000)IQSAV
G0T01101
IY(NRUN,INT(T*1.5+1))=INT(Q1/.5)
GOTO1101
IY(NRUN,INT(T*1.5+1))=INT(Q2/.5)
GOT01101
IY(NRUN,INT(T*1.5+1))=INT(Q3/.5)
GOTO1101
IY(NRUN,INT(T*1.5+1))=INT(Q4/.5)
GOTO1101
IY(NRUN,INT(T*1.5+1))=INT(PERCNT*1000)
GOT01101
IY(NRUN,INT(T*1.5+1))=INT(BIOMSS/.5)
GOT01101
SKIP PLOTTING IN THIS VERSION
GOTO 1101
IF(FT1(T,.1).GE.DT)GOTO 1101
IF(ITCNT.LT.IPLOT)GOT01101
ITCNT=0
IX=T*7
IY=Q1
CALL GGPLT(6,IX,IY,1)
IY=Q2
CALL GGPLT(1,IX,IY,1)
IY=Q3
CALL GGPLT(2, IX,IY,1)
IY=Q4
CALL GGPLT(3,IX,IY,1)
IY=Q9/DT
CALL GGPLT(4,IX,IY,1)
IY=BIOMSS
CALL GGPLT(5,IX,IY,1)
!YELLOW FOR CLIMAX SPECIES
¡BLUE FOR TRANSITIONAL SPECIES
¡RED FOR WEEDS
¡MAGENTA FOR CONSUMERS
¡GREEN FOR PRODUCTIVITY
¡CYAN FOR BIOMASS
IY=EUSED*100/JO ¡SCALE EUSED TO 0-100
CALL GGPLT(7, IX,IY+350,1) ¡PLOT POWER USED WHITE
IY=(DRAIN/DT)* 100/JO ¡SCALE DRAIN TO 0-100
CALL GGPLT(5,IX,IY+350,1) ¡CYAN FOR DRAINS
IY=DIVERS*50
CALL GGPLT(2, IX,IY+350,1)
IF(T.LT.TIME)GOTO 5
WRITE(5,11011)JO,Q1,Q2,Q3,Q4,PERCNT*100.,BIOMSS
FORMAT(1X,7(1X,F10.4))
IF(NRUN.LT.NSLICE)GOT02
CALL ASSIGN(2,'SUCMANY',6)
ENCODE(40,25001,DES)IQSAV,FILE
FORMAT(1X,' TANK Q',11,' FOR DATA FILE ',16A1)
WRITE(2)(DES(I),1=1,40),NSLICE,NCNTS,
((IY(JCNT,KCNT),KCNT=1,NCNTS),JCNT=1,NSLICE)
CLOSE(UNIT=2)
WRITE(5,1061)IQSAV

205
1061 FORMAT(* END OF RUN # ,14)
1062 CONTINUE
C ESC=27
C IF (ICOPY.EQ.O)GOT01102
C WRITE(3/11021)
C11021 FORMAT(' S(H)')
C CALL GGERA
C1102 CALL GGOFF
C WRITE(5/1009)ESC,ESC/ESC
C1009 FORMAT( 1X,A1 ,'Prtml' ,A1 <' ,A1 ,' [H'//
C & INITIAL VALUES OF VARIABLES')
C WRITE(5,114)
C WRITE(5/112)Q1INIT,Q2INIT,Q3INIT,Q4INIT,0, 0
C WRITE(5,111)
C 111 FORMAT(' MAXIMUM VALUES OF VARIABLES ')
C WRITE(5,114)
C 114 FORMAT(6X,'Q1',7X,'Q2',7X,'Q3',7X,'Q4', 6X,'PROD',5X,'BIOMASS')
C WRITE(5#112)M1,M2,M3,M4,M9/DT,BMAX
C 112 FORMAT(' ',6F9.3)
C WRITE( 5,113)
C 113 FORMAT(' FINAL VALUES OF VARIABLES')
C WRITE( 5,114)
C WRITE(5,112)Q1,Q2,Q3,Q4,Q9/DT,BIOMSS
C WRITE( 5, 115)PAVAIL,PAUSED, ( PUSED/PAVAIL) *100 ,DUSED
C 115 FORMAT(
C &1X,' ENERGY AVAILABLE = ',F12.4/
C &1X,' TOTAL PRODUCTIVITY = ',F12.4/
C &1X,' ENERGY USED = ',F12.4,' PERCENT USED = ',F12.4/
C &1X,' ENERGY DRAINED = ',F12.4)
C WRITE(5,116)
C 116 FORMAT( 6X,' K0',8X, K1',8X,'K2',8X,'K3',8X,'K7',8X #'K8',8X,
C +'K9',8X,'F1')
C WRITE( 5,117) K0 K1,K2,K3,K7,K8,K9,F1
C 117 FORMAT(' ',8(2X,F8.4))
C WRITE(5,118)
C 118 FORMAT(6X,'D1',8X,'D2', 8X,'D3',8X,'D4',8X, L1', 8X,'L2',8X,
C +'L3',8X,'JO')
C WRITE(5,117)D1,D2,D3/D4,L1,L2,L3,JO
C WRITE(5,119)DT,TIME
C 119 FORMAT(' DT THIS RUN = ',F6.4,' TOTAL T= ',F6.2)
C WRITE(5,120)(FILE(I),1=1,16)
C 120 FORMAT(' DATA FILE DESIGNATION FOR THIS RUN 'r16A1)
C WRITE(5,121)Q1SIZE,Q2SIZE,Q3SIZE,Q4SIZE
C121 FORMAT(3X,'Q1SIZE Q2SIZE Q3SIZE Q4SIZE'/1X,4F71)
C IF(ICOPY.EQ.O)GOTO1201
C CALL GGON
C WRITE(3/11021)
C CALL GGOFF
1201 END

1 REM THREPATH MODEL VERSION 7/1/84
2 REM WITHINPUT COEFFICIENTS CHANGED
3 REM NAME=> TPMOD7.BAS
90 PRINT "WHAT IS VALUE FOR ENERGY INPUT (J0-J1)"
91 INPUT J1
100 REM AND PRINT FNV$(C,X,Y) FOR PLOTTING VECTORS. C=COLOR(1-7)
110 PRINT "DO YOU WANT GRAPHICS ON '9 INPUT Q$
120 IF Q$<>"Y" GO TO 150
130 PRINT CHR$(27)+"PpS(E)W(R,I(G),P1,N0 ,A0 ,S0) S( A[ 0,479] [767,0])
140 DEF FNV$(C,X,Y)="W(I"+STR$(C)+")V["+STR$(X)+","+STR$(Y)+"]"
150 DEF FNP$(C,X,Y)="W(I"+STR$(C)+")P["+STR$(X)+","+STR$(Y)+"]V[]
160 DEF FNT$(C,N,A$)="W(I"+STR$(C)+")T(S"+STR$(N)+") '"+A$+"
170 DEF FNB$((5,X,Y,X1,Y1)=FNP$(C,X,Y)+FNV$(C,X,Y1)+FNV$(C,X1,Y1)
+FNV$(C,X1,Y)+FNV$(C,X,Y)
180 A$=CLK$
190 A9=TTYSET(255,132)
200 N9=-1.8
210 T9=1
220 W=0
230 £=100
240 K1=.5
250 K2=1.00000E-03
260 K3=1.00000E-06
270 K4=.2
275 K6=K1
276 K7=10*K2
277 K8=10*K3
280 T=0
290 A$="Threepath Model"
300 PRINT FNP$(7,626,475);FNT$(7,1,A$)
310 PRINT FNB$(7,620,456,767,479)
330 PRINT FNB$(7,0,0,767,479)
340 PRINT FNP$(7,0,270);FNV$(7,767,270)
350 PRINT FNP$(7,0,380);FNV$(7,767,380)
360 PRINT FNP$(7,0,244);FNV$(7,767,244)
370 JO=J1/2+COS(W*T/57.2958)*J1/2
380 J9=J0/(1+K6+K7*Q+K8*Q*Q)
390 P1=P1+J0-J9
400 J7=J7+J0
410 R1=K1*J9
420 R2=K2*Q*J9
430 R3=K3*Q*Q*J9
440 R4=K4*Q
450 Q9=T9*(R1+R2+R3-R4)
460 Q=Q+Q9
470 X0=R1+R2+R3
480 X1=100*R1/X0
490 X2=100*R2/X0
500 X3=100*R3/X0
510 IF Q$="Y" THEN 580
520 PRINT "J0=";J0,"J9=";J9,"T=";T
530 PRINT "Q=";Q,"Q9=";Q9
540 PRINT "R1=";R1,"R2=";R2,"R3=";R3,"R4=";R4
550 PRINT "X1=";X1,"X2=";X2,"X3=";X3

560 PRINT
570 IF Q$<>" Y" THEN 640
580 PRINT FNP$(1,T*2,X1+275),FNP$(2,T*2,X2+275)
,FNP$(3,T*2,X3+275),FNP$(4,T*2,5+J9/1)
590 PRINT FNP$(5,T*2/JO/100+5)
600 PRINT FNP$(7,T*2,Q/1000+385)
610 PRINT FNP$( 4,600,270) ;"T(S1) JR=" ;J9?"'"
620 PRINT FNP$(6,T*2,170+(100*J9/J0))
630 Q7=Q7+Q REM TOTAL Q TO GET AVERAGE
640 T=T+T9 IF T<360 THEN 370
650 PRINT FNP$(4,600,270);"T(S1)' "?"
660 J5=P1/J7*100
670 PRINT FNP$(7,0,270);"T(S1)'POW USED=";P1
;"POW AVAIL=";J7;"PERCENT USED=";J5;"AVE Q=";Q7/T
680 INPUT X
690 PRINT CHR$(27)+""
700 PRINT P1/J7;"FRACTION OF TOTAL POWER USED"
710 END

o o
208
C
C
C GIGI GRAPHICS SUBROUTINE PACKAGE
C WRITTEN BY JOHN R. RICHARDSON
C
C SEPTEMBER 1982
C
C VERS 11: ALL UPDATES AND CURRENT TO SEPTEMBER 1982
C
C VERS 12: FEBRUARY 28 1984 ADDITIONS
C ADDED GGPLOT (CALL TO GGPLT)
C ADDED GGDMP (HARDCOPY DUMP)
C ADDED GGVERS
ALL I/O IS TO LOGICAL UNIT 3
C
(2******************************** ******************************* ********
C
C
C
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
THE NORMAL CALLING SEQUENCE TO SET UP THE GIGI WOULD BE C
AS FOLLOWS: C
C
CALL GGON ¡TURNS GRAPHICS ON C
CALL GGINIT ¡SENDS NORMAL INITIALIZATION C
CALL GGERA ¡ERASE THE SCREEN C
CALL GGAXIS(0,0,767,479) ¡SETS NORMAL AXIS WITH ORIGIN C
! AT THE LOWER LEFT CORNER C
***********************************************************************
TASKBUILDING USING THE GIGI ROUTINES C
RUN THE TASK BUILDER (TKB ) C
TKB>MYPROG=MYPROG,LB:[1,1]GGLIB/LB C
TKB>/
ENTER OPTIONS
TKB>ASG=TTn:3 ¡WHERE n EQUALS GIGI TERMINAL NUMBER C
TKB>// I COULD USE TI: INSTEAD C
***********************************************************************
SUBROUTINE GGVERS(IVERS)
C CALL TO THIS WILL GIVE THE CURRENT VERSION OF THE GGLIB
IVERS=12
RETURN
END
C
C
C
SUBROUTINE GGON
C THIS WILL SEND THE ESC Pp SEQUENCE TO THE GIGI TO ENABLE THE
C GRAPHICS
BYTE ESC
ESC=27
WRITE(3/100)ESC
100 FORMAT('+*,1A1,'Pp')
RETURN
END
O O

209
C
C
C
SUBROUTINE GGOFF
C THIS WILL SEND THE ESC NEEDED
C TO TURN OFF THE GIGI GRAPHICS MODE
BYTE ESC
ESC=27
WRITE(3/100)ESC
100 FORMAT('+', 1A1,' ')
RETURN
END
C
C
C
SUBROUTINE GGERA
C ROUTINE TO PERFORM SCREEN ERASE
WRITE (3,100)
100 FORMAT(,+',,S(E)')
RETURN
END
C
C
C
SUBROUTINE GGDMP
C ROUTINE TO PERFORM SCREEN DUMP TO LA34/LA100 PRINTER
WRITE(3,100)
100 FORMAT('+','S(H) )
RETURN
END
C
C
C
SUBROUTINE GGINIT
C ROUTINE TO INITIALIZE THE GIGI
WRITE(3,100)
100 FORMAT('+',"W(R,14,P1,NO,SO,A0)')
RETURN
END
C
C
C
SUBROUTINE GGAXIS(IX,IY,IFX,IFY)
C
C ROUTINE TO INITIALIZE THE AXIS OF THE GIGI
C WHERE IX = LOWER LEFT CORNER X VALUE
C IY = LOWER LEFT CORNER Y VALUE
C IFX= UPPER RIGHT CORNER X VALUE
C IFY= UPPER RIGHT CORNER Y VALUE
WRITE(3,100)IX,IFY,IFX,IY
100 FORMAT('+','S(A[',15,',,15,'][',15,',',15,'])')
RETURN
END
C

o o
210
C
C
SUBROUTINE GGPLT(COLOR,IX,IY,IFLAG)
C SUBROUTINE TO POSITION GRAPHICS CURSOR ON THE GIGI
C SET IFLAG TO NUMBER > 0 TO PLOT POINT AND TO 0
C TO MOVE CURSOR TO POINT WITHOUT PLOTTING POINT
C COLOR = BYTE variable 0-7 for color
C IX = Integer value of X
C IY = Integer value of Y
C IPT = Integer flag >1 plot a point
BYTE COLOR
IF(IFLAG.GT.0)GOTO10
WRITE(3,100)COLOR,IX,IY
100 FORMAT('+ W(I',11,')P[',14,',',14,'] ')
GOTO20
10 WRITE(3,101)COLOR,IX,IY
101 FORMAT('+ W(I',11,')P[',14,',',14,']V[]')
20 CONTINUE
RETURN
END
C
C
C
SUBROUTINE GGPLOT(COLOR,IX,IY,IFLAG)
C ROUTINE TO ALLOW FOR VARIATION IN SPELLING OF GGPLT ROUTINE
C ADDED IN VERS 12
BYTE COLOR
CALL GGPLT(COLOR,IX,IY,IFLAG)
RETURN
END
C
C
C
SUBROUTINE GGVEC(COLOR,IX,IY)
C SUBROUTINE TO DRAW A VECTOR ON THE GIGI FROM ITS PRESENT POSITION
C TO THE IX,IY POSITION IN THE PARAMETER LIST. USE GGPLT FOR
C INITIAL COORDINATES IF NEEDED.
C COLOR = BYTE variable 0-7 for color
C IX = Integer value of X
C IY = Integer value of Y
BYTE COLOR
WRITE(3,100)COLOR,IX,IY
100 FORMAT(*+ W(I',11,')V[',14,',',14,']')
RETURN
END
C
SUBROUTINE GGBOX(COLOR,IX,IY,IX1,IY1)
C SUBROUTINE TO DRAW A BOX ON THE GIGI GIVEN THE OPPOSITE COORDINATE
C PAIRS FOR THE RECTANGLE.
C COLOR = BYTE variable 0-7 for color
C 1X0 = Integer value of X
C IY0 = Integer value of Y

211
C 1X1 = Integer value of X opposite
C IY1 = Integer value of Y opposite
C
C IF THE FILL IS TURNED ON THE BOX WILL BE FILLED AUTOMATICALLY
C
BYTE COLOR
CALL GGPLT(COLOR,IX,IY,1)
CALL GGVEC(COLOR,1X1,IY)
CALL GGVEC(COLOR,IX1,IY1)
CALL GGVEC(COLOR,IX,IY1)
CALL GGVEC(COLOR,IX,IY)
RETURN
END
C
C
C
SUBROUTINE GGCIRC(COLOR,IX,IY,IRAD)
C SUBROUTINE TO DRAW A CIRCLE AT POINT IX,IY WITH A RADIUS OF IRAD
C COLOR = BYTE variable 0-7 for color
C IX = Integer value of X
C IY = Integer value of Y
C IRAD = Integer radius of circle
C
C IF THE FILL IS TURNED ON THE CIRCLE WILL BE FILLED AUTOMATICALLY
C
BYTE COLOR
CALL GGPLT(COLOR,IX,IY,0)
WRITE(3,100)COLOR,IX,IY+IRAD
100 FORMAT('+ W(I',11,')C[',14,',',14,']')
CALL GGPLT(COLOR,IX,IY,1) !LEAVE CURSOR AT CENTER
RETURN
END
C
C
c
SUBROUTINE GGTEXT(COLOR,IX,IY,TEXT,ISIZE,ITILT)
C SUBROUTINE TO WRITE TEXT AT IX,IY ON SCREEN
C Writes text at ix,iy with size and rotation of
C characters given
C COLOR = BYTE variable 0-7 for color
C IX = Integer value of X
C IY = Integer value of Y
C TEXT = BYTE array containing 80 char or less
C ISIZE = Integer value for text size 0-8
C IROT = Integer value of degress of rotation for
C line of text (multiple of 45)
C
C THIS SUBROUTINE CALLS LENGTH TO DETERMINE
C THE LENGTH OF THE STRING
C
BYTE COLOR,TEXT(1)
N=0
CALL GGPLT(COLOR,IX,IY,0)
CALL LENGTH(TEXT,N)

212
WRITE (3/100)ITILT,ISIZE,ITILT,(TEXT(I),I=1,N)
100 FORMAT('+ T(D',14,')(S',12,')(D',14,')',1H',A1,1H')
RETURN
END
C
C
C
SUBROUTINE LENGTH(TEXT,N)
BYTE TEXT(80)
IFL AG= 0
DO 20 1=80,1,-1
IF(TEXT(I).GT.32)IFLAG=1
N=I
IF (IFLAGEQ.0)G0T020
G0T099
20 CONTINUE
99 RETURN
END
C
C
C
SUBROUTINE GGFILL(IFLAG)
C SUBROUTINE TO TURN ON/OFF COLOR FILL CHARACTERISTIC
C IFLAG=0 NOFILL, IFLAG=1 FILL
C
WRITE(3,100)IFLAG
100 FORMATW(S',11,')')
RETURN
END

213
PROGRAM PLOTZ
C
C
C **6/27/83
C CHANGED AXIS ROUTINE FOR THREECORNERED ORIGIN**
C
C VERSION 1.6
C WRITTEN BY JOHN RICHARDSON
C APRIL 27, 1983
C
C SURFACE PLOTTING PROGRAM
C
DIMENSION IY(50,200),IOUT(200),IOUTY(200)
DIMENSION IX(200),MASK(800)
BYTE FNAM(20),DES(40),GON(3),GOFF(2),COLOR
BYTE BLACK,BLUE,RED,MAGENT,GREEN,CYAN,YELLOW,WHITE,ESC
INTEGER DELTAX,DELTAY
COMMON /IAREA/MASK
DATA FNAM/15*0,'.','D','A','T',0/
DATA MASK/800*0/,DELTAX/6/,DELTAY/6/
BLACK= 0
BLUE=1
RED=2
MAGENT=3
GREEN=4
CYAN=5
YELLOW=6
WHITE=7
COLOR=GREEN
IXSCLE=3
ESC=27
WRITE(5,499)ESC,ESC
499 FORMAT('+',A1,1PrTM11,A1,' ')
C !SET TERMINAL TO ANSII MODE
TYPE 500
500 FORMAT(1X,'INPUT FILE NAME ')
ACCEPT 501,(FNAM(I),1=1,15)
501 FORMAT(15A1)
OPEN(UNIT=1,NAME=FNAM,FORM='UNFORMATTED',TYPE='OLD')
READ(1),(DES(I),1=1,40),NRUN,NCNTS
+,((IY(J,K),K=1,NCNTS),J=1,NRUN)
CLOSE(UNIT=1)
TYPE 5
5 FORMAT(' REVERSE THE SLICES? (1-YES, 0-NO) ')
ACCEPT 6,NSLICE
6 FORMAT(15)
WRITE(5,61)
61 FORMAT(' SHIFT SUCCESSIVE SLICES (1 = LEFT, -1 = RIGHT) '$)
READ (5,62)ISHFT
62 FORMAT(13)
C
C
WRITE(5,621)
621 FORMAT(' WHAT ARE THE VALUES FOR DELTAX, DELTAY, IXSCLE [I] '$)

214
READ(5,622)DELTAX,DELTAY,IXSCLE
622 FORMAT(314)
C
C
WRITE(5,623)
623 FORMAT(' WHAT IS CROSS HATCH INTERVAL [I] '$)
READ(5,624)NXHTC
624 FORMAT(13)
C
C
DELTAX=DELTAX*ISHFT
171 CALL GGON
CALL GGINIT
CALL GGAXIS(0,0,767,479)
CALL GGERA
C CALL GGBOX(7,0,0,767,479)
C
C SET UP DATA POINTS FOR X AXIS
C
DO 19 NPOINT=1,NCNTS
IX(NPOINT)=NPOINT*IXSCLE
19 CONTINUE
nl.ine=1
DO 11 NPOINT=1,NCNTS
IOUTY(NPOINT)=IY(IABS(NLINE-NSLICE*NRUN),NPOINT)/4
11 CONTINUE
DO 20 NLINE=1,NRUN,1
DO 10 NPOINT=1,NCNTS
IOUT(NPOINT)=IY(IABS(NLINE-NSLICE*NRUN),NPOINT)/4
10 CONTINUE
CALL GG3DX(COLOR,IX,IOUT,IOUTY,NCNTS,NLINE,DELTAX,DELTAY,NXHTC)
DO 30 NPOINT=1,NCNTS
IOUTY(NPOINT)=IOUT(NPOINT)
30 CONTINUE
20 CONTINUE
CALL AXIS(COLOR,NRUN,NCNTS,DELTAX,DELTAY,IXSCLE)
C
C
CALL GGOFF
WRITE(5,2000)ESC,FNAM
2000 FORMAT(' + ',A1,1[ H' /' ',20A1)
WRITE(5,2001)ESC
2001 FORMAT('+',A1,'[H 0-QUIT, 1-SCREENDUMP, 2-SCRDMP NO LABEL '$)
READ(5,2002)IANS
2002 FORMAT(12)
IF(IANS.EQ.0)GOTO2100
WRITE(5,2004)ESC
2004 FORMAT(,+',Al,,[H,,80X)
IF (IANS.NE.2)GOTO 20035
WRITE(5,2005)ESC
2005 FORMAT('+'A1,'[H',80X/80X)
20035 CALL GGON
WRITE(5,2003)

o o
215
2003 FORMAT(' S(H)')
CALL GGOFF
2100 END
C
C
C
C
SUBROUTINE GG3DX(COLOR,IX,IY,IYX,NPNTS,N,DELTAX,DELTAY,NXHTC)
BYTE COLOR
DIMENSION IX(1),IY(1),MASK(1),IYX(1)
COMMON /IAREA/MASK
INTEGER DELTAY,DELTAX
IXOFF=200
IF(DELTAX.LT.O)IXOFF=50
IF(N.NE.1)GOTO10
C
C SET UP MASK FOR FIRST SLICE
DO 5 1=1,NPNTS
MASK(IX(I)+(IXOFF-N*DELTAX))=IY(I)+N*DELTAY
5 CONTINUE
C
C
10 CONTINUE
DO 20 1=1,NPNTS
IXOUT=IX(I)+(IXOFF-N*DELTAX)
IYOUT=IY(I)+N*DELTAY+20
IF(IYOUT.GEMASK(IXOUT))GOTO50
GOTO 20
50 MASK(IXOUT)=IYOUT
CALL GGPLT(COLOR,IXOUT,IYOUT,1)
IF(N.LE.1)GOTO20
IF(I.EQ.1)GOTO19
IF(IMOD(I,NXHTC).NE.0)GOTO20
19 IX2=IXOUT+DELTAX
IY2=IYX(I)+(N-1)*DELTAY+20
IF(DELTAX.LT.0)GOT0190
IF(IY2.LT.MASK(1X2))GOTO20
190 CALL GGVEC(COLOR,1X2,IY2)
20 CONTINUE
GOTO 33
IF(N.EQ.1)G0T033
M1=MASK(IXOFF-(N-1)*DELTAX)
C DO 33 1=1 ,DELTAX
C MASK(IXOFF(N*DELTAX)+I)=M1
33 CONTINUE
RETURN
END
C
C
SUBROUTINE AXIS(COLOR,NLINE,NPNTS,DELTAX,DELTAY,IXSCLE)
BYTE COLOR
INTEGER DELTAX,DELTAY,MASK(1)
INTEGER X0,Y0,X1,Y1,X2,Y2,X3,Y3,X4,Y4,X5,Y5,X6,Y6,XORG,YORG
INTEGER X7,Y7

216
COMMON /IAREA/MASK
IXOFF=200
IF(DELTAX.LT.0)IXOFF=50
ISIGN=DELTAX/1AB S(D ELTAX)
C
C
XO=IXOFF
Y0=20
C
X1=IX0FF+IXSCLE*NPNTS
Y1=20
C
X2=IXOFF-NLINE*DELTAX
Y2=20+NLINE*DELTAY
C
X3=X2
Y3=Y2+250
C
X4=X2+NPNTS*IXSCLE
Y4=Y2
C
X5=X0
Y5=Y0+250
C
X6=X4
Y6=Y3
C
XORG=X2
YORG=Y2
C
X7=X1
Y7=Y5
C
CALL GGPLT(COLOR,X6,Y6,1)
IF(DELTAX.LT.0)GOTO15
IYTEMP=MASK(X4)
CALL GGVEC(COLOR,X6,IYTEMP)
GOTO16
15 CALL GGVEC(COLOR,X4,Y4)
CALL GGVEC(COLOR,X1,Y1)
16 CALL GGPLT(COLOR,X1,Y1,1)
CALL GGVEC(COLOR,X0,Y0)
IF (DELTAX.GT.0)CALL GGVEC(COLOR,X2,Y2)
CALL GGPLT(COLOR,X2,Y2,1)
161 IF(DELTAX.GT.0)GOTO191
IYTEMP=MASK(X2)
CALL GGPLT(COLOR,X2,IYTEMP,1)
191 CALL GGVEC(COLOR,X3,Y3)
IF(ISIGN.GT.0)GOTO200
C SURFACE FOR LEFT SHIFT
CALL GGPLT(COLOR,X3,Y3,1)
CALL GGVEC(COLOR,X5,Y5)
CALL GGVEC(COLOR,X0,Y0)
199 GOTO201

217
C SURFACE FOR RIGHT SHIFT
200 CALL GGPLT(COLOR,X6,Y6,1)
CALL GGVEC(COLOR,X7,Y7)
CALL GGVEC(COLOR,X1,Y1)
201 CONTINUE
C
C VERTICAL AXIS TICS
C
DO 20 1=0,10
IY0=(I*25)+Y2
IF(DELTAX.LT.0)GOTO19
CALL GGPLT(COLOR,X2,IY0,1)
CALL GGVEC(COLOR,X2-6,IY0)
GOTO20
19 CALL GGPLT(COLOR,X4,IY0,1)
CALL GGVEC(COLOR,X4+6,IY0)
20 CONTINUE
C
C HORIZONTAL AXIS TICS
C
DO 25 1=0,10
IX0=I*NPNTS*IXSCLE/10+X0
CALL GGPLT(COLOR,1X0,Y0,1)
CALL GGVEC(COLOR,1X0,Y0-6)
25 CONTINUE
C
C ANGLE AXIS TICS
C
ZLINE=NLINE
DO 35 ZI=0.,ZLINE,ZLINE/10.
IF(DELTAX.LT.0)G0T027
IX0=X0-DELTAX*ZI
IY0=Y0+DELTAY*ZI
GOTO28
27 IX0=X1-DELTAX*ZI
IY0=Y1+DELTAY*ZI
28 CONTINUE
IC4=ISIGN*6
CALL GGPLT(COLOR,IX0,IY0,1)
CALL GGVEC(COLOR,IX0-IC4,IYO-6)
35 CONTINUE
C
C BACK AXIS LINE
C
C CALL GGPLT(COLOR,X2,Y2,1)
C CALL GGVEC(COLOR,X4,Y4)
C
C TOP AXIS LINE
C
CALL GGPLT(COLOR,X3,Y3,1)
CALL GGVEC(COLOR,X6,Y6)
RETURN
END

218
1 PROGRAM MEASURE3 WRITTEN BY JOHN R. RICHARDSON
2 VERSION=2!
3 VERSION 1.0 BASELINE SET 6/1/85
5 MAIN PROGRAM BEGINS AT LINE 1000
6 VERSION 1.5 6/3/85
7 CLEANED UP OLD FORTRAN CODE, ADDED DOUBLE OUTPUT FILE MODE
8 SAVE TRUE DIGITIZER VALUES FOR PLOTTER FILE OUTPUT
9 CALCULATE AREAS BASED ON SCALED DATA
10 VERSION 1.6 6/7/85
11 FIXED ERROR IF NO FILES OF B DRIVE,
12 CHANGED DATA ARRANGEMENT IN OUTPUT FILES
13 VERSION 1.7 6/24/85
14 ADDED ERROR OUTPUT ROUTINE FOR ERRORS OTHER THAN NO FILES
15 VERSION 2.0 ADDED SCALE3 SUB FOR DIFFERENT XSCALE AND YSCALE
16 7/2/85
99 *****************************************************************
100 SUBROUTINE DIGINI (LINE 3000-3490) OPENS DIGITIZER
110 AND SETS INITIAL PARAMETERS FOR PROGRAM
130 SUBROUTINE STREAM MODE (3800-3899) TURNS ON STREAM MODE
150 SUBROUTINE POINT MODE (3900-3999) TURNS ON POINT MODE
170 SUBROUTINE FILE HANDLER (4000-4220) OPENS DATA FILE FOR OUTPUT
190 SUBROUTINE SCALE2 (5000-5610) HANDLES SETTING UP USER
200 COORDINATES AND ORIGINS AND OFFSETS
220 SUBROUTINE DELAY (6000-6010) ARE TIMING ROUTINES THAT
230 MAY BE NEEDED FOR SENDING SETUP INFORMATION TO DIGITIZER
250 SUB INPUT (8000-8070) GETS DATA SENT FROM DIGITIZER
270 SUB DIGURU (9000-9070) SCALES DIGITIZER INPUT TO REAL WORLD
290 COORDINATES
300 SUBROUTINE AREAP (1660-2030) GETS INPUT POINTS FOR AN AREA
330 SUBROUTINE AREAX (2070-2580) CALCULATES THE AREA
350 SUBROUTINE PERIX (2610-2810) CALCULATE THE CLOSED AND OPEN
370 PERIMETERS FROM A SET OF GIVEN POINTS
1000 '**************************************************************
1010 '************************* MAIN PROGRAM START *****************
1020 '**************************************************************
1040 ON ERROR GOTO 20000
1060 CLS:PRINT:PRINT:PRINT:PRINT
1070 PRINT "AREA MEASUREMENT PROGRAM VERSION ";VERSION
1120 GOSUB 4000:' CALL FILE HANDLER
1130 GOSUB 3000:' CALL DIGINI
1140 GOSUB 5000:' CALL SCALE2
1260 'CONTINUE
1270 GOSUB 1660:' CALL AREAP (X,Y,AREA,NPOINT,IERR,RESOL)
1280 IF (IERR= 0) THEN GOTO 1290 ELSE PRINT "ERROR ";IERR;"
HAS OCCURRED NOT ENOUGH POINTS FOR AN AREA":GOTO 1270
1290
1340
1350
1351
1370
1380
1390
1400
1410
GOSUB 2070:' CALL AREAX(X,Y,AREA,NPOINT)
GOSUB 2610:' CALL PERIX(X,Y,PERI1,PERI2,NPOINT)
AREA=AREAIO
FOR ";NPOINT;"
PRINT THE MEASURED AREA IS [";AREAIO;"]
PRINT CLOSED PERIMETER= ";PERI2
PRINT OPEN PERIMETER= ";PERI1
PERIOUT=PERI2
PRINT BELL$;" KEEP THIS AREA OR RE-MEASURE [K or R]
POINTS"

219
1420 INPUT ANS$
1440 IF (ANS$ ="R")GOTO 1260
1480 PRINT #2,XT(1), YT(1),PENDUM;AREA;PERIOUT;NPOINT
1481 PRINT #3,AREA;PERIOUT
1490 PRINT #2/XT( 1) ,YT(1)/PENUP
1500 FOR 1=2 TO NPOINT
1510 PRINT #2,XT(I), YT(I),PENDWN
1520 NEXT I:'200 CONTINUE
1525 IF WET5=99 THEN 1590
1530 PRINT #2,XT(1),YT(1),PENDWN
1590 PRINT M DO YOU WANT TO INPUT ANOTHER AREA (Y OR N)
1600 INPUT ANS$
1620 IF (ANS$ = "Y") GOTO 1260
1630 CLOSE:PRINT "TYPE SYSTEM TO EXIT FROM BASIC OR RUN TO RERUN":END
1640 ****************************************************************
1641 ************************** END OF MAIN PROGRAM *****************
1642 '***************************************************************
1660 PRINT "SUBROUTINE AREAP":IERR=0
SUBROUTINE AREAP(X,Y,AREA,NPOINT,IERR,RESOL)
1700 IERR=0
1710 NPOINT=0
1720 AREA=0
1740 PRINT ENTER FIRST POINT BY PRESSING THE '1* KEY "
1745 PRINT ENTER REMAINING POINTS BY PRESSING ANY KEY BUT '2'"
1750 PRINT THEN QUIT ENTERING POINTS BY PRESSING '2' "
1760 GOSUB 9000:' CALL DIGURU (XIN,YIN/CODE)
1770 XOLD=XIN
1780 YOLD=YIN
1790 IF (CODE$ <> "1") GOTO 1760
1800 NPOINT=1
1810 X(NPOINT)=XOLD
1811 XT(NPOINT)=XTRUE
1820 Y(NPOINT)=YOLD
1821 YT(NPOINT)=YTRUE
1830 IF (CODE$ = "2") GOTO 1950
1840 GOSUB 9000:'200 CALL DIGURU (XIN,YIN,CODE)
1850 IF (CODE? = "2" ) GOTO 1950
1870 NPOINT=NPOINT+1
1880 X(NPOINT)=XIN
1881 XT(NPOINT)=XTRUE
1890 Y(NPOINT)=YIN
1891 YT(NPOINT)=YTRUE
1900 XOLD=XIN
1910 YOLD=YIN
1930 PRINT XIN,YIN,"CODE =";CODE$:BEEP
1940 GOTO 1840
1950 IF (NPOINT < 3) THEN IERR =1: '300
1970 IF (IERR <> 0)THEN RETURN
2020 IERR= 0
2025 'GOSUB 3900
2030 RETURN
2070 PRINT SUBROUTINE AREAX( X, Y,AREAIO,NPOINT)"
2130 PRINT **** DIGITIZER AREA CALCULATION ****"
2140 NUMPNT=0:'200

220
2150
2160
2170
2180
2190
2200
2210
2220
2230
2240
2250
2260
2270
2280
2290
2300
2310
2320
2330
2340
2350
2360
2370
2380
2390
2400
2410
2420
2430
2440
2450
2460
2470
2480
2490
2500
2510
2520
2530
2540
2550
2560
2570
2580
2590
2600
2610
2620
2630
2640
2650
2660
2670
2680
A1=0!
A2=0!
AREAIOO!
NUMPNT=NUMPNT+1
'C READ FIRST PAIR
XF=X(NUMPNT)
YF= Y(NUMPNT)
XP=XF
YP=YF
'300 CONTINUE
'C
NUMPNT=NUMPNT+1
XC=X(NUMPNT)
YC=Y(NUMPNT)
A1=A1+(XP*YC)
XP=XC
IF(NUMPNT = NPOINT)GOTO 2330
GOTO 2240
IF(NUMPNT > 2)GOTO 2370:'400
' WRITE(LUNO,401)
PRINT ?NOT ENOUGH DATA POINTS FOR AREA CALCULATION"
RETURN
A1=A1+(XC*YF):'450
NUMPNT=1
XF=X(NUMPNT)
YF=Y(NUMPNT)
XP=XF
YP=YF
NUMPNT= NUMPNT+1:'500
YC=Y(NUMPNT)
XC=X(NUMPNT)
A2=A2+(YP*XC)
YP=YC
IF (NUMPNT = NPOINT)GOTO 2500
GOTO 2430
A2=A2+(YC*XF):'600
AREAIO=ABS((A1-A2)*5)
RETURN
'700 AREAIO=0.
' RETURN
'800 CONTINUE
' AREA10=0 *
RETURN
' END
'C
'C
PRINT SUBROUTINE PERIX(X,Y/PERI1,PERI2,NPOINT)"
' DIMENSION X(1),Y(1)
PER11=0!
PERI2=0!
NUMPNT=1
XIN=X(1)
YIN=Y(1)
XL=XIN

221
2690
2700
2710
2720
2730
2740
2750
2760
2770
2780
2790
2800
2810
2820
2830
3000
3001
3002
3003
3005
3006
3007
3008
3009
3010
3011
3020
3025
3030
3031
3040
3041
3050
3051
3060
3061
3070
3071
3080
3081
3090
3091
3100
3110
3111
3310
3320
3330
3340
3350
3360
3370
3380
YL=YIN
NUMPNT=NUMPNT+1:'100
XN= X(NUMPNT)
YN=Y(NUMPNT)
PERI1=PERI1+FNRDIST(XN,YN,XL,YL)
XL=XN
YL=YN
IF (NUMPNT = NPOINT)GOTO 2780
GOTO 2700
PERI2=FNRDIST (XIN,YIN,XL,YL):'300
PERI2=PERI2+PERI1
RETURN
' END
PRINT "SUBROUTINE DIGINI"' SUBROUTINE DIGINI
LCB=13
DEF FNRDIST(X1,Y1,X2,Y2)=ABS((X1-X2)2+(Y1-Y2)2).5
DEF FNANGLER(X1,Y1,X2,Y2)=ATN((Y2-Y1)/(X2-Xl))
+ (SGN(ABS(X2-X1))-SGN((X2-X1)))* 1.570796
PRINT "INITIALIZATION SEQUENCE FOR DIGITIZER":BEEP
PRINT" PLEASE MAKE SURE DIGITIZER IS ON"
INPUT HIT RETURN WHEN READY ",DUM$
DIM X(1000),Y(1000),XT(1000),YT(1000)
PENUP=3:PENDWN=2:PENDUM=6:BELL$=CHR$(7)
OPEN"COM1:9600,E,7,2,RS,CS,DS,CD" AS #1:'OPEN AUX PORT FOR I/O
CLS
LD$="#]":QT$="/"
PRINT #1,"#]L":PRINT DIGITIZER BEING RESET ":GOSUB 6005
PRINT #1,"#](":'SET RESOLUTION TO .001
GOSUB 6001
PRINT #1,"#]6":'SET RATE TO 2/SEC
GOSUB 6001
PRINT # 1, "#]'02":'SET INCREMENT TO .01
GOSUB 6001
PRINT #1,"#]M":' TURN ON INCREMENTAL MODE
'GOSUB 6001
PRINT #1,"#]J":1 TURN ON STREAM MODE
GOSUB 6001
PRINT # 1,"#]9":' SET SERIAL TAG AS LAST CHARACTER
GOSUB 6001
PRINT #1 ,"#]>": SET NO FIELD DELIMITERS
GOSUB 6001
PRINT #1,"#]I":' RESET FOR POINT MODE FOR BEGINNING SETUP
PRINT #1,"#]/":' SEND END OF REMOTE FORMATTING
GOSUB 6001
XSCALE=1!:'USER X AXIS SCALE FACTOR
XOFF=0!:' USER X AXIS OFFSET
YSCALE=1!:'USER Y AXIS SCALE FACTOR
YOFF=0!: 'USER Y AXIS OFFSET
ANGLE=0!:' USER SKEW CORRECTION FACTOR
' XROUND=0.
' YROUND=0.
UXSCAL=1!:'USER PLOTTER X SCALE FACTOR

222
3390 UYSCAL=1!:'USER PLOTTER Y SCALE FACTOR
3400 UROT=0! :'USER PLOTTER ROTATION ANGLE
3410 ARRLEN=.15
3420 ARRWID= .07
3430 ARROFF=.03
3440 IARRTY=3
3450 PDLEN=.1
3460 PULEN=.05
3470 XLAST=0!:'LAST X COORD CALCULATED
3480 YLAST=0!:'LAST Y COORD CALCULATED
3490 RETURN
3500 'CLOSE FILE THEN REOPEN IT
3510 CLOSE #1
3520 OPEN"COM1:9600,E,7,2,RS,CS,DS,CD" AS #1:'OPEN AUX PORT FOR I/O
3530 RETURN
3800 PRINT "STREAM MODE SUBROUTINE"
3810 PRINT #1,"#]J":' TURN ON STREAM MODE
3811 GOSUB 6001
3820 PRINT #1,"#]M":' TURN ON INCREMENTAL MODE
3821 GOSUB 6001
3899 RETURN
3900 PRINT SUBROUTINE FOR SETTING POINT MODE"
3910 PRINT #1,"#]I":' RESET FOR POINT MODE FOR BEGINNING SETUP
3911 GOSUB 6001
3999 RETURN
4000 PRINT "FILE HANDLING SUBROUTINE":' SUBROUTINE FOR FILE HANDLING
4004 PRINT "CURRENT DATA FILES: ":PRINT
4005 FILES "B:*.*"
4006 PRINT:PRINT:PRINT:
4010 'PRINT "FILE HANDLING"
4020 PRINT "WOULD YOU LIKE TO OPEN A NEW FILE OR APPEND TO AN EXISTING"
4030 INPUT FILE (1-NEW, 2-OLD)",FILEMODE
4040 IF FILEMODE <1 OR FILEMODE >2 THEN 4020
4050 IF FILEMODE =2 THEN 4100
4060 INPUT "WHAT IS THE NAME FOR THE FILE (1-8 CHARACTERS) "y FILENAME$
4061 IF LEN (FILENAME$)>8 THEN 4060
4062 IF INSTR(FILENAME$:") <>0 THEN PRINT "INPUT FILENAME
ONLY WITHOUT DRIVE SPECIFIER":GOTO 4000
4070 NTEMP=INSTR(FILENAME$
4075 IF NTEMP=0 THEN 4085
4078 NLEN=LEN(FILEAME$)
4080 FILENAME$=MID$(FILENAME$,1,NTEMP-1)
4085 FILENAME$="B:"+FILENAME$
4090 OPEN "O",2,FILENAME$+".DAT"
4091 OPEN "O",3,FILENAME$+"PRN"
4093 INPUT "WHAT IS DESCRIPTION OF THIS DATA SET" ,DESC$
4094 PRINT #3,CHR$ (34)+DESC$
4095 GOTO 4220
4100 'OPEN FOR APPEND
4110 INPUT "WHAT IS THE NAME OF THE EXISTING FILE ",FILENAME$
4120 IF LEN(FILENAME$)>8 THEN 4110
4130 IF INSTR(FILENAME$/":") <>0 THEN PRINT "INPUT FILENAME
ONLY WITHOUT DRIVE SPECIFIER":GOTO 4100
4140 NTEMP=INSTR(FILENAME$

223
4150
4160
4170
4200
4202
4204
4206
4210
4211
4220
4250
5000
5030
5040
5050
5060
5070
5080
5100
5110
5120
5130
5140
5150
5160
5170
5180
5190
5200
5210
5220
5240
5242
5244
5246
5247
5248
5249
5250
5252
5260
5270
5280
5290
5300
5310
5320
5330
5340
5350
5351
5352
5353
5360
IF NTEMP=0 THEN 4200
NLEN=LEN(FILENAME?)
FILENAME$=MID$(FILENAME$,1,NTEMP-1)
FILENAME$*"B:"+FILENAME$
OPEN "I",3,FILENAME$+".PRN"
LINE INPUT #3,DUM$:PRINT:PRINT DUM$:PRINT:PRINT
CLOSE #3
OPEN"A",2/FILENAME$+".DAT"
OPEN"A",3,FILENAME$+".PRN"
'INPUT "BASIN NUMBER = ",WET1
RETURN
'PRINT"SUBROUTINE SCALE3" : SUBROUTINE SCALE3
GOSUB 3900: GO SET POINT MODE FIRST !
PRINT ****** DIGITIZER THREE-POINT SCALING ******"
PRINT :PRINT:PRINT :BEEP:GOSUB 6001
PRINT DIGITIZE THE ORIGIN OF THE GRAPH >>":BEEP
GOSUB 8000:XORG=XIN:YORG=YIN:' CALL DIGDRU(XORG,YORG,IBTN)
IF(VAL(CODE$)=10)GOTO 5530
PRINT DIGITIZE ANY OTHER KNOWN"
PRINT POINT ON THE SAME HORIZONTAL (X-AXIS) LINE>>"
GOSUB 8000:XHZ=XIN:YDUMM=YIN:' CALL DIGDRU(XHZ,YHZ,IBTN)
IF(VAL(CODE$)= 10)THEN RETURN
XSCALE=1!
XOFF=0!
YSCALE=1!
YOFF=0!
ANGLE=0!
XROUND=0!
YROUND=0!
XD=FNRDIST(XORG,YORG XHZ,YDUMM)
IF(XD = 0!)THEN RETURN:'GOSUB 3800:RETURN:'STREAM MODE
'PRINT TWO-POINT X DISTANCE IN DIGITIZER REAL UNITS: ";XD
PRINT "DIGITIZE A THIRD KNOWN POINT ON THE VERTICAL (Y-AXIS) LINE"
GOSUB 8000:XDUMM=XIN:YHZ=YIN
ANG1=FNANGLER(XORG,YORG,XHZ,YDUMM)
ANG2=FNANGLER(XORG,YORG,XDUMM,YHZ)
YANG=1.570796-(ANG2-ANG1)
YD=FNRDIST(XORG,YORG,XDUMM,YHZ)*COS(YANG)
IF YD=0 THEN RETURN
PRINT "X-DISTANCE= ";XD?" Y-DISTANCE= ";YD
PRINT ENTER USER COORDINATES"
PRINT OF THE FIRST POINT (REAL) [0.0,0.0]: "7
INPUT X1U,Y1U
'5 FORMAT(2F10.0)
XDEF=XD+X1U:YDEF=YD+Y1U
' WRITE(LUNO,6)XDEF
PRINT ENTER USER X COORDINATE"
PRINT OF THE SECOND POINT (REAL) ";XDEF
INPUT X2U
IF(X2U = 01)THEN X2U=XDEF
PRINT ENTER USER Y COORDINATE"
PRINT OF THE THIRD POINT (REAL) ";YDEF
INPUT Y3U
XU-X2U-X1U

224
5365 YU=Y3U-Y1U
5370 ANGLE=ANG1
5380 IF(XU O 01)THEN XSCALE=XU/XD
5390 IF(YU <> 0)THEN YSCALE=YU/YD
5400 IF(X1U = 0! AND Y1U = 0!)GOTO 5510
5402 X1U=X1U/XSCALE:Y1U=Y1U/YSCALE
5410 ANGLU=FNANGLER(0!,01,X1U,Y1U)
5420 ANGLU=ANGLE+ANGLU
5430 DISTU=FNRDIST(0!,0!,X1U,Y1U)
5440 1 DISTX=DISTU/XSCALE
5450 DISTY=DISTU/YSCALE
5460 XROT=DISTU*COS(ANGLU)
5470 YRT=DISTU*SIN(ANGLU)
5480 XOFF=XORG-XROT
5490 YOFF=YORG-YROT
5500 GOTO 5530
5510 XOFF=XORG:'100
5520 YOFF=YORG
5530 '200 WRITE(LUNO,201)
5540 PRINT ENTER X-AXIS ROUNDOFF (REAL) [0.0]: ;
5550 INPUT XROUND
5560 '202 FORMAT(F6.0)
5570 WRITE(LUNO,203)
5580 PRINT ENTER Y-AXIS ROUNDOFF (REAL) [0.0]:
5590 INPUT YROUND
5600 RETURN:'GOSUB 3800:RETURN :'RESET STREAM MODE FIRST!
5610 END
6000 TIMER LOOPS
6001 BEEP :FOR IDUM=1 TO 375 :NEXT IDUM:PRINT TIME$:RETURN:'1 SEC
6002 BEEP :FOR IDUM=1 TO 750 :NEXT IDUM:PRINT TIME$:RETURN:'2 SEC
6003 BEEP :FOR IDUM=1 TO 1125:NEXT IDUM:PRINT TIME$:RETURN:'3 SEC
6005 BEEP :FOR IDUM=1 TO 1875:NEXT IDUM:PRINT TIME$:RETURN:'5 SEC
6010 BEEP :FOR IDUM=1 TO 3750:NEXT IDUM:PRINT TIME$:RETURN:'10SEC
8000 REM ++++++ GET INPUT FROM COM BUFFER ++++++++
8010 WHILE LOC(1) < LCB
8020 WEND
8030 'IF LOF(1) < 24 THEN BEEP:BEEP: BEEP
8040 DZ$=INPUT$(LCB/#1)
8041 'PRINT DZ$
8050 X$=LEFT$(DZ$,5) : Y$=MID$(DZ$,6,5) : CODE$= MID$(DZ$,11,1)
8060 XIN=VAL(X$)/1000 : YIN=VAL(Y$)/1000
8061 'PRINT XIN,YIN,CODE$:BEEP
8070 RETURN
9000 SUBROUTINE DIGURU(X,Y,IBTN)
9010 GOSUB 8000
9020 DISTU=FNRDIST(XOFF,YOFF,XIN,YIN)
9030 ANGU=FNANGLER(XOFF,YOFF,XIN,YIN)-ANGLE
9040 XIN=DISTU*COS(ANGU)*XSCALE
9050 YIN=DISTU*SIN(ANGU)*YSCALE
9055 XTRUE=XIN/XSCALE:YTRUE=YIN/XSCALE
9070 RETURN
20000 IF ERR=53 AND ERL=4005 THEN PRINT "NO FILES ON B:":RESUME 4006
20030 PRINT "ERROR NUMBER ";ERR;" HAS OCCURRED AT LINE ";ERL

225
PULSING MODEL GIGI VERSION
MODIFIED TO WRITE DATA FILE FOR 3-D GRAPHICS PROGRAM
PULSEGGM.FTN VERSION
BASELINE MODEL 2.0 PREVIOUS TO AUG 29,1984
VERS 2.1 CHANGED FORMAT OF HARDCOPY PRINTOUT OF VARIABLES
VERS 2.11 ADDED TOTALS AND DESCRIPTION TO OUTPUT LIST
VERS 2.12 ADDED K8 TO OUTPUT LIST
VERS 2.121 (1/24/85) ADDED TRACKING COEFFICIENTS ON K2,K9,K11
AND CHANGED FORMAT 11003 FOR VARS(IVAL) FROM F6.1 TO G15.4
PROGRAM PULSE
BYTE XTEXT(80),DES(40),YTEXT(80)
DIMENSION VARS(20),ALPHA(20)
DIMENSION FILE(3)
REAL M1,M2,M3,M4,M9,K10,K11,K12,K13
REAL K1,K2,K3,K4,K5,K6,K7,K8,K9,J,JO,JNORM
EQUIVALENCE (VARS(1),K1),
(VARS(2),K2),
(VARS(3),K3),
(VARS(4),K4),
(VARS(5),K5),
(VARS(6),K6),
(VARS(7),K7),
(VARS(8),K8) ,
(VARS( 9) ,K9) ,
(VARS(10),K10) ,
(VARS(11),K11),
(VARS(12),K12),
(VARS(13),K13),
(VARS(14),XJ0INI),
(VARS(15),Q1IC),
(VARS(16),Q2IC),
(VARS(17),Q3IC),
(VARS(18),Q4IC)
VIRTUAL IY(6,25,180)
DATA FILE/3*' '/
DATA XTEXT/ ',P','u','1','s','e',' ,'M','o','d','e'
'1',' ',67*0/
DATA DES/40*0/
DATA YTEXT/80*0/
DATA ALPHA/K1 ','K2 ','K3 ','K4 ','K5 ','K6 ','K7 '
,'K8 ','K9 ','K10 ','K11 ,'K12 ','K13 ','JOIN','Q1IC'
, 'Q2IC 'Q3IC Q4IC ,2* '/
FT 1(A,3)=ABS(AINT(A/B)-A/B)
VERS=2.121 19/7/84 ; 1/24/85
WRITE(5,100)
FORMAT(' WHAT IS THE DATA FILE FOR THIS MODEL RUN ?')
READ(5,101)(FILE(I),1=1,3)
FORMAT(3A4)

226
WRITE(5,1011)FILE
1011 FORMAT(1X,3A4)
C WRITE( 5,1016)
C1016 FORMAT(' DO YOU WANT HARDCOPY (1-YES, O-NO) '$)
C READ(5,1017)ICOPY
ICOPY=0
C1017 FORMAT(12)
C WRITE (5,1018)
C1018 FORMAT(1 WHICH Q TO SAVE (1,2,3,4,5=JR,6=%POW USED)'$)
C READ (5,1019)IQSAV
IQSAV=1
C1019 FORMAT(13)
C WRITE(5,1020)
C1020 FORMAT(1 DO YOU WANT TO PLOT THE GRAPHS (1-YES, 0-NO)'$)
C READ( 5,102DIPLOT
IPLOT=0
C1021 FORMAT(11)
C WRITE(5,99)
C99 FORMAT(' HOW LONG TO RUN? ')
C READ(5,98)TIME
C98 FORMAT(G6.0)
CALL ASSIGN(1,FILE)
READ(1)E1,E2,E3,E4,E5,E6,E7,E8,E9,E10,E11,E12,E13,E14,E15,
C COEFFICIENTS **********************************************
+NUM,K1,K2,K3,K4,K5,K6,K7,K8,K9,K10,K11,K12,K13
+,XJ0INI,Q1,Q2,Q3,Q4
C INITIAL CONDITIONS*****************************************
CLOSE (UNIT=1)
Q1IC=Q1
Q2IOQ2
Q3IC=Q3
Q4IC=Q4
C********************************************************
C
C
WRITE(5,1012)K1,K10,K2,K11,K3,K12,K4,K13,K5,XJ0INI,K6,Q1,
+K7,Q2,K8,Q3,K9,Q4
1012 FORMAT(1X,'1-K1 1,G12.6,' 10-K10',G12.6/
+ 1X,
'2-K2 '4
,G12 6,'
11-K11',G12.6/
+ 1X,
'3-K3 4
, G12.6 '
12-K12',G12.6/
+ 1X,
'4-K4 '4
,G12.6,'
13-K13',G12.6/
+ 1X,
'5-K5 i
,G12.6,'
14-XJ0INI',G12.6/
+ 1X,
'6-K6 '4
,G12.6,'
15-Q1IC ,G12.6/
+ 1X,
'7-K7 '4
G12.6,'
16-Q2IC ,G12.6/
+ 1X,
'8-K8 '4
,G 12.6, '
17-Q3IC',G12.6/
+ 1X,
'9-K9 '4
,G12.6 '
18-Q4IC,G12.6/
+ 1X,
' INPUT VARIABLE
NUMBER TO VARY => '$)
READ ( 5,10 13) IVAL
1013 FORMAT(12)
WRITE (5,1014)ALPHA(IVAL),VARS(IVAL)
1014 FORMAT(' VARIABLE ',A4,' = ',G12.6/
+ HOW MUCH TO INCREMENT? '$)
READ(5,1015)XINC
1015 FORMAT(G15.6)

227
c
c
c
£********************************************************
CALL ASSIGN (4,'TTO:')
IF (IPLOT.EQ.0)G0T0499
CALL GGON
CALL GGINIT
CALL GGAXIS(0,0,767,479)
499 Q1IC=Q1
Q2IC=Q2
Q3IC=Q3
Q4IC=Q4
NSLICE=25
NCNTS=150
NRUN=0
500 CONTINUE
C J0=XJ0INI+NRUN*4.
IF(NRUN.EQ.0)GOTO501
VARS(IVAL)=VARS(IVAL)+XINC !INCREMENT VALUE WE ARE VARYING
501 NRUN=NRUN+1
J0=XJ0INI
J=J0/(1+K13*Q1*Q4) ¡GIVE JR (J) INITIAL VALUE
T=0.
q**********************************************************
C SET K VALUES TO TRACK FOR MULTIRUN MODEL
C
K3=. 1 *K2
K4=.9*K2
C
K7=.1*K9
K8=.9*K9
C
K5=1*K11
K6=.9*K11
q***********************************************************
Q1=Q1IC
Q2=Q2IC
Q3=Q3IC
Q4=Q4IC
EUSED=0.0
M1=0.
M2=0.
M3=0.
M4=0.
R1=0
R2=0.
R3=0.
R4=0.
R5=0.
R6=0.
R7=0.
RS=0 .
R9=0.

228
R10=0
R11=0.
R12=0
EUSED=0 0
PAVAIL=0.0
C
C WRITE(5,1081)
C1081 FORMAT(' SCALE FACTOR FOR Q2200. OR 1000. [R] )
C READ(5,1082)SFACT
C1082 FORMAT(G7.2)
C WRITE( 5,108)
C108 FORMAT(' WHAT IS THE TIME INTERVAL DT [R] ')
C READ(5,109)DT
C109 FORMAT(G5.3)
TIME=750.
DT=. 1
SFACT=100.
NTIME=TIME
XDT=DT/10
C START OF LOOP ***********************************************
IF (IPLOT.EQ.0)GOTO5
CALL GGERA
CALL GGBOX(7,0,0,767,479)
CALL GGTEXT(7,626,475,XTEXT,1,0)
CALL GGBOX(7,620,452,767,479)
<2 PRINT INITIAL CONDITIONS*************************************
5 T=T+DT
C RATE EQUATIONS***********************************************
J=J0/(1+K13*Q1*Q4)
POWUSE=10 0.*(JO-J)/JO
PAVAIL=PAVAIL+J0
EUSED=EUSED+JO-J
R1=DT*K1*Q1*Q4*J
R2=DT*K2*Q1
R3=DT*K3*Q1
R4=DT*K4*Q1
R5=DT*K5*Q2
R6=DT*K6*Q2
R11=DT*K11*Q2
R7=DT*K7*Q2*Q3*Q3
R8=DT*K8*Q2*Q3*Q3
R9=DT*K9*Q2*Q3*Q3
R10=DT*K10*Q1*Q4*J
R12=DT*K12*Q3
C LEVEL EQUATIONS *********************************************
1091 CONTINUE
Q1=Q1+R1-R2
Q2=Q2+R3-R9-R11
Q3=Q3+R5+R7-R12
Q4=Q4+R4+R6+R12+R8-R10
IF(FT1(T,1.)GTDT)GOTO 110
IF(IPLOT.EQ.0)G0T02 0000
ITIME=T
IXC=Q1/10

CALL GGPLT(4,ITIME,IXC, 1)
IXOQ2/SFACT
CALL GGPLT(3,ITIME,IXC,1)
IXOQ3/10.0
CALL GGPLT(1,ITIME,IXC,1)
IXC=Q4/100.
CALL GGPLT(5,ITIME,IXC,1)
IXC=(J0-J)*(250./J0)
CALL GGPLT(2,ITIME,IXC,1)
C
C FIND WHICH Q TO SAVE IN ARRAY
20000 CONTINUE !GOTO(21000,22000,23000,24000,25000,26000)IQSAV
C GOTO110
21000 IY(1,NRUN,INT(T/5.)+1)=INT(Q1/2.)
C GOTO110
22000 IY(2,NRUN,INT(T/5)+1)=INT(Q2/20)
C GOTO110
23000 IY(3,NRUN,INT(T/5.)+1)=INT(Q3/2.)
C GOT0110
24000 IY(4,NRUN,INT(T/5.)+1)=INT(Q4/40.)
C GOTO 110
25000 IY(5,NRUN,INT(T/5)+1)=INT((J0-J)*5.)
C GOTO110
26000 IY(6,NRUN,INT(T/5)+1)=INT((POWUSE-80)*50)
110 CONTINUE
M1=AMAX1(Ml,Q1)
M2=AMAX1(M2,Q2)
M3=AMAX1(M3,Q3)
M4=AMAX1(M4,Q4)
IF(T.LT.TIME)GOTO 5
C
ENCODE(80,11003,YTEXT)NRUN,ALPHA(IVAL),VARS(IVAL)
+,EUSED,100*EUSED/PAVAIL
11003 FORMAT(2X,12,' VARIABLE ',A4,' = ',G15.4,' POWER USED ',
+G12.6,' PPU: ',G12.6)
IF (IPLOT.EQ.0)WRITE(4,11004)YTEXT
11004 FORMAT(1X,80A1)
IF (IPLOT.EQ.1)CALL GGTEXT(7,0,460,YTEXT,1,0)
IF (ICOPY.EQ.1)WRITE(3,11001)
11001 FORMAT('+S(H)')
IF(NRUN.LT.NSLICE)GOT050 0
IF (IPLOT.EQ.0)GOTO24999
CALL GGERA
CALL GGOFF
24999 IQSAV=1
24995 CONTINUE
CALL ASSIGN(2,1PULSAV*,6)
ENCODE(40,25001,DES)IQSAV,FILE
25001 FORMAT(1X,' TANK Q',11,' FOR DATA FILE ',3A4)
WRITE(2)(DES(I),1=1,40),NRUN,NCNTS,
+ ((IY(IQSAV,J1,K),K=1,NCNTS),J1=1,NRUN)
CLOSE(UNIT=2)
IQSAV=IQSAV+1
IF (IQSAV.LT.7)G0T024995
!Q1 GREEN
!Q2 MAGENTA
!Q3 BLUE
!Q4 CYAN
¡POWER USED RED

230
c
c
WRITE(4,114)
114 FORMAT(//1 Q1 Q2 Q3 Q4 TOTAL')
WRITE(4,1121)0110,2210,0310,04IC,Q4IC+Q3IC+Q2IC+Q1IC
WRITE(4,1122)M1,M2,M3,M4
WRITE(4,1123)Q1,Q2,Q3,Q4,Q4+Q3+Q2+Q1
1121 FORMAT(' INIT ',4(2X,G8.2),2X,G12.6)
1122 FORMAT(' MAX ',4(2X,G3.2))
1123 FORMAT(' FINAL',4(2X,G8.2),2X,G12.6)
119
120
WRITE (4,116) K1 ,K2,K3,K4,K5,K6,.K7 ,K8 ,K9 ,K10 ,K11 ,K12 ,K13 JO J
F0RMAT(/1X,'K1= ',G12.6,' K2= ',G12.6,' K3= ',G12.6/
+' K4= ',G12.6,' K5= ',G12.6,' K6= ',G12.6/
+ K7= ',G12.6,' K8= ',G12.6,' K9= ',G12.6/
+' K10=',G12.6,' K11=',G12.6,' K12=',G12.6/
+' K13=',G12.6,' J0= ',G12.6,' JR= ',G12.6)
EUSED=EUSED*DT
PAVAIL=PAVAIL*DT
PPU=100.*EUSED/PAVAIL
WRITE(4,119)VERS,DT,EUSED,PAVAIL,PPU
FORMAT(/' PULSE MODEL VERS',F6.3,' TIME STEP(DT) = ',F6.4/
+' TOTAL POWER USED = ',G15.6,' POWER AVAILABLE =',G15.6/
+' PERCENT POWER USED = ',G15.6)
WRITE(4,120)(FILE(I),1=1,3),E1,E2,E3,E4,E5,E6,E7,E8,E9,E10,
+E11,E12,E13,E14,E15
FORMAT(' DATA FILE NAME = ',3A4,1X,15A4)
IF(ICOPY.EQ.0)G0T0999
CALL GGON
WRITE(3,11001)
CALL GGOFF
999
END

231
C SURFACE PULSING MODEL PROGRAM 3/24/83
C ADDITION OF CONSUMER CEILING TO ALLOW UP TO 100 TOTAL
C CONSUMERS
C
C DIFFUSION ADDED 7/21/83 TO NUTRIENT TANK Q4
C
C VERS 3.01 ADDED STARTING CONDITION TO FILE OUTPUT
C VERS 3.02 ADD TIME AND DATE TO BEGINNING OF PROG
C
C
PROGRAM SURPUL
C Q1=PRODUCER
C Q2=STORAGE (PRODUCER)
C Q3=CONSUMER
C Q4=NUTRIENTS
DIMENSION Q1(12,12),Q4(12,12),E(12,12),Q3(100)
DIMENSION Q4T(12,12)
DIMENSION ETYPE(3),IX(144)
DIMENSION Q2(12,12),IXYZ(100)
INTEGER*4 ICNT(12,12)
BYTE TITLE(10),ICON(12,12),BUF1(9),BUF2(8)
REAL M,K1,K2,K3,K4,K5,K6,K7,K8,K9,K10,MTOT
REAL K11,K12,K13,JO,JR
BYTE ESC,TEXT(80),COLOR,ICOLOR,CHAR
INTEGER X1(100),Y1(100),T1,T2,XTEMP,YTEMP
FT1(A,3)=ABS(AINT(A/B)-A/B)
IXY(I,J)=(I-1)*12+J
DATA TITLE/'D','S','P','1' ,'O','O',' '/
DATA Q4T/144*0.0/ !DK
DATA ICON/144*0/
DATA ETYPE(1)/'HIER'/
DATA ETYPE(2)/'EVEN'/
DATA ETYPE(3)/'RAND'/
ESC=27
VERS=3.02
CALL TIME(BUF2)
CALL DATE(BUFI)
WRITE(5,5)ESC,ESC,ESC,ESC,TITLE,VERS,BUF2,BUF1 !3.0
5 FORMAT(1X,A1,'PrTM1',A1,1 ',A1,'[2J',A1,'[H', 13.0
&' SURFACE MODEL ',10A1,' VERSION ',F5.2/ 13.0
&1X,8A1,1X,9A1/ 13.02
&' DO YOU WANT GRAPHICS ON (1-YES, 0-NO) '$)
READ (5,6)IOFLAG
6 FORMAT(11)
C
C
WRITE (5,7)
7 FORMAT(' PLOTTING INTERVAL FOR PRODUCER AND CONSUMER [I] '$)
READ (5,8)ITINT,ITINTC
8 FORMAT(13,13)
WRITE(5,808)
808 FORMAT(' HARDCOPY AT PLOTTING INTERVAL (1-YES,0-NO)'$)
READ(5,8081)IPTR
8081 FORMAT(13)

8082
81
82
83
84
841
842
85
86
9
11
91
92
121
122
12
13
C
14
15
150
232
TINT=ITINT
TINTC=ITINTC
WRITE (5,81)
CONTINUE
FORMAT(1 HOW LONG TO RUN? [ R] '$)
READ (5,82)TTIME
FORMAT(G6.0)
WRITE(5,83)
FORMAT(' WHAT IS DT [R] '$)
READ (5,84)DT
FORMAT(G10.6)
XDT=DT/TINT
X DTC= DT/TINTC
WRITE(5,841)
FORMAT(' WHAT IS NUTRIENT CONC. OF OUTER NONREACTIVE'/
& RING ( 39000 IC; 0.0 TO ? ) [R] '$)
READ (5,842)Q40IC
FORMAT(G16.5)
WRITE(5,85)
FORMAT(' WHAT IS DIFFUSION COEFFICIENT? [R] '$)
READ (5,86)DK
FORMAT(F8.5)
WRITE( 5,9)
FORMAT(' INPUT THE SEARCH LENGTH FOR PREDATOR [I] ',$)
READ (5,11)N
FORMAT(12)
WRITE(5,91)
FORMAT( 1X, FEEDING AND DOUBLING THRESHOLD [R,R] '$)
READ(5,92)PTHRSH,THRESH
FORMAT(2G8.2)
WRITE(5,121)
FORMAT(1X,'INPUT (0-SUCCESSION; 1-STEADY STATE) [I] '$)
READ(5,122)ISSUC
FORMAT(12)
WRITE(5,12)
FORMAT(' WHAT ENERGY TYPE WOULD YOU LIKE'/
+ '1: STD INPUT'/
+ '2: CONSTANT INPUT'/
+ '3: RANDOM INPUT',20X,'ENERGY TYPE [I] '$)
READ(5,13) IETYP
FORMAT(11)
IF(IETYP.EQ.1)GOTO150
WRITE(5,14)
FORMAT(' WHAT IS THE MEAN VALUE OF ENERGY '$)
READ(5,15)XMEAN
FORMAT(F5.2)
CONTINUE
GPP=0.
CNSUMP=0.
Q1TOT=0.
Q2TOT=0.
Q3TOT=0.
Q4TOT=0.
PROD=0.
!GPP COUNTER (TOTAL)
!CONSUMPTION BY CONSUMERS (TOTAL)
!AMOUNT OF PRODUCERS
¡AMOUNT OF STORAGE
¡AMOUNT OF CONSUMERS
¡AMOUNT OF NUTRIENTS
¡TOTAL PRODUCTION (GPP+CNSUMP)
! DK
! DK
! DK
! DK
! DK
! DK
! DK
! DK
! DK
! 3.0
¡3.0
¡3.0
¡3.0

233
147
C
151
C
C
51
511
551
552
52
521
C
C
C
C
ETOT=0.
TOTPOW=0.0
T=0.
T1= 1
T2=T1
¡TOTAL ENERGY INPUT (SUM OF MATRIX)
¡MEASURE TOTAL POWER USED: SUM OF EUSED
¡
¡NUMBER OF CONSUMERS
Q1IC=1000. ¡1C CONDITION FOR Q1
Q4IC=39000.
Q2IC=1000.
IF(ISSUC.EQ.O)GOTO147
Q4IC=30000. !IC CONDITION FOR NUTRIENT TANK
Q2IC=10000. !IC FOR PRODUCER STORAGE
CONTINUE
Q3IC=50
THRESH=500. ¡DOUBLING THRESHOLD FOR CONSUMER
IF (IOFLAG.EQ.O)GOT05 21
CALL GGON
IF (IPTR.EQ.0)GOTO151
WRITE(3,20171)
CALL GGINIT
CALL GGAXIS(0,0,767,479)
CALL GGERA
CALL GGBOX(7,0,0,767,479)
WRITE(3,51)
¡3.0
¡3.0
¡3.0
I 3.0
CLEAR MACROS AND DEFINE ONE TO DRAW BOXES
FORMAT( +', @. @:A P[+0,+0]W(S1)V[,+24]V[+24,]V[,-24]V[-24,]
+ W( SO) @ ?')
WRITE (3,511)
FORMAT ( +' @ : B T(A1) P[+0 ,+0] V[,+24] V[+24 ] V[,-24] V[-24 ]
+ W(SO)T(A0) <3;')
WRITE(3,551)
FORMAT('+L(A1)'/
+'+LM 7"FFFFFFFFFFFFFFFFFFFF;'/
+'+L"6"AA55AA55AA55AA55AA55;'/
+'+L"5"92492492492492492492;'/
+'+L"3"84210842103421084210;')
WRITE(3,552)
FORMAT('+L"4"88442211884422118844;'/
+,+L"2"42009100240091004200; '/
+'+L"1"20000840021000042000;'/
+ '+L"0"00002000000200002000; '/
+'+L"B"00000000000000000000;')
DO 52 1=0,7
CALL GGPLT(1,735,(1+1)*24-16,0)
CHAR=I+48
WRITE(3,398)CHAR
CALL GGBOX(7,725,0,767,248)
CALL GGBOX(7,0,0,767,248)
CALL GGBOX(7,575,0,725,248)
CONTINUE

234
C
SET UP SURFACE OF FORCING ENERGY
DATA TEXT/80*0/
DATA E 70,0,0,0,0,0,0,0,0,0,0,0,
+ 0,0.8,0.8,0.8,0.8,0.8,0.8,0.8,0.8,0.8,0.8,0,
+ 0,0.8,1,1,1,1,1,1,1,1,0.8,0,
+ 0,0.8,1,1,1,1,1,1,1,1,0.8,0,
+ 0,0.8,1,1,1.2,1.2,1.2,1.2,1,1,0.8,0,
+ 0,0.8,1,1,1.2,1.2,1.2,1.2,1,1,0.8,0,
+ 0,0.8,1,1,1.4,1.4,1.4,1.4,1,1,0.8,0,
+ 0,0.8,1,1,1.2,1.2,1.2,1.2,1,1,0.8,0,
+ 0,0.8,1,1,1,1,1,1,1,1,0.8,0,
+ 0,0.8,1,1,1,1,1,1,1,1 ,0.8,0,
+ 0,0.8,0.8,0.8,0.8,0.8,0.8,0.8,0.8,0.8,0.8,0,
+ 0,0,0,0,0,0,0,0,0,0,0,07
C
X1(1)=5
Y1(1)=5
Q3(1)=Q3IC
ICON(5,5)=1
IXYZ(1)=IXY(5,5)
!SET PREDATOR CLOSE TO CENTER
!FIRST PREDATOR LOCATION
!CODED LOCATION
K1=.417E-5
K2= .5
K3=.05
K4=.45
K5=.5E-4
K6=.45E-3
K7=.2E-6
K8=.18E-5
K9=.2E-5
K10=.417E-5
Kl1=.5E-3
K12=.05
K13=.7833E-6
DO 200 1=2,11
DO 200 J=2,11
Q1(I,J)=Q1IC
Q2(I,J)=Q2IC
Q4(I,J)=Q4IC
E(I,J)=E(I,J)*100.
200 CONTINUE
DO 201 IK1,12 DK
Q4(1,IK)=Q40IC SDK
Q4(12,IK)=Q40IC SDK
Q4(IX,1)=Q40IC SDK
Q4(IK,12)=Q40IC SDK
201 CONTINUE !DK
C
C
C
CHANGE ENERGY LEVEL?
IF(IETYP.EQ.1)G0T0221
DO 220 1=2,11
DO 220 J=2,11
E(I,J) = (XMEAN*RAN(-1) + .5)* 100
SCALE RANDOM FUNCTION

235
IF (IETYP.EQ.3)GOTO220
E(I,J)=XMEAN*100.
220 CONTINUE
221 DO 250 1=2,11
DO 250 J=2,11
ETOT=ETOT+E(I,J)
250 CONTINUE
IF (IETYP.EQ.2)GOTO300 ¡CHANGED 3 TO 2 IN 3.0
SF=ETOT/(100.*100.)
ETOT=0.0
DO 270 1=2,11
DO 270 J=2,11
E(I,J)=E(I,J)*XMEAN/SF
ETOT=ETOT+E(I,J)
270 CONTINUE
C
C
C>>>> LOOP START <<<<<<
C
300 CONTINUE LOOP START
IT=T ¡GET INTEGER VALUE OF TIME FOR MOD FUNCTION
EUSED=0.0 ¡ENERGY USED PER TIME I.E. POWER
SPTEMP=0
PTEMP=0.
DO 400 1=2,11
DO 400 J=2,11
C.... RATE EQUATIONS
C
17=0
INUM=0
XEQ=00
C
IF (ICON(I,J).NE.0)XEQ=1.0
J0=E(I,J)
JR=J0/(1+K13*Q1(I,J)*Q4(I,J))
R1=DT*K1*Q1(I,J)*Q4(I,J)*JR
R2=DT*K2*Q1(I,J)
R3=DT*K3*Q1(I,J)
R4=DT*K4*Q1(I,J)
R5=DT*K5*Q2(I,J)
R6=DT*K6*Q2( I, J)
R10=DT*K10*Q1(I,J)*Q4( I,J)*JR
R11=DT*K11*Q2( I,J)
EUSED=EUSED+(J0-JR)*DT
C.... LEVEL EQUATIONS ....
C
PTEMP=PTEMP+R1 ¡PRIMARY PRODUCTION
Q1(I,J)=Q1(I,J)+R1R2
IF(Q1(I,J).LT.0.0)Q1(I,J)=0.0
Q2(I,J)=Q2(I,J)+R3-R11
Q4(I,J)=Q4(I,J)+R4+R6-R10
Q4(I,J)=Q4(I,J)+R5 *(1-XEQ)
C ADD LINEAR FLOW TO Q4 IF Q3 NOT THERE

236
c
c...
c
c.
c
CONSUMER CHECKING ROUTINE
IF(XEQ.EQ.0)G0T0350
IXYLOC=IXY(I,J)
INUM=1
DO 217 17=1,T2
IF (IXYZ(17).NE.IXYLOC) G0T0217
RATE EQUATIONS FOR CONSUMERS
¡SKIP IF NO CONSUMER PRESENT
¡GET CODED LOCATION
¡AT LEAST ONE CONSUMER PRESENT
¡GET CONSUMER NUMBER
¡WRONG CONSUMER GOTO 217
XDT=DT
...IF RATIO OF Q2/Q3 IS TOO LOW THEN ITERATE MORE SLOWLY
IF(Q2(I/J)/Q3(17).LT.5.0)XDT=.01
DO 650 DDT=XDT,DT/XDT
R7=XDT*K7*Q2(I,J)*Q3(17)*Q3(17)*XEQ
R8=XDT* K8 Q2 (I, J) Q3 (17) Q3 (17) XEQ
R9=XDT*K9*Q2(I,J)*Q3(17)*Q3(17)*XEQ
R12=XDT*K12*Q3(17)*XEQ
SPTEMP=SPTEMP+R5*(XDT/DT)+R7 1SPTEMP = CONSUMP
C
C... LEVEL EQUATIONS FOR CONSUMERS
Q3(17)=Q3(17)+R5/(ICON(I,J))*(XDT/DT)+R7-R12
Q2(I,J)=Q2(I#J)-R9 ¡UPDATE PREY CONSUMED
Q4(I,J)=Q4(I,J)+R8+R12 ¡UPDATE NUTRIENTS
650 CONTINUE
C
C
C
IF((T-XTIME).LT.1.)G0T0217 ¡SKIP MOVEMENT IF NOT WHOLE DT
XTIME=T
C CHECK PRESENT POSITION FOR VALUE OF Q2
QMAX=0
IF(Q2(I,J).LT.PTHRSH)GOTO 1457
XTEMP=X1(17)
YTEMP=Y1(17)
GOTO600
C IF Q2 IS STILL CONSUMABLE DON'T MOVE, JUST EAT SOME MORE
1457 DO 600 I2=I-N,I+N
DO 600 J2=JN,J+N
IF (I2.LT.1) GO TO 600
IF (I2.GT.12) GO TO 600
IF (J2.LT.1) GO TO 600
IF (J2.GT.12) GO TO 600
IF(Q2(I2,J2).LT.QMAX)GOTO580
QMAX=Q2(12, J2)
XTEMP=I2
YTEMP=J2
580 CONTINUE
600 CONTINUE
ICON(I,J)=ICON(I,J)-1
Cl REMOVE CONSUMER FROM PRESENT LOCATION
ICON(XTEMP,YTEMP)=ICON(XTEMP,YTEMP)+1
C!MOVE CONSUMER TO NEW LOCATION

o o
237
IXYZ(17)=IXY(XTEMP,YTEMP)
C!CODE NEW LOCATION
C
C... CHECK TO SEE IF IT IS TIME TO REPRODUCE
C
IF(Q3(17).LTTHRESH)G0T02000 !IF GREATER THAN REPRODUCTION
T1=T1+1 !INCREASE NUMBER OF CONSUMERS
IF(T1.GT.100)GOT02000 ¡ALLOW NO MORE THAN 100
Q3(I7)=Q3(I7)/2
Q3(T1)=Q3(I7)
X1 (T1)58 XTEMP
Y1 ( T1 )*YTEMP
IXYZ(T1)=IXY(XTEMP,YTEMP)
ICON(XTEMP,YTEMP)=ICON(XTEMP,YTEMP)+1
2000 CONTINUE
XI(I7)=XTEMP
Y1(17)=YTEMP !REM REMEMBER WHERE TO START NEXT TIME
C
C
2001 CONTINUE
C
217 CONTINUE
350
C
C
400
C
C
C
C
C
C
C
453
C
C
CONTINUE
CONTINUE
GPP=GPP+PTEMP ¡ACCUMULATE TOTAL GPP
T2=T1
IF (T2.GE.100)T2=100
END CONSUMER LOOP AND DO BOOKEEPING
CNSUMP=CNSUMP+SPTEMP
PROD=GPP + CNSUMP
TOTPOW=TOTPOW+EUSED
COUNT UP THE CONSUMERS
NPROD=100
IF(T2LT10 0)NP ROD=T2
Q3TOT=Q.
DO 453 1=1,NPROD
Q3TOT=Q3TOT+Q3(I)
ICNT(X1(I), Y1 (I))=ICNT(X1(I),Y1(I))+1
CONTINUE
COUNT UP PRODUCERS AND NUTRIENTS
Q1TOT=0.
Q4TOT=0.

238
Q2TOT=0.
DO 4531 1X1=2,11
DO 4531 1X2=2,11
Q1T0T=Q1T0T+Q1(1X1,1X2)
Q2TOT=Q2TOT+Q2(1X1,1X2)
Q4TOT=Q4TOT+Q4(1X1,1X2)
4531 CONTINUE
Q4TOUT=Q4(1,1)*44.
TOT=Q1TOT+Q2TOT+Q3TOT+Q4TOT
IF(IOFLAG.EQ.0)GOTO5000
C
C
C WRITE TEMPORARY INFORMATION AND PLOT GPP, POWER (EUSED)
C
C
IF ((T-PTIME1).LT.1.0)GOTO 5000
PTIME1=T
ENCODE(80,2006,TEXT)T,T2,EUSED/(100.*DT),ETOT/100.,VERS,TITLE
&,ETYPE(IETYP)
2006 FORMAT(1X,'T=',F6.2,' CONS=',I3,' POW USED=',F6.2,
& AVAIL POW= ',F62,' VER:',F5.2,1X,10A1,1A4)
CALL GGTEXT(7,0,475,TEXT,1,0)
IPT=PTEMP/(2*DT*1000.)
CALL GGPLT(4,IT,IPT+250,1) ¡GREEN = GPP
IPOWER=((EUSED/(100*XMEAN*DT))80)*5. ¡OUTPUT 80 TO 100
CALL GGPLT(2,IT,IPOWER+250,1) ¡RED = POWER
C
ENCOD E(80,4532,TEXT)Q1TOT,Q2TOT,Q4TOT,Q3TOT,TOT
4532 FORMAT(1X,'Q1= ',F10.2,' Q2= ',F10.2,' Q4= ',F10.2,
& Q3TOT= ',F10.2,' TOT= 1,F10.2)
CALL GGTEXT(6,0,460,TEXT,1,0)
IYT=Q2TOT/20000
CALL GGPLT(3,IT,250+IYT,1) ¡MAGENTA = PRODUCERS
IYT=Q3TOT/2 0 0.+2 5 0.
CALL GGPLT(7,IT,IYT,1) ¡WHITE =CONSUMERS
C
C DRAW PRODUCERS
C
C l
IF((T-PTIME).LT.TINT)GOTO4500 !
PTIME=T
DO 4050 1=2,11 !
DO 4050 J=2,11 1
COLOR=Q2(I,J)/(2*1000.) ¡3.0 CHANGED 4*1000 TO 2*1000
IF(COLOR.GT.7)COLOR=7 !
1X9=1*24-40 !
JY=J*24-44 !
CHAR=48+COLOR ¡
IF(CHAR.GT.57)CHAR=57 1
CALL GGPLT(COLOR,1X9,JY,0) ¡POINT TO LOWER LEFT
WRITE(3,398)CHAR 1
398 FORMAT('+','T(A1)W(S', 1H',A1,1H',') @B') ¡
399 FORMAT(' @A') ¡DRAW BOX (MACRO)
4050 CONTINUE !

239
CHAR=' B'
WRITE (3,3985)CHAR l
CALL GGPLT(0/600,10,0)
CALL GGBOX(0,600,10,700,230)
CALL GGFILL(O)
CALL GGBOX(7,600,10,700,110)
CALL S0RT1(Q2,IX,144,400.)
CALL GGPLT(7,601,IX(1)+10,1)
DO 3991 IT1=2,100
CALL GGVEC(3,600+ITl,IX(IT1)+10)
3991 CONTINUE
C 1
C
C
4500 CONTINUE
C
C>>>> PLOT CONSUMERS <<<<<<
C !
C !
ICOLOR=0 !
IF((T-PTIMEC).LT.TINTC)GOTO 5000 !
PTIMEC=T
CHAR='B' !
WRITE (3,3985)CHAR !
3985 FORMAT(' + ','T( A1) W( S',1H',A1,1H',')') !
CALL GGBOX(ICOLOR,284,4,560,244) !
CALL GGFILL(O) !
CALL GGPLT(7,0,0,1)
CALL GGVEC(7,767,0)
DO 2005 11=1,T2 1
ICOLOR=Q3(I1)/50. !
IF(ICOLOR.GT.7)ICOLOR=7 !
CHAR=ICOLOR+48 !
CALL GGPLT(ICOLOR,X1(11)*24-4+284,Y1(11)*24-44,0) !
WRITE(3,398)CHAR !
2005 CONTINUE !
CALL GGBOX(7,600,130,700,230)
CALL SORT1(Q3,IX,T2,40)
CALL GGPLT(7,601,IX(1)+130,1)
DO 2017 IT1=2,T2
CALL GGVEC(7,600+IT1,IX(IT1)+130)
2017 CONTINUE
IF(IPTR.EQ.0)GOT05 000
WRITE (3,20171)
20171 FORMAT('+S(H)')
C !
C !
C
C SEE IF ITS TIME TO QUIT
5000 CONTINUE
C !DK
C DIFFUSION IDK
C SDK
IDK
QXT=0 0

240
DO 5002 1=2,11
I DK
DO 5002 J=2,11
! DK
DO 5001 IT*11,1+1
! DK
DO 5001 JT-J*1,J+1
! DK
QXT=QXT+DK*(Q4(IT,JT)-Q4(I,J))*DT
SDK
5001
CONTINUE
! DK
Q4T(I,J)=Q4(I,J)+QXT
! DK
QXT=0.0
! DK
5002
CONTINUE
! DK
DO 5003 1=2,11
!DK
DO 5003 J=2,11
SDK
Q4(I,J)=Q4T(I,J)
SDK
5003
CONTINUE
T=T+DT
IF(T.LT.TTIME)GOTO300
SDK
C>>>>>> END OF MAIN LOOP <<<<<<
C
CALL GGOFF
DO 439 1=1,12
DO 439 J= 1,12
ICNT(I,J)=ICNT(I,J)*DT
439 CONTINUE
CALL ASSIGN(4,'SURF4')
WRITE(4,440)VERS,TITLE,BUF1,BUF2
440 FORMAT('1','SURFACE MODEL VERSION NO. ',F6.2,1X,10A1,
& 1X,9A1,1X,8A1)
WRITE(4,454)ETOT,ETYPE(IETYP),PROD,TOTPOW,TOTPOW/(TTIME*100.),
& GPP,CNSUMP,N,ISSUC,DK
454 FORMAT(1X,' INPUT ENERGY TOTAL= ',F10.2,' ENERGY TYPE ',A4/
& 1X,' TOTAL PRODUCTION =',G15.6/
& 1X,' TOTAL POWER USED =',G15.5,' AVE POWER/CELL = ',G15.6/
& 1X,' GPP= ',G15.6,' TOTAL CONSUMPTION* *,G15.6/
& 1X,' SEARCH LENGTH =',I3,' STARTING CONDITION = ',12/
& 1.X,' DIFFUSION COEFFICIENT = 'F7.5)
WRITE(4,455)TTIME,DT,Q4TOT/1000,Q4TOUT/1000.,
& (Q4TOUT+Q4TOT)/10 0 0.
455 FORMAT(1X,' FOR ',F10.0,'ITERATIONS DT= ',F7.3/
& 1X,' TOTAL NUTRIENTS (KG) = ',F10.2/
& 11X,'Q4 OUTER TOTAL (KG) = 'F10.2/
& 1 1X, 'TOTAL INNER AND OUTER (KG) = ',F10.2/
& 1X,10X,' NUTRIENT MATRIX Q4(I,J)'/)
WRITE(4,456)((Q4(I,J)/1000.,1=1,12),J=12,1,-1)
456 FORMAT(1X,12F7.2)
WRITE(4,457)PTHRSH,THRESH,Q2TOT/1000.
457 FORMAT(//1X,'VALUES FORQ2(I,J) PRODUCERS'/
& 1X,'PRODUCER THRESHOLD FOR CONSUMER MOVING* ',F10.2/
& 1X,'CONSUMER THRESHOLD FOR DIVIDING INTO = ',F10.2/
& 1X,' TOTAL PRODUCERS (KG) = *,F10.2)
WRITE(4,4561)((Q2(I,J)/1000.,1=2,11),J=11,2,-1)
4561 FORMAT(8X,10F7.2)
WRITE(4,4581)
4581 FORMAT(//' CONSUMER VISITATION MATRIX '/)
WRITE(4,4582)((ICNT(I,J),1=1,12),J=12,1,-1)
4582 FORMAT(12(1X,I6))

WRITE(4,4584)
4584 FORMAT(//' FINAL CONSUMER DISTRIBUTION')
WRITE(4,4585)((ICON(I,J),1*1,12),J=12,1,-1)
4585 FORMAT(12(1X,I6))
WRITE(4,4571)Q3TOT
4571 F0RMAT(//1X,'VALUES FOR CONSUMERS TOTAL CONSUMERS =',G15
WRITE(4,458)((I,Q3(I),X1(I),Y1(I),IXYZ(I)),1=1,NPROD)
458 FORMAT(1X,14,' Q3= ',F8.2,' X=',I2,' Y =',12,1X,I4)
CALL CLOSE(1)
END
C
C
C
SUBROUTINE SORT1(X,IX,N,SF)
DIMENSION X(1),IX(1)
DO 20 1=1,N
IX(I)=X(I)/SF
20 CONTINUE
DO 40 1=1,N
DO 40 J=I,N
IF(IX(J).LT.IX(I))G0T04 0
ITEMP=IX(I)
IX(I)=IX(J)
IX(J)=ITEMP
40 CONTINUE
RETURN
END

242
PROGRAM GRAPH2
C vers RGRF
C
C WRITTEN BY JOHN RICHARDSON
C
C CALL TO REGLIN ADDED 6/15/83
C
C MAXIMUM NUMBER OF DATA POINTS SET TO 250
C
C
C MODIFIED 4/11/83 FOR RGL LIBRARY
C
BYTE XTEXT(80),YTEXT(80),TITLE(80),ESC
BYTE FNAME(16),IFNAM(16)
LOGICAL IRFLAG,SMOOTH,SHADE
DIMENSION X(250),Y(250),Y1(250)
DATA TITLE/80*0/
DATA XTEXT/80*0/
DATA YTEXT/80*0/
DATA FNAME/16*0/
DATA IFNAM/16*0/
ESC=27
TYPE 10
10 FORMAT(' GRAPHING PROGRAM FOR GIGI'
&/' COMPLIMENTS OF JOHN RICHARDSON'/)
WRITE(5,105)ESC,ESC
105 FORMAT( 2X, A1, PrTM 1' A1, )
TYPE 111
111 FORMAT(' PROGRAM REQUIRES THE TT: BUFFER BE SET TO NOWRAP'//
&' SET /NOWRAP=TI:'//
&' IF THIS IS NOT DONE PLEASE EXIT PROGRAM AND CORRECT THIS'//)
TYPE 15
15 FORMAT(' IS DATA IN A DATA FILE? (1=YES, 0=NO, -1=EXIT) '$)
ACCEPT 16,IANSI
16 FORMAT(12)
IF (IANSI.LT.0)STOP'MAKE CHANGES AND RERUN'
IF (IANS1.EQ.0) GOTO 11
TYPE 161
161 FORMAT(' FILE NAME FOR DATA: '$)
ACCEPT 162, (FNAME(I),1*1,16)
162 FORMAT(16A1)
OPEN(UNIT=1,NAME=FNAME,TYPE*'OLD',FORM*'FORMATTED')
READ (1,1621)TITLE
1621 FORMAT(1x,80A1)
WRITE(5,1621)TITLE
READ(1,1622)NPAIRS
1622 FORMAT(1X,I3)
DO 1630 1=1,NPAIRS
READ(1,*)X(I),Y(I)
C1625 FORMAT(2G15 6)
1630 CONTINUE
N=NPAIRS
GOTO 51
11
TYPE 20

243
20
30
35
36
59
60
50
601
602
603
604
606
608
611
511
51
5001
5002
501
502
503
504
FORMAT(' HOW MANY PAIRS OF POINTS TO PLOT $)
ACCEPT 30,N
FORMAT(13)
IF(I.GT.500)GOTO11
TYPE 35
FORMAT(1X,'DESCRIPTION OF DATA (UP TO 80 CHARACTERS '/)
ACCEPT 36,(TITLE(K),K=1#80)
FORMAT(80A1)
DO 50 1=1,N
TYPE 60,1
FORMAT(' X AND Y VALUES FOR POINT ',13,
' SEPARATED BY COMMAS [R] ')
READ (5,*,ERR=9911)X(I),Y(I)
CONTINUE
CLOSE(UNIT=1)
WRITE(5,601)
F0RMAT(//1X,'DO YOU WANT TO SAVE DATA (1-YES, 0
READ(5,602)ISAVE
FORMAT(11)
IF (ISAVE.NE.1)GOT051
WRITE( 5,603)
FORMAT(1 WHAT IS THE FILE NAME FOR THE DATA '$
READ(5,604)IFNAM
FORMAT(16A1)
CALL ASSIGN(2,IFNAM)
WRITE(2,606)TITLE
FORMAT(1X,80A1)
WRITE(2,608)N
FORMAT(1X,13)
DO 511 1=1,N
WRITE(2,611)X(I),Y(I)
FORMAT(1X,G15.6,' ',G15.6)
CONTINUE
CLOSE(UNIT=2)
XMAX=0.
YMAX=0.
A=0.
3=0.
R2=0.
CEE=0.
TYPE 5001
FORMAT(' DO YOU WANT TO RUN REGRESSION ON DATA? (1-YES,
ACCEPT 5002,IRGS
FORMAT(11)
IF (IRGS.NE.0)CALL REGLIN(N,X,Y,A,3,R2,CEE)
TYPE 501
FORMAT(' WHAT IT THE X- AXIS DESCRIPTION')
ACCEPT 502,XTEXT
FORMAT(80A1)
CALL STRIP(XTEXT,80)
TYPE 503
FORMAT(' WHAT IS THE Y-AXIS DESCRIPTION')
ACCEPT 504,YTEXT
FORMAT(80A1)
-NO)'$)
)
0-NO)')

244
CALL STRIP(YTEXT,80)
TYPE 70
70 FORMAT(' WANT TO INPUT MINIMUMS AND MAXIMUMS (1-yes, 0-no)'$)
ACCEPT 72,IMIN
72 FORMAT(11)
IF (IMIN.EQ.0)GOTO85
TYPE 74
74 FORMAT(1X,'WHAT ARE XMIN AND XMAX [R] '$)
ACCEPT *,XMIN,XMAX
TYPE 76
76 FORMAT(1X,'WHAT ARE YMIN AND YMAX [R] '$)
ACCEPT *,YMIN,YMAX
85 CONTINUE
TYPE 89
89 FORMAT(1X,' LINE TYPE (0-9) '$)
ACCEPT 891,ILIN
891 FORMAT(I1)
8910 TYPE 892
892 FORMAT(1X,'VALUE FOR DATA MARKER (0-9, -1 TO SEE LIST) '$)
ACCEPT 893,IMARX
IF(IMARK.GE0)G0T089 3 0
WRITE(5,8921)
8921 FORMAT(/' 0- POINT'
&
/'
1- SQUARE'
&
/'
2- OCTAGEN'
&
/
3- TRIANGLE'
St
/'
4- CROSS'
&
/
5- X'
&
/
6- Y'
&
/'
7- DIAMOND'
&
/'
8- ARROWHEAD
&
/'
9- HOURGLASS
&
/'
10-POINT IN -
G0T08910
8930 CONTINUE
893 FORMAT(15)
IF (IMIN.EQ.1)IROUND=0
IF (IMIN.EQ.1)G0T09910
TYPE 90
90 FORMAT(' ROUND MAX AND MIN VALUES? (1-YES, 0-NO) '$)
ACCEPT 99,1 ROUND
99 FORMAT(11)
IF(IROUND.GT.1)GOTO51
IF(IROUND.LT.0)GOTO51
9910 I RFLAG=.FALSE
IF(IROUND.EQ.1)IRFLAG=.TRUE.
WRITE (5,9901)
9901 FORMAT(' CURVEFIT THE DATA LINE (1-YES; 0-NO)'$)
READ(5,9902)ISM
9902 FORMAT(11)
SMOOTH=.FALSE.
IF(ISM.EQ.1)SMOOTH=.TRUE.
WRITE(5,991)ESC
991 FORMAT('+'1A1,'[H*)
!SEND CURSOR HOME

SHADE=.FALSE.
CALL INITGR( 5)
CALL CLRSCR
CALL CLRTXT
CALL SCOLOR('GRAYO' 0)
CALL SCOLOR('GRAY1'/1)
CALL SCOLOR('GRAY2',2)
CALL SCOLOR('GRAY3',3)
WRITE(5,991)ESC
CALL DPAPER('LIN',10,2,'LIN',10,2/'GRAY3')
IF (IMINEQ.0)GOTO1211
CALL LNAXIS('YL',YTEXT,YMIN,YMAX,IRFLAG)
CALL LNAXIS('XB',XTEXT,XMIN,XMAX,IRFLAG)
GOTO1212
1211 CALL LNAXIS('YL',YTEXT,,,IRFLAG)
CALL LNAXIS(1XB',XTEXT,,,IRFLAG)
WRITE(5,991)ESC
1212 CALL PDATA(N,X,Y,'L','GRAY2',IMARK,ILIN,SMOOTH,SHADE,0.0)
IF(IRGS.EQ.0)GOTO1234
DO 1277 11=1,N
Y1(I1)=B*X(11)+A
1277 CONTINUE
CALL PDATA(N,X,Y1, L' ,'GRAY3',0,1,.FALSE.,.FALSE.,0.0)
TYPE 121,ESC
WRITE (5,1278)B,A,R2,CEE
1278 FORMAT(/1X,' Y= ',G12.4,'*X + ',G12.4,' :R92 = ',G12.4,
+'STD ERR = ',G12.4)
1234 TYPE 121,ESC
121 FORMAT('+',1A1,'[H 0-QUIT; 1- REPLOT; 2-SCREENDUMP',$)
ACCEPT 122, IANS
122 FORMAT(12)
IF(IANS.EQ.1)GOTO 51
IF (IANS.NE2)GOTO 2550
WRITE(5,1221)ESC
1221 FORMAT('+',1A1,'[H',80X)
CALL CPYSCR
GOTO1234
2550 STOP 'END'
9911 WRITE(5,9912)
9912 FORMAT(' ERROR IN ENTRY PLEASE RE-ENTER')
GOTO 59
END
C
C
C
SUBROUTINE STRIP (TEXT,N)
BYTE TEXT(1)
DO 20 I=N,1,-1
IF (TEXT(I).EQ.32.OR.TEXT(I).EQ.0)GOTO20
TEXT(1+1)=0
RETURN
20 CONTINUE
RETURN
END

SUBROUTINE REGLIN(N,X,Y,A,B,R2,CEE)
BASED ON PROGRAM IN 'COMMON BASIC PROGRAMS'
LON POOLE AND MARY BORCHERS P. 145
DIMENSION X( 1),Y( 1)
REAL J,K,L,M
J=0.0
K=0 0
L=0.0
M=0 0
A=0.0
B=0.0
R2=0.0
CEE=00
DO 100 1=1,N
J=J+X(I)
K=K+Y( I)
L=L+X(I)*X(I)
M=M+Y(I)*Y(I)
R2=R2+X(I)*Y(I)
CONTINUE
XN=N
B=(XN*R2-K*J)/(XN*L-J*J)
A=(K-B*J)/XN
J=B*(R2-J*K/XN)
M=M-(K**2)/XN
K=MJ
R2=J/M
CEE=SQRT(K/(XN-2))
RETURN
END

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Alexander, J. F. 1978. Energy basis of disasters and the
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Anderson, D. J. 1986. Ecological sucession. In J. Kikkawa
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BIOGRAPHICAL SKETCH
John R. Richardson was born in Monett, Missouri, on
April 30, 1945. He attended school in Monett and graduated
from Monett High School in 1963. After four years at the
Missouri School of Mines and Metallurgy in Rolla, Missouri,
he graduated with a B.S. degree in chemistry. He attended
the University of Missouri in Columbia for two years study
ing biochemistry. In 1970 he began working at the Molecular
Virology Institute in St. Louis, Missouri.
In 1973 he started at the University of Florida and
graduated with a M.S. degree in environmental engineering
sciences. After working for several years with the Missouri
Department of Natural Resources, he returned to the Univer
sity of Florida to pursue a Ph. D. in 1973. He graduated
with a Doctor of Philosophy degree in 1988.
254

I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
//V / fyjLlWL-'
Howard T. Odum, Chairman
Graduate Research Professor of
Environmental Engineering
Sciences
X certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Assistant Research Professor
of Environmental Engineering
Sciences

t certify that X have read this study and that in ray
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Professor of Forest Resouces
and Conservation
I certify that I have read this study and that in ray
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Clay L. Montagi^
Assistant Professor of
Environmental Engineering
Sciences
This dissertation was submitted to the Graduate Faculty
of the College of Engineering and to the Graduate School and
was accepted as partial fulfillment of the requirements for
the degree of Doctor of Philosophy
April 1988
Dean, College of Engineering
Dean, Graduate School

^ | F | O RID A
3 1262 08554 8567^



250
Meinhardt, H. 1982. Models of biological pattern formation.
Wiley. New York. pp. 230.
Mueller-Dombois, D. 1980. The 'ohai'a dieback phenomenon
in the Hawaiian rain forest. In J. Cairns (ed). The
Recovery Process in Damaged Ecosystems. Ann Arbor
Science. Ann Arbor, pp. 153-161.
Nicolis, G., and I. Prigogine. 1977. Self-organization in
Non-equilibrium Systems. Wiley. New York. pp. 478.
Odum, E. P. 1969. The strategy of ecosystem development.
Science 164:262-270.
Odum, H. T. 1971. Environment, Power, and Society. John
Wiley. New York. pp. 331.
Odum, H. T. 1982. Pulsing, power and hierarchy. In W. J.
Mitsch, R. K. Ragade, R. W. Bosserman, and J. A. Dillon
Jr. (eds). Energetics and Systems. Ann Arbor Science
Publishers. Ann Arbor. pp. 33-59.
Odum, H. T. 1983a. Systems Ecology. Wiley. New York. pp.
644.
Odum, H. T. 1983b. Maximum power and efficiency: a rebuttal.
Ecological Modelling 20:71-82.
Odum, H. T. 1986. Enmergy in ecosystems. In N. Polunin
(ed). Ecosystem Theory and Application. John Wiley &
Sons Ltd. New York. pp. 337-369.
Odum, H. T., and E. C. Odum. 1981. Energy Basis for Man
and Nature. Second edition. McGraw-Hill. New York,
pp. 337.
Odum, H. T., and R. F. Pigeon. 1970. A Tropical Rain
Forest: A study of irradiation an ecology at El Verde,
Puerto Rico. NTIS. Springfield, Va.
Odum, H. T., and R. T. Pinkerton. 1955. Time's speed
regulator, the optimum efficiency for maximum output in
physical and biological systems. American Scientist
43:331-343.
Oliver, C. D. 1981. Forest development in North America
following major disturbances. Forest Ecology and
Management 3:153-168.
Ostwald, W. 1892. Lehrbadr der allgemeinen. Chemie 2:37.
Pacault, A. 1977. Chemical evolution far from equilibrium.
In H. Haken (ed) Synergetics-A Workshop. Springer-
Verlag. New York. pp. 133-154.


225
PULSING MODEL GIGI VERSION
MODIFIED TO WRITE DATA FILE FOR 3-D GRAPHICS PROGRAM
PULSEGGM.FTN VERSION
BASELINE MODEL 2.0 PREVIOUS TO AUG 29,1984
VERS 2.1 CHANGED FORMAT OF HARDCOPY PRINTOUT OF VARIABLES
VERS 2.11 ADDED TOTALS AND DESCRIPTION TO OUTPUT LIST
VERS 2.12 ADDED K8 TO OUTPUT LIST
VERS 2.121 (1/24/85) ADDED TRACKING COEFFICIENTS ON K2,K9,K11
AND CHANGED FORMAT 11003 FOR VARS(IVAL) FROM F6.1 TO G15.4
PROGRAM PULSE
BYTE XTEXT(80),DES(40),YTEXT(80)
DIMENSION VARS(20),ALPHA(20)
DIMENSION FILE(3)
REAL M1,M2,M3,M4,M9,K10,K11,K12,K13
REAL K1,K2,K3,K4,K5,K6,K7,K8,K9,J,JO,JNORM
EQUIVALENCE (VARS(1),K1),
(VARS(2),K2),
(VARS(3),K3),
(VARS(4),K4),
(VARS(5),K5),
(VARS(6),K6),
(VARS(7),K7),
(VARS(8),K8) ,
(VARS( 9) ,K9) ,
(VARS(10),K10) ,
(VARS(11),K11),
(VARS(12),K12),
(VARS(13),K13),
(VARS(14),XJ0INI),
(VARS(15),Q1IC),
(VARS(16),Q2IC),
(VARS(17),Q3IC),
(VARS(18),Q4IC)
VIRTUAL IY(6,25,180)
DATA FILE/3*' '/
DATA XTEXT/ ',P','u','1','s','e',' ,'M','o','d','e'
'1',' ',67*0/
DATA DES/40*0/
DATA YTEXT/80*0/
DATA ALPHA/K1 ','K2 ','K3 ','K4 ','K5 ','K6 ','K7 '
,'K8 ','K9 ','K10 ','K11 ,'K12 ','K13 ','JOIN','Q1IC'
, 'Q2IC 'Q3IC Q4IC ,2* '/
FT 1(A,3)=ABS(AINT(A/B)-A/B)
VERS=2.121 19/7/84 ; 1/24/85
WRITE(5,100)
FORMAT(' WHAT IS THE DATA FILE FOR THIS MODEL RUN ?')
READ(5,101)(FILE(I),1=1,3)
FORMAT(3A4)


43
Multi-run simulation of pulse model (Figure
14) with variation in quadratic pathway (K9
varied from 0.5E-6 to 0.53E-5 with K7 and K3
varied proportionately) 135
44 Multi-run simulation of pulse model (Figure
14) with variation in linear pathway (Kll
varied from 0.0 to 0.12E-2 and K5 and KS
varied proportionately) with quadratic
pathway held at zero 137
45 Simulation of pulse model with prey-predator
sectors (Figure 15) 139
46 Simulation of pulse model (Figures 14 and 18)
with cells in a linear ring and diffusion
between consumers of each cell in ring
(DK=.l). Initial conditions of consumers
were set to near zero except for one "seed"
consumer at lower left corner of matrix which
was set to 100 143
47 Simulation of pulse model (Figures 14 and 18)
with cells in a linear ring but without dif
fusion. Initial conditions of producers and
consumers were set to random distribution
around ring 146
48 Simulation of pulse model (Figures 14 and 18)
with cells in a linear ring and a high level
of diffusion between consumers of each cell
(DK=.1) and random distribution of producers
and consumers around ring 148
49 Simulation of pulse model (Figures 14 and 18)
with cells in a linear ring and a low level
of diffusion between consumers of each cell
(DK=.001) and random distribution of
producers and consumers around ring 150
50 Simulation of pulse model (Figure 14) with
cells arranged in two dimensions (Figure 18)
without diffusion and with a constant energy
source 153
51 Simulation of pulse model (Figure 14) with
cells arranged in two dimensions (Figure 18).
Energy source hierarchically is distributed
from center outward and no diffusion between
ce 11s
155


221
2690
2700
2710
2720
2730
2740
2750
2760
2770
2780
2790
2800
2810
2820
2830
3000
3001
3002
3003
3005
3006
3007
3008
3009
3010
3011
3020
3025
3030
3031
3040
3041
3050
3051
3060
3061
3070
3071
3080
3081
3090
3091
3100
3110
3111
3310
3320
3330
3340
3350
3360
3370
3380
YL=YIN
NUMPNT=NUMPNT+1:'100
XN= X(NUMPNT)
YN=Y(NUMPNT)
PERI1=PERI1+FNRDIST(XN,YN,XL,YL)
XL=XN
YL=YN
IF (NUMPNT = NPOINT)GOTO 2780
GOTO 2700
PERI2=FNRDIST (XIN,YIN,XL,YL):'300
PERI2=PERI2+PERI1
RETURN
' END
PRINT "SUBROUTINE DIGINI"' SUBROUTINE DIGINI
LCB=13
DEF FNRDIST(X1,Y1,X2,Y2)=ABS((X1-X2)2+(Y1-Y2)2).5
DEF FNANGLER(X1,Y1,X2,Y2)=ATN((Y2-Y1)/(X2-Xl))
+ (SGN(ABS(X2-X1))-SGN((X2-X1)))* 1.570796
PRINT "INITIALIZATION SEQUENCE FOR DIGITIZER":BEEP
PRINT" PLEASE MAKE SURE DIGITIZER IS ON"
INPUT HIT RETURN WHEN READY ",DUM$
DIM X(1000),Y(1000),XT(1000),YT(1000)
PENUP=3:PENDWN=2:PENDUM=6:BELL$=CHR$(7)
OPEN"COM1:9600,E,7,2,RS,CS,DS,CD" AS #1:'OPEN AUX PORT FOR I/O
CLS
LD$="#]":QT$="/"
PRINT #1,"#]L":PRINT DIGITIZER BEING RESET ":GOSUB 6005
PRINT #1,"#](":'SET RESOLUTION TO .001
GOSUB 6001
PRINT #1,"#]6":'SET RATE TO 2/SEC
GOSUB 6001
PRINT # 1, "#]'02":'SET INCREMENT TO .01
GOSUB 6001
PRINT #1,"#]M":' TURN ON INCREMENTAL MODE
'GOSUB 6001
PRINT #1,"#]J":1 TURN ON STREAM MODE
GOSUB 6001
PRINT # 1,"#]9":' SET SERIAL TAG AS LAST CHARACTER
GOSUB 6001
PRINT #1 ,"#]>": SET NO FIELD DELIMITERS
GOSUB 6001
PRINT #1,"#]I":' RESET FOR POINT MODE FOR BEGINNING SETUP
PRINT #1,"#]/":' SEND END OF REMOTE FORMATTING
GOSUB 6001
XSCALE=1!:'USER X AXIS SCALE FACTOR
XOFF=0!:' USER X AXIS OFFSET
YSCALE=1!:'USER Y AXIS SCALE FACTOR
YOFF=0!: 'USER Y AXIS OFFSET
ANGLE=0!:' USER SKEW CORRECTION FACTOR
' XROUND=0.
' YROUND=0.
UXSCAL=1!:'USER PLOTTER X SCALE FACTOR




Number per Class Number per class
173
a Size Distribution of Cecropia Gaps
0.02.02.040-4 .0606 .0808 .1 01 0.1 21 2.141 4.1 61 6.1 8 >.18
Size Class (sq. inches)
/I Combined data
b Size Distribution of Cecropia Gaps
from Aerial Photographs
Size Class (sq. inches)
pic2 yz7A pics
PIC8
CT71 PIC1


96
whatever power may be left over (particularly the linear
path) .
Frequency Studies
The basic three path model (Figure 10) was also used to
test the effects of different frequencies of input power on
the model at three different power levels. At the lowest
power level (J0=500, Figure 25) the differences between
pathways in percent power used was the greatest. The great
est frequency response occurred at low frequencies. The
frequency response was flat with only the linear path en
abled. When all pathways were present, the percent power
used was highest with a peak at approximately 2 cycles. A
peak of power utilization also occurred with the combin
ations of J1+J2 and J1+J3. The pathways that showed a min
imum in the frequency response were composed of J2+J3 (the
two nonlinear pathways combined) and J2. The quadratic
pathway alone did nothing since no power was used (compare
with Figure 21).
When the input power was increased to 2000 (Figure 26),
the linear pathway showed no change in output with change in
frequency and the quadratic pathway had no output. The
combination of J1+J2 here again had a slight maximum at
about 2 cycles while J2 alone had a maximum at zero cycles.
The combination of all of the pathways (J1+J2+J3) and J1+J3
had a slight minimum of power utilization at about 8 cycles,
while the combination of J2+J3 showed a slight minimum at
about 2 cycles.


Figure 7. Mite predator prey experiment (HufEaker 1958)
(a) Spatial distribution.
Prey concentration is shown by intensity of small
blocks (darker is higher density) and predator
locations are marked with small circles.
(b) Time series of total predators and prey in spatial
area. Letters on graph refer to the time series
for the spatial display next to the letter.


158


144
ducers and consumers was constant and the same as the homo
geneous initial conditions) and diffusion set to zero
(Figure 47). The output was based entirely on the random
field from the initial conditions. Each individual cell
model was producing and consuming at the same rate but there
was no spatial synchronization of the cells. The pattern
repeated itself over time (compare T=50, 47A with T=700,
47N) .
When diffusion was set at a high level (0.1) between
the consumers, with the same random initial distribution of
producers and consumers, the resulting pattern was quite
different (Figure 48). The pulsing consumers moved in a
wave around the ring followed by a wave of production (T=50,
100, 150, 200, 250, 300; Figure 48A-F) followed by another
wave of consumption beginning just prior to T=350 (48G).
This was similar to the simulation in Figure 46 that began
with a homogeneous initial distribution of producers and
consumers and had waves of consumption and production around
the ring.
When the model was run with random distribution of
producers and consumers (Figure 49) and a low value of
diffusion (DK=0.001), the spatial pattern that developed had
some properties of both of the two previous runs. Because
speed of movement was less with a lower value of diffusion,
a number of focal points for pulse waves were generated
which then run into each other and stop. The production
follows the pattern of consumption with multiple foci.


Figure 1. Spatial patterns based on chemical reaction
mechanisms.
(a) Spatial patterns generated by 3elousev-Zhabotinski
chemical reaction (Prigogine 1980).
(b) Spatial patterns generated by simulation model
used to describe morphogenesis (Meinhard (1932).


in
r-


234
C
SET UP SURFACE OF FORCING ENERGY
DATA TEXT/80*0/
DATA E 70,0,0,0,0,0,0,0,0,0,0,0,
+ 0,0.8,0.8,0.8,0.8,0.8,0.8,0.8,0.8,0.8,0.8,0,
+ 0,0.8,1,1,1,1,1,1,1,1,0.8,0,
+ 0,0.8,1,1,1,1,1,1,1,1,0.8,0,
+ 0,0.8,1,1,1.2,1.2,1.2,1.2,1,1,0.8,0,
+ 0,0.8,1,1,1.2,1.2,1.2,1.2,1,1,0.8,0,
+ 0,0.8,1,1,1.4,1.4,1.4,1.4,1,1,0.8,0,
+ 0,0.8,1,1,1.2,1.2,1.2,1.2,1,1,0.8,0,
+ 0,0.8,1,1,1,1,1,1,1,1,0.8,0,
+ 0,0.8,1,1,1,1,1,1,1,1 ,0.8,0,
+ 0,0.8,0.8,0.8,0.8,0.8,0.8,0.8,0.8,0.8,0.8,0,
+ 0,0,0,0,0,0,0,0,0,0,0,07
C
X1(1)=5
Y1(1)=5
Q3(1)=Q3IC
ICON(5,5)=1
IXYZ(1)=IXY(5,5)
!SET PREDATOR CLOSE TO CENTER
!FIRST PREDATOR LOCATION
!CODED LOCATION
K1=.417E-5
K2= .5
K3=.05
K4=.45
K5=.5E-4
K6=.45E-3
K7=.2E-6
K8=.18E-5
K9=.2E-5
K10=.417E-5
Kl1=.5E-3
K12=.05
K13=.7833E-6
DO 200 1=2,11
DO 200 J=2,11
Q1(I,J)=Q1IC
Q2(I,J)=Q2IC
Q4(I,J)=Q4IC
E(I,J)=E(I,J)*100.
200 CONTINUE
DO 201 IK1,12 DK
Q4(1,IK)=Q40IC SDK
Q4(12,IK)=Q40IC SDK
Q4(IX,1)=Q40IC SDK
Q4(IK,12)=Q40IC SDK
201 CONTINUE !DK
C
C
C
CHANGE ENERGY LEVEL?
IF(IETYP.EQ.1)G0T0221
DO 220 1=2,11
DO 220 J=2,11
E(I,J) = (XMEAN*RAN(-1) + .5)* 100
SCALE RANDOM FUNCTION


248
Brokaw, N. V. L. 1985a. Treefalls, regrowth, and community
structure in tropical forests. In S. T. A. Pickett and
P. S. White (eds). The Ecology of Natural Disturbances
and Patch Dynamics. Academic Press. Orlando, pp. 53-
69.
Brown, M. T. 1980. Energy basis for hierarchies in urban
and regional landscapes. Ph.D. dissertation.
University of Florida. Gainesville, Fla. pp. 360.
Brown, S., A. E. Lugo, S. Silander, and L. Liegel. 1983.
Research History and Opportunities in the Luquillo
Experimental Forest. General Technical Report SO-44.
Southern Forest Experiment Station. U. S. Dept, of
Agriculture. New Orleans. pp. 128.
Calow, P. 1978. Life Cycles. Chapman and Hall. London,
pp. 164.
Costanza, R. 1979. Embodied energy for economic-ecologic
systems. Ph.D. dissertation. University of Florida.
Gainesville, Fla. pp. 254.
Crow, T. R. 1980. A rainforest chronicle: A 30-year record
of change in structure and composition at El Verde,
Puerto Rico. Biotropica 12:42-55.
Doyle, T. W. 1982. A description of FORICO, a tropical gap
dynamics model of the lower montane rain forest of
Puerto Rico. ORNL/TM-8102. Oak Ridge National
Laboratory. Oak Ridge, Tenn. pp. 47.
Doyle, T. W., H. H. Shugart, and D. C. West. 1982. FORICO:
gap dynamics model of the lower montane rain forest in
Puerto Rico. ORNL/TM-8115. Oak Ridge National
Laboratory. Oak Ridge, Tenn. pp. 57.
Emanuel, W. R., D. C. West, and H. H. Shugart. 1978.
Spectral analysis of forest model time series.
Ecological Modelling 4:313-323.
Field, R. J. 1985. Chemical organization in time and space.
American Scientist 73:142-150.
Field, R. J., and R. M. Noyes. 1974. Oscillations in
chemical systems. V. Quantitative explanation of band
migration in the Belousov-Zhabotinskii reaction. J. Am.
Chem. Soc. 96:2001-2006.
Gardner, M. 1970. The fantastic combinations of John
Conway's new solitaire game 'Life'. Scientific
American 223:120-123.


0009
UJ!i
0
0009


PERCE NI
62
EDGE EFFECT
Prim*tr and Cntr
PERIMETER -* CENTER



109
Tim
Time
100


251
Pearlstine, L., H. McKellar, and W. Kitchens. 1985.
Modelling the impacts of a river diversion on
bottomland forest communities in the Santee River
floodplain, South Carolina. Ecological Modelling 29:
283-302.
Peet, R. K. 1981. Changes in biomass and production during
secondary forest succession. In D. C. West, H. H.
Shugart, and D. B. Botkin (eds) Forest Succession
Concepts and Application. Springer-Verlag. New York,
pp. 324-338.
Peet, R. K., and N. L. Christensen. 1980. Succession: a
population process. Vegetatio 43:131-140.
Phipps, R. L. 1979. Simulations of wetlands forest
vegetation dynamics. Ecological Modelling 7:257-288.
Pickett, S. T. A., and P. S. White. 1985. The Ecology of
Natural Disturbance and Patch Dynamics. Academic Press.
Orlando, pp. 472.
Platt, T., and K. L. Denman. 1975. Spectral analysis in
ecology. Annual Review Ecol. Syst. 6:189-210.
Poundstone, W. 1985. The Recursive Universe. William
Morrow and Company. New York. pp. 252.
Prigogine, I. 1978. Time, structure, and fluctuations.
Science 201:777-785.
Prigogine, I. 1980. From Being to Becoming. W. H. Freeman.
San Francisco, pp. 272.
Prigogine, I. 1982. Order out of chaos. In W. J. Mitsch,
R. K. Ragade, R. W. Bosserman, and J. A. Dillon Jr.
(eds) Energetics and Systems. Ann Arbor Science
Publishers. Ann Arbor. pp. 13-32.
Prigogine, I., and I. Stengers. 1984. Order Out of Chaos.
Bantam Books, Inc. New York. pp. 349.
Prigogine, I., and J. M. Wiaume. 1946. Biology et
thermodynamique des phenomenes irreversibles.
Experientia 2:451-453.
Ransom, R. J. 1981. Computers and embryos: models in
develpomental biology. Wiley. New York. pp. 212.


Number Number
177
Size Class Distribution
Size Class Distribution
Random Energy Input
0.0.2.02.0404.0606.0808.1 01 O. 1 21 21 41 4. 1 61 6.1 8 >.18
Class Size (Percent of simulation area)


Figure
LIST OF FIGURES
Page
1 Spatial patterns based on chemical reaction
mechanisms 6
2 Hilborn (1979) model 8
3 The spatial development of cells based on
simple r-pentamino initial condition 12
4 Basic autocatalytic model with flow-limited
energy source 17
5 Basic multiple path model with three input
pathways representing differing feedback
regimes, linear, autocatalytic, and quadratic... 20
6 Size class distribution of gaps formed in
tropical forest at Barro Colorado (Brokaw
1982) 26
7 Mite predator prey experiment (Huffaker
1958) 30
8 Size class distribution over time of plot of
trees in tropical forest at El Verde (Crow
1980) 36
9 Energy circuit language symbols (Odum 1983) 39
10 Three pathway model used to test effects of
various energy inputs on kinetic mechanisms 43
11 Three pathway model with multiple drain path
ways 47
12 Three pathway model with individual competing
units having single input pathways similar to
combined model 49
13 Parallel production-consumption model 52
14 Pulse model of tropical forest ecosystem
model 55
15 Pulse model with additional prey-predator
sector 58
16 Number of edge and center cells as a function
of total number of cells in a given square
area 62
vi


13
the possible mechanisms and energy consequences of pattern
formation in ecosystems.
Concepts of Pulsing, Patterns and Power
Maximum Power in Systems
Although in the last century Podalinsky, Ostwald and
Boltzman suggested energy use controlled system performance
(Martinez-Alier 1987), Lotka (1922) made a more definitive
statement. He stated that evolution proceeded in such a
direction as to make the total energy flux through the
system a maximum compatible with the constraints on the
system. He related this to Ostwald's (1892) idea of all
possible energy transformations, that one takes place which
brings about the maximum transformation in a given time.
A theory of minimum entropy generation was put forth by
Prigogine (Prigogine and Wiaume 1946) that a system evolved
toward a stationary state characterized by the minimum
entropy production compatible with the constraints on the
system. He has since called this a failure and probably a
special case of systems near equilibrium (Prigogine 1984).
Prigogine (1978, 1980, 1982; Prigogine and Stengers 1984)
now deals with systems far from equilibrium that have
dynamic and oscillatory behavior. He has not postulated any
definite theory about the energetic consequences of these
types of systems.
Odum and Pinkerton (1955) proposed that natural sys
tems tend to operate at that efficiency which produces a


Figure 26. Simulation of the three path model in Figure
10. Percent power used as a function of frequency of the
input power (J0=2000).
Linear pathway:
Autocatalytic pathway:
Quadratic pathway:
J1=K1*R
J2=K2*Q*R
J3=K3*Q*Q*R


33
systems with different production and consumption kinetics
and different combinations of components.
Second, a generalize production-consumption minimodel
calibrated with tropical rainforest data was used to study
the energetics of pulsing behavior.
Third, spatial pattern formation was investigated using
the pulsing production-consumption model as subunits in a
spatially distributed format. The spatial effects and
energy implications of various patterns of energy inputs,
edges, and lateral connectivity were determined.
These spatial simulations included several types of
inter-block exchange. Hierarchical relationships are rep
resented in these models when each consumer component inter
acts with more than one producer unit. The distribution of
gaps developed by simulations was compared with gaps in the
tropical rainforest in Puerto Rico.
Finally, insights and hypotheses were developed about
behavior of ecological systems.
Data site: Luquillo Rainforest, Puerto Rico
Data from the Lower Montane Rainforest in the Luquillo
Mountains of Puerto Rico were used to compare some of the
spatial simulations of pulsing and patches. Extensive
studies on this forest were published previously (Odum and
Pigeon, 1970).
Changes in structure and composition of a plot of
tropical rain forest near El Verde in Puerto Rico over a
period of 30 years were reported by Crow (1980). Data


164
with a hierarchical energy input and a consumer search
length of 1 cell (Figure 55) and a search length set to 5
cells (Figure 56). The simulations are quite different in
both the spatial patterns generated and in the time series
graph of the simulation.
When limited to a search length of 1 cell, the con
sumption pattern moved like a wave from left to right across
the producers after starting in the center. With a longer
search length (Figure 56), the consumption began in the
center and spread out in a circular pattern over the pro
ducers. There are two of these waves of consumption during
the time of the simulation for the search length of 5. The
run with a search length of 1 cell has slower consumption
and only moves across the field once.
Rain Forest Gaps and Hierarchies
Size Class Distributions
Three different size class distributions (Figure 57)
were generated from the data set from the radiation site at
El Verde to characterize the hierarchical patterns in the
vegetation. Figure 57a represents the distribution of
plants by diameter. This can be compared to the data from
Crow (1980) in Figure 8. The distribution of plants by
crown diameter (Figure 57b) and by height (Figure 57c) was
hierarchical. The sampling technique affected the results
in the lowest size classes.


TABLE OF CONTENTS
ACKNOWLEDGEMENTS iii
LIST OF FIGURES vi
ABSTRACT xi
CHAPTER 1 INTRODUCTION 1
Historical Perspective 2
Previous Models of Pulsing Patterns in Time and Space.2
Pattern Formation 3
Concepts of Pulsing, Patterns and Power 13
Maximum Power in Systems 13
Design for Maximum Power 14
Pathway Configuration 15
Pulsing and Patterns in Ecosystems 21
Succession and Disturbance 21
Edges 23
Hierarchies and Patches 24
Mode Is 25
Gap Models and Patch Dynamics 27
Spatial Systems and Models 28
Plan of Study 32
Objectives 32
Data Site: Luquillo Rain Forest, Puerto Rico 33
CHAPTER 2 METHODS and MODELS 37
Simulation Procedures and Programs 40
Simulation Models 41
Minimodel Tests 41
Pulse Model 53
Pulse Model with Prey-Predator Sectors 56
Spatial Model 59
Format for Spatial Graphs 67
Measurement of Hierarchies at El Verde Site 67
CHAPTER 3 RESULTS 71
Simulation of Three Path Model 71
Individual Pathway Tests 71
Frequency Studies 86
Simulation of Parallel Production-Consumption Model 91
Single Run Simulations 91
Multiple Run Simulations 100
i v


249
Garwood, N. C., D. P. Janos, and N. V. L. Brokaw. 1979.
Earthquake-caused landslides: A major disturbance to
tropical forests. Science 205:997-999.
Godel, K. 1931. Uber formal unentscheidbare Satze der
Principia Mathematica und verwandter System I.
Monatshefte fur Mathematik und Physik 38:173-198.
Haken, H. 1977a. Synergetics, a workshop. Springer-Verlag.
Berlin, pp. 267.
Haken, H. 1977b. Synergetics, an introduction. Springer-
Verlag. Berlin. pp. 355.
Haken, H. 1979. Synergetics and bifurcation theory.
Annals New York Academy of Sciences 316:357-375.
Hilborn, R. 1979. Some long term dynamics of predator-prey
models with diffusion. Ecological Modelling 6:23-30.
Horn, H. S. 1976. Succession. In R. M. May (ed).
Theoretical Ecology Principles and Applications. W. B.
Saunders. Philadelphia, pp. 187-204.
Huffaker, C. B. 1958. Experimental studies on predation:
dispersion factors and predator-prey oscillations.
Hilgardia 27:343-383.
Jantsch, E. 1980. The Self-Organizing Universe. Pergamon
Press. Oxford, pp. 343.
Jordan, C. F., and G. E. Drewry. 1969. Secondary
succession in the irradiated area. The Rain Forest
Project Annual Report. PRNC-129 :65-86.
Runo, E. 1987. Principles of predator-prey interaction in
theoretical, experimental, and natural population
systems. Advances in Ecological Research 16:249-337.
Leigh Jr, E. G., A. S. Rand, and D. W. Windsor. 1982. The
Ecology of a Tropical Forest. Smithsonian Institution
Press. Washington, D.C. pp. 468.
Lotka, A. J. 1920. Undampened oscillations derived from
the law of mass action. J. Amer. Chem. Soc. 42: 1595-
1598.
Lotka, A. J. 1922. Contribution to the energetics of
evolution. Proc. N. A. S. 8:147-151.
Martinez-Alier, J. 1987. Ecological Economics. Blackwell.
New York.


Figure 27. Simulation of the three path model in Figure
10. Percent power used as a function of frequency of the
input power (J0=10000).
Linear pathway:
Autocatalytic pathway:
Quadratic pathway:
J1=K1*R
J2=K2*Q*R
J3=K3*Q*Q*R


Figure 34. Simulation of
consumption model in Figu
for entire run as a funct
consumer (Q4). This repr
33e.
the parallel
re 13. Total
ion of the in
esents a cros
production-
percent power
itial value of
s section of F
used
the
igure


Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
SPATIAL PATTERNS AND MAXIMUM POWER IN ECOSYSTEMS
by
John R. Richardson
April, 1988
Chairman: Howard T. Odum
Major Department: Environmental Engineering Sciences
Studies of dynamic systems have shown that oscillations
in time and space are related, both being generated by non
linear, pulsing behavior that is derived from the mathema
tics of energy processing. Similar mathematics exist in
chaos theory, bifurcation theory, and catastrophe theory.
Production-consumption models that simulate pulsing proper
ties of ecological systems are of this class. This dis
sertation examines the spatial patterns and energetics of
autocatalytic and pulsing models as a paradigm for ecolog
ical and general systems. Configurations were tested with
steady or varying resource availability for ability of the
model systems to maximize power as the criterion for utility
and success. The spatial distribution of gaps generated by
simulations was compared to that observed in rain forests.
Models studied included (a) aggregated, single
compartment autocatalytic designs; (b) parallel production-
consumption design; (c) production-consumption-recycle
xi


191
kinetics of these pathway configurations are similar to
others studied by chaos theory, bifurcation theory and
catatastrophe theory.
2) Hierarchical structure is expressed in kinetics as
increasing turnover times with increasing territory. Path
ways of control of production-consumption systems must match
the turnover time of the appropriate hierarchical level in
order to cause reinforcement.
3) In early successional systems there may be critical
minimum stocks of producers and consumers for a system to
grow.
4) Similar maximum power processing may be achieved by
a wide variety of spatial patterns.
5) Connectivity in systems has a greater role in
pattern formation at higher levels of the hierarchy.
Control of patterns and patchiness through consumer control
is highly dependent on the spatial connectivity of the
consumers.
6) Patch size m
the consumer and the
7) Some of the
simplified for human
be grouped according
ay be related to the turnover time of
spatial connectivity of the consumers,
great complexity of ecosystems may be
comprehension if varied mechanisms can
to the basic kinetics, energetics and
hierarchical roles they perform.


NUMBER OF PREDATORS
8
SPATIAL LOCATION


10
Figure 3 is an example of the patterns generated from a
simple five cell seed (R-pentomino) during 512 iterations.
This pattern stabilizes (no more deaths and no more births)
after 1103 iterations, although it is an oscillating steady
state. Individual subsets of the final stable pattern
oscillate.
The 'game of life' model has some of the features of
autocatalysis (or cooperative behavior). Two or three live
cells are required for survival or birth of new cells. It
also has the feature of diffusive inhibition because indivi
dual cells that move out from a population center can become
isolated and die. This rule-based system has no energy
constraint that governs development and thus gives no energy
basis for pattern formation.
The common theme that runs through these examples is
one of combined interactions of autocatalytic growth with
some form of inhibition, diffusion or other mechanism for
preventing the autocatalytic growth from spreading too
rapidly. A concept that is sometimes misunderstood or mis
interpreted is that the terms fluctuation (Prigogine 1980,
1984) and bifurcation theory (Pacault 1977) refer to a
change in the kinetics of reacting components of a system.
This change in kinetics gives rise to the oscillations or
pulses in the output.
The models in this dissertation also use combinations
of autocatalytic and diffusion (linear) pathways to study


Figure 47. Simulation of pulse
with cells in a linear ring but
time unit (e.g. A=0) density of
matrix is shown along with size
time series from A to 0 summari
totals in matrix.Initial condit
sumers were set to random distr
model (Figures 14 and 18)
without diffusion. For each
producer and consumer in the
class distribution. The
zes the temporal pattern of
ions of producers and con-
ibution around ring.
I
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12000
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10000
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6000
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900
260
200 9/a
160
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163


Figure 11. Three pathway
ways. Used to test effects
threepath model.
Linear input:
Autocatalytic input:
Quadratic input:
Linear drain:
Autocata lytic drain:
Quadratic drain:
multiple drain path-
order drain pathways on
J1=K1*JR
J2=K2*Q*JR
J3=K3*Q*Q*JR
J4=k4*Q
J5=K5*Q*Q
J6=K5*Q*Q*Q
model with
of higher
dQ=J1+J2+J3-J4-J5-J6
JR=J0-J1-K2'*J2-K3'*J3


i


184
with this structure may be at a disadvantage and could be
eliminated through consumption by a higher level of the
hierarchy or competition by other systems at the same level
of the hierarchy.
If a system was not materials conservative (feedbacks
from the consumer to the nutrient storage cut off or di
verted, Figure 39) then the system ceased pulsing and ran
down. The system had no feedback pathways and so did not
capture all of the available energy.
Implications for Succession
Role of Individual Units
Early successional producers can be thought of as pre
paring the way for succession to occur. Although early
successional species may have other roles, in the parallel
production-consumption model (Figure 13) they can be seen as
providing an energy source to the consumer level of the
model as the rest of the system builds up. When the early
successional species was at a low level, the consumer level
(Q4) remained low (Figure 35). This low consumer level did
not feed back enough to the producers to stabilize the
system and the system crashed. As the early successional
species (Q3) reached a threshold initial condition, suf
ficient structure was built and the system progressed to a
steady state.


185
If the consumer level in a successional system is too
low then the system may not be stable. In the parallel
production-consumption model, the consumer provided a feed
back on the producers through the input production multi
plier and through consumption on the producers. When the
consumer was at a level that was too low, succession as
depicted by the model (Figure 33 and 36) did not begin. At
some initial threshold level of consumers, the model pro
ceeded through a successional sequence.
In developing management plans for revegetating sites
disturbed by mining, intensive agriculture or natural dis
turbances, it is imperative that careful attention be paid
to the whole structure of the ecosystem that is being re
built. Without the proper mix of early, middle, and late
successional producers along with a set of consumers that
match the producers, the restablishment of a natural suc
cessional sequence may be retarded or destroyed.
Succession and Pulsing
The role of pulsing in succession may be that in some
systems it is necessary to have the pulsed recycle to main
tain energy flows near maximum levels. Several cases of the
pulsing model (Figures 38 and 39) showed that when recycling
was disturbed power use dropped. Certain types of sucession
may need an alternation of production and consumption at a
frequency that allows the maximum use of available energy.
Systems in which available nutrients become bound in the


52
Simulation of pulse model (Figure 14) with
cells arranged in two dimensions (Figure 18).
Energy source is hierarchically distributed
from center outward and diffusion is between
consumers of each cell (DK=.001) 158
53 Simulation of pulse model (Figure 14) with
cells arranged in two dimensions (Figure 18).
Energy source is randomly distributed and
diffusion is between consumers of each cell
(DK= .001) 160
54 Simulation of pulse model (Figure 14) with
cells arranged in two dimensions (Figure 18).
Energy source is randomly distributed and
diffusion between nutrient storages (Q4) of
each cell is set to high level (DK=.l) 163
55 Simulation of the pulse model (Figure 14)
with cells arranged in two dimensions (Figure
18). Moving consumer model with search length
set to one cell, no diffusion and hierarchi
cal energy distribution 166
56 Simulation of the pulse model (Figure 14)
with cells arranged in two dimensions (Figure
18). Moving consumer model with search length
set to five cells, no diffusion and
hierarchial energy distribution 163
57 Size class distribution of trees at El Verde
radiation site (November 1964) 170
58 Size distribution of Cecropia gaps in tropi
cal rainforest at El Verde 173
59 Size distribution of gaps in tropical rain
forest pulsing model simulation (Figures 14
and 18) at time =760 177
60 Character set for displaying spatial graphs
on GIGX computer terminal for use with
screencopy to printer 196
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biomass may benefit by the fast release from a pulse of
consumption and recycle.
186
Spatial Pattern Formation
Synchronous vs. Asynchronous Systems
When a spatially organized system is totally synchron
ized (all subunits behaving as one) the system may be like
a monoculture with little pattern formation other than that
of the local source inputs. In this state, pattern diver
sity is low. Where cells are not all synchronized with each
other, patterns can develop that are dependent on the
asynchronous nature of the individual subunits as well as
the local energy sources.
When the spatial model was simulated with all of the
individual cells uncoupled (not linked through intercell
diffusion processes) and totally synchronized (all cells
begun with the same initial conditions and an even energy
distribution), no pattern was generated (Figure 50). The
level of producers and consumers was the same in each cell
at every point in time.
Any variation in the energy input over the matrix area
lead to individual cells pulsing at frequencies depending on
the energy level local to that area (Figure 51). Although
the pattern was quite different from the synchronized one,
the energy use is the same (Table 5 in Appendix).


Figure 15. Pulse model with additional prey-predator
sector.
Individual rate equations:
Rate equations for state variables
R1 =
K1*Q1*Q4*JR
dQl = R1 R2
R2 =
K2*Q1
dQ2 = R3 R9 Rll
R3 =
K3*Q1
dQ3 = R7 + R5 R12 R18 -
R14
R4 =
K4*Q1
dQ4 = R4 + R6 + R12 + R12 -
RIO
R5 =
K5*Q2
+ R16 + R17 + R20
R6 =
K6*Q2
dQ5 = R15 R17 + R19
R7 =
K7*Q2*Q3*Q3
JR = J0/(1 + K13*Q1*Q4)
R8 =
K8*Q2*Q3*Q3
R9 =
K9*Q2*Q3*Q3
RIO
= K10*Q1*Q4*JR
Rll
= K11*Q2
R12
= K12*Q3
R13
= K13*Q1*Q4*JR
R14
= K14*Q3*Q5*Q5
R15
= K15*Q3*Q5*Q5
R16
= K16*Q3*Q5*Q5
R17
= K17*Q5
R18
= K18*Q3
R19
= K19*Q3
R20
= K20*Q3


190
continuities. These small differences act as information
storage for future pattern development.
Power Use and Edge Effects
No system exists in an infinite plane without edges.
Edges were manipulated in the spatial models to understand
their role in pattern formation. Some of the simulations
allowed consumers to diffuse into or out of the spatial
matrix at high and low levels of diffusion.
When the consumer level on the outside ring was kept at
a low value (0.0), the percent power used decreased (Table 8
in Appendix) with increasing rates of diffusion. If the
outside buffer had a high value for the consumer (Q3 equal
100) then just the reverse was seen. With increasing rates
of diffusion there was an increase in the percent power
used. This implies that consumer exchange can act as an
energy source or a drain in a system depending on the
relationship of the system to its surrounding area through
its edges.
General Principles
The following are some general principles suggested by
model studies, which may be useful hypotheses in future
experimental studies.
1) Multiple pathways increase efficiencies and enable
better use of fluctuating energy sources. Multiple steady
states can result from one basic configuration. The


150


Figure 55. Simulation of the pulse model (Figure 14) with
cells arranged in two dimensions (Figure 18). Moving
consumer model with search length set to one cell, no
diffusion and hierarchical energy distribution. For each
time unit (e.g. A=0) density of producer and consumer in the
matrix is shown along with size class distribution. The
time series from A to 0 summarizes the temporal pattern of
totals in matrix.
4000
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208
C
C
C GIGI GRAPHICS SUBROUTINE PACKAGE
C WRITTEN BY JOHN R. RICHARDSON
C
C SEPTEMBER 1982
C
C VERS 11: ALL UPDATES AND CURRENT TO SEPTEMBER 1982
C
C VERS 12: FEBRUARY 28 1984 ADDITIONS
C ADDED GGPLOT (CALL TO GGPLT)
C ADDED GGDMP (HARDCOPY DUMP)
C ADDED GGVERS
ALL I/O IS TO LOGICAL UNIT 3
C
(2******************************** ******************************* ********
C
C
C
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
THE NORMAL CALLING SEQUENCE TO SET UP THE GIGI WOULD BE C
AS FOLLOWS: C
C
CALL GGON ¡TURNS GRAPHICS ON C
CALL GGINIT ¡SENDS NORMAL INITIALIZATION C
CALL GGERA ¡ERASE THE SCREEN C
CALL GGAXIS(0,0,767,479) ¡SETS NORMAL AXIS WITH ORIGIN C
! AT THE LOWER LEFT CORNER C
***********************************************************************
TASKBUILDING USING THE GIGI ROUTINES C
RUN THE TASK BUILDER (TKB ) C
TKB>MYPROG=MYPROG,LB:[1,1]GGLIB/LB C
TKB>/
ENTER OPTIONS
TKB>ASG=TTn:3 ¡WHERE n EQUALS GIGI TERMINAL NUMBER C
TKB>// I COULD USE TI: INSTEAD C
***********************************************************************
SUBROUTINE GGVERS(IVERS)
C CALL TO THIS WILL GIVE THE CURRENT VERSION OF THE GGLIB
IVERS=12
RETURN
END
C
C
C
SUBROUTINE GGON
C THIS WILL SEND THE ESC Pp SEQUENCE TO THE GIGI TO ENABLE THE
C GRAPHICS
BYTE ESC
ESC=27
WRITE(3/100)ESC
100 FORMAT('+*,1A1,'Pp')
RETURN
END
O O


Figura 19. Steady state power utilization oE units in the
three path model (Figure 10) as a function of input power
(JO) .


205
1061 FORMAT(* END OF RUN # ,14)
1062 CONTINUE
C ESC=27
C IF (ICOPY.EQ.O)GOT01102
C WRITE(3/11021)
C11021 FORMAT(' S(H)')
C CALL GGERA
C1102 CALL GGOFF
C WRITE(5/1009)ESC,ESC/ESC
C1009 FORMAT( 1X,A1 ,'Prtml' ,A1 <' ,A1 ,' [H'//
C & INITIAL VALUES OF VARIABLES')
C WRITE(5,114)
C WRITE(5/112)Q1INIT,Q2INIT,Q3INIT,Q4INIT,0, 0
C WRITE(5,111)
C 111 FORMAT(' MAXIMUM VALUES OF VARIABLES ')
C WRITE(5,114)
C 114 FORMAT(6X,'Q1',7X,'Q2',7X,'Q3',7X,'Q4', 6X,'PROD',5X,'BIOMASS')
C WRITE(5#112)M1,M2,M3,M4,M9/DT,BMAX
C 112 FORMAT(' ',6F9.3)
C WRITE( 5,113)
C 113 FORMAT(' FINAL VALUES OF VARIABLES')
C WRITE( 5,114)
C WRITE(5,112)Q1,Q2,Q3,Q4,Q9/DT,BIOMSS
C WRITE( 5, 115)PAVAIL,PAUSED, ( PUSED/PAVAIL) *100 ,DUSED
C 115 FORMAT(
C &1X,' ENERGY AVAILABLE = ',F12.4/
C &1X,' TOTAL PRODUCTIVITY = ',F12.4/
C &1X,' ENERGY USED = ',F12.4,' PERCENT USED = ',F12.4/
C &1X,' ENERGY DRAINED = ',F12.4)
C WRITE(5,116)
C 116 FORMAT( 6X,' K0',8X, K1',8X,'K2',8X,'K3',8X,'K7',8X #'K8',8X,
C +'K9',8X,'F1')
C WRITE( 5,117) K0 K1,K2,K3,K7,K8,K9,F1
C 117 FORMAT(' ',8(2X,F8.4))
C WRITE(5,118)
C 118 FORMAT(6X,'D1',8X,'D2', 8X,'D3',8X,'D4',8X, L1', 8X,'L2',8X,
C +'L3',8X,'JO')
C WRITE(5,117)D1,D2,D3/D4,L1,L2,L3,JO
C WRITE(5,119)DT,TIME
C 119 FORMAT(' DT THIS RUN = ',F6.4,' TOTAL T= ',F6.2)
C WRITE(5,120)(FILE(I),1=1,16)
C 120 FORMAT(' DATA FILE DESIGNATION FOR THIS RUN 'r16A1)
C WRITE(5,121)Q1SIZE,Q2SIZE,Q3SIZE,Q4SIZE
C121 FORMAT(3X,'Q1SIZE Q2SIZE Q3SIZE Q4SIZE'/1X,4F71)
C IF(ICOPY.EQ.O)GOTO1201
C CALL GGON
C WRITE(3/11021)
C CALL GGOFF
1201 END


224
5365 YU=Y3U-Y1U
5370 ANGLE=ANG1
5380 IF(XU O 01)THEN XSCALE=XU/XD
5390 IF(YU <> 0)THEN YSCALE=YU/YD
5400 IF(X1U = 0! AND Y1U = 0!)GOTO 5510
5402 X1U=X1U/XSCALE:Y1U=Y1U/YSCALE
5410 ANGLU=FNANGLER(0!,01,X1U,Y1U)
5420 ANGLU=ANGLE+ANGLU
5430 DISTU=FNRDIST(0!,0!,X1U,Y1U)
5440 1 DISTX=DISTU/XSCALE
5450 DISTY=DISTU/YSCALE
5460 XROT=DISTU*COS(ANGLU)
5470 YRT=DISTU*SIN(ANGLU)
5480 XOFF=XORG-XROT
5490 YOFF=YORG-YROT
5500 GOTO 5530
5510 XOFF=XORG:'100
5520 YOFF=YORG
5530 '200 WRITE(LUNO,201)
5540 PRINT ENTER X-AXIS ROUNDOFF (REAL) [0.0]: ;
5550 INPUT XROUND
5560 '202 FORMAT(F6.0)
5570 WRITE(LUNO,203)
5580 PRINT ENTER Y-AXIS ROUNDOFF (REAL) [0.0]:
5590 INPUT YROUND
5600 RETURN:'GOSUB 3800:RETURN :'RESET STREAM MODE FIRST!
5610 END
6000 TIMER LOOPS
6001 BEEP :FOR IDUM=1 TO 375 :NEXT IDUM:PRINT TIME$:RETURN:'1 SEC
6002 BEEP :FOR IDUM=1 TO 750 :NEXT IDUM:PRINT TIME$:RETURN:'2 SEC
6003 BEEP :FOR IDUM=1 TO 1125:NEXT IDUM:PRINT TIME$:RETURN:'3 SEC
6005 BEEP :FOR IDUM=1 TO 1875:NEXT IDUM:PRINT TIME$:RETURN:'5 SEC
6010 BEEP :FOR IDUM=1 TO 3750:NEXT IDUM:PRINT TIME$:RETURN:'10SEC
8000 REM ++++++ GET INPUT FROM COM BUFFER ++++++++
8010 WHILE LOC(1) < LCB
8020 WEND
8030 'IF LOF(1) < 24 THEN BEEP:BEEP: BEEP
8040 DZ$=INPUT$(LCB/#1)
8041 'PRINT DZ$
8050 X$=LEFT$(DZ$,5) : Y$=MID$(DZ$,6,5) : CODE$= MID$(DZ$,11,1)
8060 XIN=VAL(X$)/1000 : YIN=VAL(Y$)/1000
8061 'PRINT XIN,YIN,CODE$:BEEP
8070 RETURN
9000 SUBROUTINE DIGURU(X,Y,IBTN)
9010 GOSUB 8000
9020 DISTU=FNRDIST(XOFF,YOFF,XIN,YIN)
9030 ANGU=FNANGLER(XOFF,YOFF,XIN,YIN)-ANGLE
9040 XIN=DISTU*COS(ANGU)*XSCALE
9050 YIN=DISTU*SIN(ANGU)*YSCALE
9055 XTRUE=XIN/XSCALE:YTRUE=YIN/XSCALE
9070 RETURN
20000 IF ERR=53 AND ERL=4005 THEN PRINT "NO FILES ON B:":RESUME 4006
20030 PRINT "ERROR NUMBER ";ERR;" HAS OCCURRED AT LINE ";ERL


2
Historical Perspective
Previous Models of Pulsing Patterns in Time and Space
In many fields from chemistry, physics, and biology to
astronomy, there are a variety of models, methods and tech
niques to describe and study systems that have discon
tinuities or other rapid fluctuations in their behavior.
Some of these are catastrophe theory (Thom 1975), bifurca
tion theory, synergetics (Haken 1977a,1977b,1979), dynamical
system theory (Rosen 1970), chaos and order (Prigogine
1980,1984, and Schaffer and Kot 1985), pulsing (Lotka 1920
and Odum 1982), pattern recognition, and morphogenesis
(Meinhardt 1982). In all of these, processes being de
scribed are parts of nonlinear thermodynamically open sys
tems. Energy constraints on these types of systems have not
previously been well studied.
In the past, efforts to describe systems using clas
sical thermodynamics centered on closed systems near equili
brium or open systems near steady state. In such systems,
available energy is small. These approaches using equili
brium thermodynamics could not account for the behavior of
many systems (Odum 1983, Prigogine 1984, Schaffer and Kot
1985).
Data with statistical anomalies are often difficult to
analyze and methods are sometimes used to minimize fluc
tuations (Platt and Denman 1975). Systems that have aperi
odic behavior, a great deal of noise, or time dependent
changes in variance are not well suited to the normal




Figure 36. Steady state values of percent power used as a
function of input energy and state variable initial
conditions for multiple simulation runs of parallel
production-consumption model (Figure 13).
(a) Vary input energy and
(b) Vary input energy and
(c) Vary input energy and
(d) Vary input energy and
Q1 (Climax species)
Q2 (Intermediate producer)
Q3 (Weed Species)
Q4 (Consumer)


40
describe the changes in storage compartment (tank) values
over time.
Simulation Procedures and Programs
The majority of the simulations in this dissertation
were done in FORTRAN-4-PLUS on a Digital Equipment Corpora
tion (DEC) PDP 11/34 with RSX-11M operating system. The
graphical outputs of the simulations were displayed on a DEC
VK-100 graphics terminal (General Image Generator and Inter
preter or GIGI) connected to a Barco color monitor and DEC
.LA-34 Decwriter. The GIGI terminal has a 760x240 pixel
resolution and can display up to eight colors on a color
monitor. In order to facilitate the graphics programming
needed in my simulation models, I developed a set of FORTRAN
subroutines with a more natural calling sequence to execute
the ReGIS (Remote Graphics Instruction Set use by the GIGI
terminal) commands from the programs. This library of rou
tines (GGLIB) is listed and documented in the Appendix.
Some of the goals of this dissertation were to examine
the structure and function of systems in time and space and
to determine how variation in coefficients may affect energy
flows and storages of the systems. Graphical display pro
grams were developed to project a simulated 3-D surface of
the output of various state variables over time and over a
range of input conditions. A special 3-D graphics display
program was written to display the output of these model
simulations (program PLOTZ, Appendix).


70
tropical rain forest at El Verde, several measurements were
made from data sets from the tropical rain forest study at
El Verde (1963-1967) in the Luquillo Mountains of Puerto
Rico (Odum and Pigeon 1970).
A data set (2048 samples) characterizing the forest at
the radiation site was generated by the U. S. Army Corps of
Engineers (Rushing 1970). At the radiation site, every
plant 1.8 m. or taller was enumerated within a radius of 30
m. from the center of the site. Each plant was recorded
with the species name, height, diameter, crown diameter,
exact location, and various other parameters.
Black and white negatives of aerial views of the radia
tion site (taken November 1963 before the radiation treat
ment) were printed as 8x10 inch photographs. Individual
gaps characterized by the presence of Cecropia peltata (an
early successional species) were digitized from the photo
graphs using a personal computer, Complot digitizer and
digitizing program written especially for this purpose
(Measure3 in Appendix).


^ | F | O RID A
3 1262 08554 8567^


Figure 30. Simulation of the parallel production-
consumption model in Figure 13. Multiple simulations of the
model with available power increasing from 50 to 300.
(a) Climax species (Ql)
(b) Intermediate producer species (Q2)
(c) Weed species (Q3)
(d) Consumer species (Q4)
(e) Percent power used (J0-Jr)/J0
(f) Total biomass


designs; and (d) multiple cell spatial models each with a
unit model but interconnected in different ways.
Models with autocatalytic feedbacks utilized more
power than the same models with only linear pathways. Per
cent power used increased with increasing available power.
Production-consumption models show multiple steady
states with pulsing behavior as a transition between two
steady states. Localized maxima of power use occur during
pulsing but the overall power use is related to input power.
Spatial patterns of production and consumption in
spatial models were related to input energy patterns, the
degree of connectivity between the individual cells in the
model, and the hierarchical level of intercell connections.
Large variations in patterns were accompanied with small
changes in power utilized.
Edges of a spatial system can act as a source or sink
for energy depending on the relationship between available
energy inside and outside the boundaries and the degree of
connectivity along the edges.
Basic autocatalytic production-consumption-recycle
models with different spatial conditions organize different
spatial patterns while generating near total utilization of
available power. The wide variety of spatial patterns
results from dynamic adaptations for maximizing power for
different spatial conditions. The simulation results
resemble patterns in nature often attributed to random
indeterminancy.


231
C SURFACE PULSING MODEL PROGRAM 3/24/83
C ADDITION OF CONSUMER CEILING TO ALLOW UP TO 100 TOTAL
C CONSUMERS
C
C DIFFUSION ADDED 7/21/83 TO NUTRIENT TANK Q4
C
C VERS 3.01 ADDED STARTING CONDITION TO FILE OUTPUT
C VERS 3.02 ADD TIME AND DATE TO BEGINNING OF PROG
C
C
PROGRAM SURPUL
C Q1=PRODUCER
C Q2=STORAGE (PRODUCER)
C Q3=CONSUMER
C Q4=NUTRIENTS
DIMENSION Q1(12,12),Q4(12,12),E(12,12),Q3(100)
DIMENSION Q4T(12,12)
DIMENSION ETYPE(3),IX(144)
DIMENSION Q2(12,12),IXYZ(100)
INTEGER*4 ICNT(12,12)
BYTE TITLE(10),ICON(12,12),BUF1(9),BUF2(8)
REAL M,K1,K2,K3,K4,K5,K6,K7,K8,K9,K10,MTOT
REAL K11,K12,K13,JO,JR
BYTE ESC,TEXT(80),COLOR,ICOLOR,CHAR
INTEGER X1(100),Y1(100),T1,T2,XTEMP,YTEMP
FT1(A,3)=ABS(AINT(A/B)-A/B)
IXY(I,J)=(I-1)*12+J
DATA TITLE/'D','S','P','1' ,'O','O',' '/
DATA Q4T/144*0.0/ !DK
DATA ICON/144*0/
DATA ETYPE(1)/'HIER'/
DATA ETYPE(2)/'EVEN'/
DATA ETYPE(3)/'RAND'/
ESC=27
VERS=3.02
CALL TIME(BUF2)
CALL DATE(BUFI)
WRITE(5,5)ESC,ESC,ESC,ESC,TITLE,VERS,BUF2,BUF1 !3.0
5 FORMAT(1X,A1,'PrTM1',A1,1 ',A1,'[2J',A1,'[H', 13.0
&' SURFACE MODEL ',10A1,' VERSION ',F5.2/ 13.0
&1X,8A1,1X,9A1/ 13.02
&' DO YOU WANT GRAPHICS ON (1-YES, 0-NO) '$)
READ (5,6)IOFLAG
6 FORMAT(11)
C
C
WRITE (5,7)
7 FORMAT(' PLOTTING INTERVAL FOR PRODUCER AND CONSUMER [I] '$)
READ (5,8)ITINT,ITINTC
8 FORMAT(13,13)
WRITE(5,808)
808 FORMAT(' HARDCOPY AT PLOTTING INTERVAL (1-YES,0-NO)'$)
READ(5,8081)IPTR
8081 FORMAT(13)


on oooaoQooooooooonnonooo o
o
U)
D >o 'd tu
w a a > iT ¡2
jo cn cn < >0 >
n M M > + X
Z D D HlO II
h3 II ¥ M O) >
ir a *o 11 s
m a a *o >
a cn cn > ><
W M M < -
g ? ? K 5
\ o h r1 s
^ a a + >
o > cn C4 ><
d >0 >o >ti T) 'O u
U> NJ £ OJ N)
II II II II II II II
> IO KD IO KD
SAWN)-*
5\\N\
X IO IO IO KD
-1 -1 ii U W j
W CO W M
O 0 H H M H
W N) -* N N N N
- M W M M
-* + W PI W
- *0 >0 I I I
*0 3 ^ U1 ui U1
DlOlOlOlOlO f JO
w to m o
II li II II < II
SS'SS P ¡3
+ +
II II
******
******
ess
ii ii ii
a q o
3 h ^
* *
io *
_ U) Kj _
O \
II
C-I -*
3 3 3 3 3 + + Jd-'WWM>iQlp^
vOifcCJto-*XZ)3at*>JdCy>lniH*iUi
lOi + Cjl I IH *
_fpo ;0Q O O XO
d ii) to -4 Z
JO I
I VO o
* -*
fO
o
OJ
ITCNT=ITCNT+1
RATE EQUATIONS'


183
frequencies (Richardson and Odum, 1981). This double
threshold behavior has also been shown in a wide variety of
prey-predator model configurations (Kuno 1987). Oscillating
chemical reactions exhibit this multiple output state be
havior (Field 1985).
At low power levels, the pulsing model supported a
constant low amount of consumers (dependent on the linear
pathway) while at high power levels the consumer was at a
constant higher level (sustained by both the linear and
quadratic pathways) with the producer at a low level. This
was also the case in the chemical reactions and prey-
predator models described above. Models with this behavior
may describe a variety of ecosystems that show various
levels of producers and consumers. A grassland ecosystem
such as the Serengeti (McNaughton 1985) may be an example of
low levels of producers supporting high levels of consumers.
A similar dependence of the highest trophic level on
the input energy was also exhibited with the parallel
production-consumption model (Figure 32) although this model
did not pulse. It should be noted with this model that the
consumer level increased and the 'climax' producer did not.
The pulsing model did not pulse when the consumer quad
ratic pathway (Figure 38) was removed, the consumer built up
to a steady state, and the percent power used declined.
There was a lot of structure in.the higher level of the
hierarchy but the system was not effective at using the
extra power that was available. Competitively, a system


Figure 37. Simulation for pulse model (Figure 14) with
base run coefficients (See Appendix).
(a) Base run of model.
(b) Input energy one-half of base run.
(c) Input energy two times the base run.
Legend:
PPU = Percent power used
Q1 = Production unit
Q2 = Stored biomass
Q3 = Pulse consumer
Q4 = Nutrient storage


41
The spatial models are broken down into cells that show
the concentration of a given parameter in the individual
cell as a color block. For display on the color monitor this
provides dramatic views of the model changes over time and
space. In order to make hardcopy printouts a display char
acter set was designed so the density of the dots in an
individual cell was correlated to the color of the cell.
This provided a way of screen-dumping the images to paper
and achieving patterns on paper that were similar to the
ones on the video screen (See Appendix for a listing of the
character set) .
Simulation Models
Minimodel Tests
First a group of minimodels were simulated to relate
energy use to basic pathway designs. Then spatial models
with these configurations were studied for energy use and
pattern formation.
Three path minimodel
In order to understand how a system processes variable
energy inputs, builds structure, and regulates or maximizes
energy flows, a simple single tank model was simulated. The
model is similar to the one described by Odum (1982) that
has parallel pathways of different types competing for
available energy (Figure 10). The model has a flow limited
source connected to a single storage (tank) by three dif
ferent pathways; a linear pathway (Jl), an autocatalytic


Figure 41. Multi-run simulation of pulse model (Figure
14) with variation in total carbon in model. (Q4 varied
from 2000 gC/m2 to 100,000 gC/m2.
(a) Production unit (Ql) .
(b) Stored biomass (Q2).
(c) Pulse consumer (Q3).
(d) Nutrient storage (Q4).
(e) Power used (JO-Jr)
(f) Percent power used 100*(J0-Jr)/(JO)


PERCENT POWER USED
V£>
O


Figure 57. Size class distribution of trees at El Verde
radiation site (November 1964)
(a) Size class distribution by diameter
(b) Size class distribution by crown diameter
(c) Size class distribution by height


238
Q2TOT=0.
DO 4531 1X1=2,11
DO 4531 1X2=2,11
Q1T0T=Q1T0T+Q1(1X1,1X2)
Q2TOT=Q2TOT+Q2(1X1,1X2)
Q4TOT=Q4TOT+Q4(1X1,1X2)
4531 CONTINUE
Q4TOUT=Q4(1,1)*44.
TOT=Q1TOT+Q2TOT+Q3TOT+Q4TOT
IF(IOFLAG.EQ.0)GOTO5000
C
C
C WRITE TEMPORARY INFORMATION AND PLOT GPP, POWER (EUSED)
C
C
IF ((T-PTIME1).LT.1.0)GOTO 5000
PTIME1=T
ENCODE(80,2006,TEXT)T,T2,EUSED/(100.*DT),ETOT/100.,VERS,TITLE
&,ETYPE(IETYP)
2006 FORMAT(1X,'T=',F6.2,' CONS=',I3,' POW USED=',F6.2,
& AVAIL POW= ',F62,' VER:',F5.2,1X,10A1,1A4)
CALL GGTEXT(7,0,475,TEXT,1,0)
IPT=PTEMP/(2*DT*1000.)
CALL GGPLT(4,IT,IPT+250,1) ¡GREEN = GPP
IPOWER=((EUSED/(100*XMEAN*DT))80)*5. ¡OUTPUT 80 TO 100
CALL GGPLT(2,IT,IPOWER+250,1) ¡RED = POWER
C
ENCOD E(80,4532,TEXT)Q1TOT,Q2TOT,Q4TOT,Q3TOT,TOT
4532 FORMAT(1X,'Q1= ',F10.2,' Q2= ',F10.2,' Q4= ',F10.2,
& Q3TOT= ',F10.2,' TOT= 1,F10.2)
CALL GGTEXT(6,0,460,TEXT,1,0)
IYT=Q2TOT/20000
CALL GGPLT(3,IT,250+IYT,1) ¡MAGENTA = PRODUCERS
IYT=Q3TOT/2 0 0.+2 5 0.
CALL GGPLT(7,IT,IYT,1) ¡WHITE =CONSUMERS
C
C DRAW PRODUCERS
C
C l
IF((T-PTIME).LT.TINT)GOTO4500 !
PTIME=T
DO 4050 1=2,11 !
DO 4050 J=2,11 1
COLOR=Q2(I,J)/(2*1000.) ¡3.0 CHANGED 4*1000 TO 2*1000
IF(COLOR.GT.7)COLOR=7 !
1X9=1*24-40 !
JY=J*24-44 !
CHAR=48+COLOR ¡
IF(CHAR.GT.57)CHAR=57 1
CALL GGPLT(COLOR,1X9,JY,0) ¡POINT TO LOWER LEFT
WRITE(3,398)CHAR 1
398 FORMAT('+','T(A1)W(S', 1H',A1,1H',') @B') ¡
399 FORMAT(' @A') ¡DRAW BOX (MACRO)
4050 CONTINUE !


146
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SS 9 'SHSi.'.i "SS
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m -S3
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t, j
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- i S-:
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3 33 dHMl ril
Ijg na
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112
Either immigration or a temporary auxiliary support system
is necessary to start a system of this class.
Similarly, when the input power was held constant
(J0=100) and the initial condition of the weed species (Q3)
was varied (Figure 35a-f) the system crashed at low levels
of Q3, but at higher levels it was stable (see Figure 29c
for a single run with Q3=0).
The system response was different with changes in the
initial conditions of Q1 and Q2 (refer to Figures 29a and
29b) because the consumer was not as dependent upon them for
its survival early in the simulation.
Initial Conditions and Total Energy Use
The behavior of the parallel production-consumption
model with different initial conditions for the state var
iables (Ql, Q2, Q3, and Q4) and input power was tested. In
this set of simulations, the total percent power used was
measured for each simulation run while varying the input
power and the initial condition of the state variables one
at a time (Figure 36).
In all four cases when JO was low, the model was unable
to utilize the energy available to it. When the input power
was above a certain point then the model was able to utilize
the input energy with two exceptions. When Q3 (weed
species) was very low, the percent power used rose to a
plateau then fell when the input energy went above a certain
level. The model was unstable under these conditions.


Figure 23. Simulation of three path model with multiple
drain pathways in Figure 11. Percent power used as a
function of energy input (JO).
Linear drain:
Autocatalytic drain:
Quadratic drain:
04=k4*Q
D5=K5*Q*Q
D6=K6*Q*Q*Q


1. It simplified programming the model because two
dimensional arrays in FORTRAN are set up in rows and
columns.
2. It simplified writing the graphics routines to dis
play the cells on a graphics terminal.
3. It reduced the edge effects of the model.
Ring model
A modified version of the two dimensional spatial model
was used to simulate a one dimensional case. The standard
spatial pulse model was connected head to tail in a ring of
36 cells.
Two dimensional models
The simplest spatial implementation was the basic pulse
model repeated over the 10x10 matrix with no interactions
between individual cells. This model (program DSP1) was
then simulated with three different energy forcing functions
1. The energy source was hierarchically distributed
(highest energy input at the center of the matrix).
2. The energy source was evenly distributed.
3. The energy source was randomly distributed.
Energy inputs were sea
whole matrix could be held
Overall energy input could
opment and energy use with
Two different initial
led so the mean input over the
constant for all energy types,
be varied to test pattern devel-
various energy levels,
conditions were tested. A suc
cessions! sequence was simulated with the initial values of


128
power was increased (similar to three path models seen
earlier) with local maxima immediately after the pulse.
The total amount of nutrients in the system also had an
important effect on the behavior of the model (Figure 41a-
f). At higher initial levels of Q4 there was little effect
on the model. At these higher ranges, the model was no
longer nutrient limited but was energy limited. At low
values for the initial concentration of Q4 the pulsing
greatly affected the labile production (Ql), the stored
biomass (Q2) and the pulsing consumer (Q3). At the lowest
level of Q4, there was no pulsing, Q2 remained at a low
steady state value, and Q3 also remained at a low steady
state value. There was a small shift in the pulsing fre
quency at the lowest initial levels of Q4 but no frequency
shift at the higher levels. The power used was greatly
affected at low initial levels of Q4 but rose only slightly
at higher values of Q4. For the same amount of change in
Q4, the variability of the power used was greater when Q4
was small than when Q4 was high. However, the percentage
change was greater in the beginning than at the end.
The turnover time of the pulsing consumer affected the
behavior of the system and use of power (Figure 42a-f). The
pulse model was simulated with the value of the drain coef
ficient (K12) of the consumer (Q3) varied with each run. As
the turnover time increased, the frequency of pulsing shift
ed to a shorter period with the amplitude decreasing until
there is no pulse at all but a continually rising consumer.


187
Coupling of Spatial Units by Diffusion Processes
In any ecosystem, spatially distributed subunits are
connected to each other through a variety of processes.
Nutrients and seeds can be carried spatially by transport
from wind, water and animal activity. Predation by con
sumers tends to reorganize the vegetation community struc
ture. The degree to which subunits are connected to one
another is strongly reflected in the patterns that may
develop.
Connectivity between subunits tends to decrease the
asynchronous behavior caused by local energy differences.
With a low level of diffusion (Dk=.001) the pulsing behavior
was propagated across cell boundaries (compare Figure 52
with Figure 51). At higher levels of diffusion (not shown),
the effect was to increase the synchronous nature of the
pulsing across the matrix. Energy use with various levels
of diffusion did not change appreciably (Table 5 in
Appendix).
In a single dimension system (ring model) the effect of
diffusion was similar. At high levels of diffusion (Figure
48) pulses were propagated around the entire ring, while at
a lower level of diffusion the propagation was confined to
local areas (Figure 49). The asynchronous pulsing (Figure
47) was thus organized into a more synchronized spatial
pattern depending on the degree of connection between the
individual cells.


Figure 49. Simulation of pulse model (Figures 14 and 18)
with cells in a linear ring and a ,low level of diffusion
between consumers of each cell (DK=.001) and random distri
bution of producers and consumers around ring. For each
time unit (e.g. A=0) density of producer and consumer in the
matrix is shown along with size class distribution. The
time series from A to 0 summarizes the temporal pattern of
totals in matrix.
1
>14000
12000
10000
000 S
I
000
4000
2000
-JO
Produoer
lx* olaatet
v
Contunar
tlx* !


32
Simulation of tile parallel production-consumption
model in Figure 13. Run with available power
increasing from 50 to 300 and the initial
value of the consumer (Q4) equal to 50 (lOx
base run in Figure 28) 107
33 Simulation of the parallel production-consumption
model in Figure 13. Multiple simulations of
the model with available power held constant
(J0=100, base run value) and the initial
value of the consumer (Q4) varied from
1 to 6 109
34 Simulation of the parallel production-consumption
model in Figure 13. Total percent power used
for entire run as a function of the initial
value of the consumer (Q4) Ill
35 Simulation of the parallel production-consumption
model in Figure 13. The initial value of weed
species (Q3) was varied from 0 to .5 and
input power was held constant (J0=100, base
run va lue) 114
36 Steady state values of percent power used as
a function of input energy and state variable
initial conditions for multiple simulation
runs of parallel production -consumption model
(Figure 13) 116
37 Simulation for pulse model (Figure 14) with
base run coefficients 119
38 Simulation of pulse model (Figure 14) without
a quadratic pathway (K7, K8, K9 = 0.0) 122
39 Simulation of pulse model (Figure 14)
without a feedbacks into Q4 (K6, K8 = 0.0) 125
40 Multi-run simulation of the pulse model
(Figure 14) with variation in input energy.
(JO varied from 0 to 250) 127
41 Multi-run simulation of pulse model (Figure
14) with variation in total carbon in model....130
42 Multi-run simulation of pulse model (Figure
14) with variation is turnover time of
pulsing consumer. (K12 varied from
.01 to .5)
132


155


o o
237
IXYZ(17)=IXY(XTEMP,YTEMP)
C!CODE NEW LOCATION
C
C... CHECK TO SEE IF IT IS TIME TO REPRODUCE
C
IF(Q3(17).LTTHRESH)G0T02000 !IF GREATER THAN REPRODUCTION
T1=T1+1 !INCREASE NUMBER OF CONSUMERS
IF(T1.GT.100)GOT02000 ¡ALLOW NO MORE THAN 100
Q3(I7)=Q3(I7)/2
Q3(T1)=Q3(I7)
X1 (T1)58 XTEMP
Y1 ( T1 )*YTEMP
IXYZ(T1)=IXY(XTEMP,YTEMP)
ICON(XTEMP,YTEMP)=ICON(XTEMP,YTEMP)+1
2000 CONTINUE
XI(I7)=XTEMP
Y1(17)=YTEMP !REM REMEMBER WHERE TO START NEXT TIME
C
C
2001 CONTINUE
C
217 CONTINUE
350
C
C
400
C
C
C
C
C
C
C
453
C
C
CONTINUE
CONTINUE
GPP=GPP+PTEMP ¡ACCUMULATE TOTAL GPP
T2=T1
IF (T2.GE.100)T2=100
END CONSUMER LOOP AND DO BOOKEEPING
CNSUMP=CNSUMP+SPTEMP
PROD=GPP + CNSUMP
TOTPOW=TOTPOW+EUSED
COUNT UP THE CONSUMERS
NPROD=100
IF(T2LT10 0)NP ROD=T2
Q3TOT=Q.
DO 453 1=1,NPROD
Q3TOT=Q3TOT+Q3(I)
ICNT(X1(I), Y1 (I))=ICNT(X1(I),Y1(I))+1
CONTINUE
COUNT UP PRODUCERS AND NUTRIENTS
Q1TOT=0.
Q4TOT=0.


Figure 39. Simulation of pulse model (Figure 14) without
feedbacks into Q4 (K6, K8 = 0.0)
U)
Simulation
for
760 years.
(b)
Simulation
for
2000 years
Legend:
PPU = Percent power used
Q1 = Production unit
Q2 = Stored biomass
Q3 = Pulse consumer
Q4 = Nutrient storage


Percent
Power Used
100
300
Percent
Power Used
116


o o
210
C
C
SUBROUTINE GGPLT(COLOR,IX,IY,IFLAG)
C SUBROUTINE TO POSITION GRAPHICS CURSOR ON THE GIGI
C SET IFLAG TO NUMBER > 0 TO PLOT POINT AND TO 0
C TO MOVE CURSOR TO POINT WITHOUT PLOTTING POINT
C COLOR = BYTE variable 0-7 for color
C IX = Integer value of X
C IY = Integer value of Y
C IPT = Integer flag >1 plot a point
BYTE COLOR
IF(IFLAG.GT.0)GOTO10
WRITE(3,100)COLOR,IX,IY
100 FORMAT('+ W(I',11,')P[',14,',',14,'] ')
GOTO20
10 WRITE(3,101)COLOR,IX,IY
101 FORMAT('+ W(I',11,')P[',14,',',14,']V[]')
20 CONTINUE
RETURN
END
C
C
C
SUBROUTINE GGPLOT(COLOR,IX,IY,IFLAG)
C ROUTINE TO ALLOW FOR VARIATION IN SPELLING OF GGPLT ROUTINE
C ADDED IN VERS 12
BYTE COLOR
CALL GGPLT(COLOR,IX,IY,IFLAG)
RETURN
END
C
C
C
SUBROUTINE GGVEC(COLOR,IX,IY)
C SUBROUTINE TO DRAW A VECTOR ON THE GIGI FROM ITS PRESENT POSITION
C TO THE IX,IY POSITION IN THE PARAMETER LIST. USE GGPLT FOR
C INITIAL COORDINATES IF NEEDED.
C COLOR = BYTE variable 0-7 for color
C IX = Integer value of X
C IY = Integer value of Y
BYTE COLOR
WRITE(3,100)COLOR,IX,IY
100 FORMAT(*+ W(I',11,')V[',14,',',14,']')
RETURN
END
C
SUBROUTINE GGBOX(COLOR,IX,IY,IX1,IY1)
C SUBROUTINE TO DRAW A BOX ON THE GIGI GIVEN THE OPPOSITE COORDINATE
C PAIRS FOR THE RECTANGLE.
C COLOR = BYTE variable 0-7 for color
C 1X0 = Integer value of X
C IY0 = Integer value of Y


220
2150
2160
2170
2180
2190
2200
2210
2220
2230
2240
2250
2260
2270
2280
2290
2300
2310
2320
2330
2340
2350
2360
2370
2380
2390
2400
2410
2420
2430
2440
2450
2460
2470
2480
2490
2500
2510
2520
2530
2540
2550
2560
2570
2580
2590
2600
2610
2620
2630
2640
2650
2660
2670
2680
A1=0!
A2=0!
AREAIOO!
NUMPNT=NUMPNT+1
'C READ FIRST PAIR
XF=X(NUMPNT)
YF= Y(NUMPNT)
XP=XF
YP=YF
'300 CONTINUE
'C
NUMPNT=NUMPNT+1
XC=X(NUMPNT)
YC=Y(NUMPNT)
A1=A1+(XP*YC)
XP=XC
IF(NUMPNT = NPOINT)GOTO 2330
GOTO 2240
IF(NUMPNT > 2)GOTO 2370:'400
' WRITE(LUNO,401)
PRINT ?NOT ENOUGH DATA POINTS FOR AREA CALCULATION"
RETURN
A1=A1+(XC*YF):'450
NUMPNT=1
XF=X(NUMPNT)
YF=Y(NUMPNT)
XP=XF
YP=YF
NUMPNT= NUMPNT+1:'500
YC=Y(NUMPNT)
XC=X(NUMPNT)
A2=A2+(YP*XC)
YP=YC
IF (NUMPNT = NPOINT)GOTO 2500
GOTO 2430
A2=A2+(YC*XF):'600
AREAIO=ABS((A1-A2)*5)
RETURN
'700 AREAIO=0.
' RETURN
'800 CONTINUE
' AREA10=0 *
RETURN
' END
'C
'C
PRINT SUBROUTINE PERIX(X,Y/PERI1,PERI2,NPOINT)"
' DIMENSION X(1),Y(1)
PER11=0!
PERI2=0!
NUMPNT=1
XIN=X(1)
YIN=Y(1)
XL=XIN


94
Table 1. Frequency response (mnimums and mximums) of
three path model (Figure 10) with varying input power.
Pathway
combination
INPUT POWER
J0=500
J0=2000
J0=10000
J1+J2+J3 (All)
MAX (2)
MIN (8)
MIN (3)
J2+J3
MIN (2)
MIN (2)
MIN (2)
J1+J3
MAX (2)
MIN (8)
MIN (3)
J1+J2
MAX (2)
MAX (2)
MAX (2)
J1
N/R
N/R
N/R
J2
MIN (2)
MAX (0)
MAX (0)
J3
N/O
N/O
MIN (2)
Numbers in parenthesis are the frequencies at which the
maximum or minimum occurs.
N/R signifies there was no frequency response for this set
of pathways.
N/O signifies there was no power uses at these inputs


Figure 13. Parallel production-consumption model.
Individual rate equations
R1 = K1*Q1* JR*Q4
R2 = K2*Q2*JR*Q4
R3 = K3*Q3*JR*Q4
R4 = D1*Q1
R3 = D2*Q2
R6 = D3*Q3
R7 = K7*Q1*Q4
R3 = K8*Q2*Q4
R9 = K9*Q3*Q4
RIO = FI* (K1*Q1*JR*Q4 + K2*Q2*JR*Q4 + K3*Q3*JR*Q)
Rll = KO* (K7*Q1*Q4 + K8*Q2*Q4 + K9*Q3*Q4)
R12 = D4*Q4
JR = J0/(1 + L1*Q1*Q4 + L2*Q2*Q4 + L3*Q3*Q4)
equations
for state va
dQl =
R1 -
R4 -
R7
dQ2 =
R2 -
R5 -
R8
dQ3 =
R3 -
R6 -
R9
dQ4 =
Rll
- R12
- RIO


114
Time
Time
100


236
c
c...
c
c.
c
CONSUMER CHECKING ROUTINE
IF(XEQ.EQ.0)G0T0350
IXYLOC=IXY(I,J)
INUM=1
DO 217 17=1,T2
IF (IXYZ(17).NE.IXYLOC) G0T0217
RATE EQUATIONS FOR CONSUMERS
¡SKIP IF NO CONSUMER PRESENT
¡GET CODED LOCATION
¡AT LEAST ONE CONSUMER PRESENT
¡GET CONSUMER NUMBER
¡WRONG CONSUMER GOTO 217
XDT=DT
...IF RATIO OF Q2/Q3 IS TOO LOW THEN ITERATE MORE SLOWLY
IF(Q2(I/J)/Q3(17).LT.5.0)XDT=.01
DO 650 DDT=XDT,DT/XDT
R7=XDT*K7*Q2(I,J)*Q3(17)*Q3(17)*XEQ
R8=XDT* K8 Q2 (I, J) Q3 (17) Q3 (17) XEQ
R9=XDT*K9*Q2(I,J)*Q3(17)*Q3(17)*XEQ
R12=XDT*K12*Q3(17)*XEQ
SPTEMP=SPTEMP+R5*(XDT/DT)+R7 1SPTEMP = CONSUMP
C
C... LEVEL EQUATIONS FOR CONSUMERS
Q3(17)=Q3(17)+R5/(ICON(I,J))*(XDT/DT)+R7-R12
Q2(I,J)=Q2(I#J)-R9 ¡UPDATE PREY CONSUMED
Q4(I,J)=Q4(I,J)+R8+R12 ¡UPDATE NUTRIENTS
650 CONTINUE
C
C
C
IF((T-XTIME).LT.1.)G0T0217 ¡SKIP MOVEMENT IF NOT WHOLE DT
XTIME=T
C CHECK PRESENT POSITION FOR VALUE OF Q2
QMAX=0
IF(Q2(I,J).LT.PTHRSH)GOTO 1457
XTEMP=X1(17)
YTEMP=Y1(17)
GOTO600
C IF Q2 IS STILL CONSUMABLE DON'T MOVE, JUST EAT SOME MORE
1457 DO 600 I2=I-N,I+N
DO 600 J2=JN,J+N
IF (I2.LT.1) GO TO 600
IF (I2.GT.12) GO TO 600
IF (J2.LT.1) GO TO 600
IF (J2.GT.12) GO TO 600
IF(Q2(I2,J2).LT.QMAX)GOTO580
QMAX=Q2(12, J2)
XTEMP=I2
YTEMP=J2
580 CONTINUE
600 CONTINUE
ICON(I,J)=ICON(I,J)-1
Cl REMOVE CONSUMER FROM PRESENT LOCATION
ICON(XTEMP,YTEMP)=ICON(XTEMP,YTEMP)+1
C!MOVE CONSUMER TO NEW LOCATION


Figure 3. The development of spatial patterns among cells
based on simple r-pentamino initial condition (a) in 'a game
of Life'
simulation.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Time = 0
Time = 8
Time = 16
Time = 32
Time = 64
Time = 128
Time = 256
Time = 512


1 REM THREPATH MODEL VERSION 7/1/84
2 REM WITHINPUT COEFFICIENTS CHANGED
3 REM NAME=> TPMOD7.BAS
90 PRINT "WHAT IS VALUE FOR ENERGY INPUT (J0-J1)"
91 INPUT J1
100 REM AND PRINT FNV$(C,X,Y) FOR PLOTTING VECTORS. C=COLOR(1-7)
110 PRINT "DO YOU WANT GRAPHICS ON '9 INPUT Q$
120 IF Q$<>"Y" GO TO 150
130 PRINT CHR$(27)+"PpS(E)W(R,I(G),P1,N0 ,A0 ,S0) S( A[ 0,479] [767,0])
140 DEF FNV$(C,X,Y)="W(I"+STR$(C)+")V["+STR$(X)+","+STR$(Y)+"]"
150 DEF FNP$(C,X,Y)="W(I"+STR$(C)+")P["+STR$(X)+","+STR$(Y)+"]V[]
160 DEF FNT$(C,N,A$)="W(I"+STR$(C)+")T(S"+STR$(N)+") '"+A$+"
170 DEF FNB$((5,X,Y,X1,Y1)=FNP$(C,X,Y)+FNV$(C,X,Y1)+FNV$(C,X1,Y1)
+FNV$(C,X1,Y)+FNV$(C,X,Y)
180 A$=CLK$
190 A9=TTYSET(255,132)
200 N9=-1.8
210 T9=1
220 W=0
230 £=100
240 K1=.5
250 K2=1.00000E-03
260 K3=1.00000E-06
270 K4=.2
275 K6=K1
276 K7=10*K2
277 K8=10*K3
280 T=0
290 A$="Threepath Model"
300 PRINT FNP$(7,626,475);FNT$(7,1,A$)
310 PRINT FNB$(7,620,456,767,479)
330 PRINT FNB$(7,0,0,767,479)
340 PRINT FNP$(7,0,270);FNV$(7,767,270)
350 PRINT FNP$(7,0,380);FNV$(7,767,380)
360 PRINT FNP$(7,0,244);FNV$(7,767,244)
370 JO=J1/2+COS(W*T/57.2958)*J1/2
380 J9=J0/(1+K6+K7*Q+K8*Q*Q)
390 P1=P1+J0-J9
400 J7=J7+J0
410 R1=K1*J9
420 R2=K2*Q*J9
430 R3=K3*Q*Q*J9
440 R4=K4*Q
450 Q9=T9*(R1+R2+R3-R4)
460 Q=Q+Q9
470 X0=R1+R2+R3
480 X1=100*R1/X0
490 X2=100*R2/X0
500 X3=100*R3/X0
510 IF Q$="Y" THEN 580
520 PRINT "J0=";J0,"J9=";J9,"T=";T
530 PRINT "Q=";Q,"Q9=";Q9
540 PRINT "R1=";R1,"R2=";R2,"R3=";R3,"R4=";R4
550 PRINT "X1=";X1,"X2=";X2,"X3=";X3


t certify that X have read this study and that in ray
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Professor of Forest Resouces
and Conservation
I certify that I have read this study and that in ray
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Clay L. Montagi^
Assistant Professor of
Environmental Engineering
Sciences
This dissertation was submitted to the Graduate Faculty
of the College of Engineering and to the Graduate School and
was accepted as partial fulfillment of the requirements for
the degree of Doctor of Philosophy
April 1988
Dean, College of Engineering
Dean, Graduate School


Figure 18. Format of spatial model display graphs.


Figure 2. Hilborn's (1979) spatial model.
(a) Energy diagram of individual cell model
Equations for simulation model.
dX(i) =a*X(i) b*X(i)*X(i) (c*X(i)*Y(i)/(d+X(i) ) )
+h*X( i +1) +h*X ( i 1) 2*h*X( i )
dY (i) =-e* Y (i) f Y (i) Y (i ) + (g*X (i) Y (i) / (d + X (i ) ) )
+k*Y(i+1) +k*Y(i 1) 2*k*Y(i)
where i is the number of the subsystem in a linear loop.
(b) Simulation results of linear series of unit models
showing level of predator vs distance around loop.


Figure 44. Multi-run simulation of pulse model (Figur
14) with variation in linear pathway (Kll varied from
to 0.12E-2 and K5 and K6 varied proportionately) with quad
ratic pathway held at zero.
(a) Production unit (Ql).
(b) Stored biomass (Q2).
(c) Pulse consumer {Q3).
(d) Nutrient storage (Q4).
(e) Power used (JO-Jr)
(f) Percent power used 100*(J0-Jr)/(JO)
O ID


Figure 40. Multi-run simulation of the pulse model (Figu
14) with variation in input energy. (JO varied from 0 to
250) .
(a) Production unit (Ql).
(b) Stored biomass (Q2).
(c) Pulse consumer (Q3).
(d) Nutrient storage (Q4).
(e) Power used (JO-Jr)
(f) Percent power used 100*(J0-Jr)/(JO)


Figure 56. Simulation of the pulse model (Figure 14) with
cells arranged in two dimensions (Figure 18) Moving
consumer model with search length set to five cells, no
diffusion and hierarchial energy distribution. For each
time unit (e.g. A=0) density of producer and consumer in the
matrix is shown along with size class distribution. The
time series from A to 0 summarizes the temporal pattern of
totals in matrix.
1X4000
? 2000
10000
000
000
4000
2000
'2
/a


Figure 31. Simulation of the parallel production-
consumption model in Figure 13. Multiple simulations of
the model with percent power used for entire run vs input
power. See Figure RS3a-f.


12


140
was hardly detectable in the simulation result (Figure 45a).
The frequency of pulsing was not changed and the power
utilized was only negligibly changed. The higher level
consumer (Q5) was near zero for the entire simulation.
When the model was run with a turnover time (K17=0.005)
of the top consumer (Q5) longer than the normal pulsing
consumer (Figure 45b), pulsing occurred at the normal fre
quency but the higher level consumer grew over time until it
began pulsing. The period of pulsing became longer and the
pulse amplitude of the stored producer and nutrient storages
became greater. The normal pulsing consumer (Q3) remained
at a low level, acting as a feeder to the higher level
consumer (Q5). The power utilized dropped slightly to 95.3.
When the turnover time of the higher level consumer
(Q5) was raised by another order of magnitude (K17= 0.0005)
the outcome was quite different (Figure 45c). The higher
level pulsing consumer (Q5) climbed toward an asymptote
while the stored production (Q2) also climbed to a steady
state value. the normal pulsing consumer (Q3) again re
mained at a low level. In this case the nutrients (Q4)
became tied up in the stored biomass (Q2) the power used
dropped to 36.1 at steady state. The percent power used for
the entire simulation was 48.2.


SHADE=.FALSE.
CALL INITGR( 5)
CALL CLRSCR
CALL CLRTXT
CALL SCOLOR('GRAYO' 0)
CALL SCOLOR('GRAY1'/1)
CALL SCOLOR('GRAY2',2)
CALL SCOLOR('GRAY3',3)
WRITE(5,991)ESC
CALL DPAPER('LIN',10,2,'LIN',10,2/'GRAY3')
IF (IMINEQ.0)GOTO1211
CALL LNAXIS('YL',YTEXT,YMIN,YMAX,IRFLAG)
CALL LNAXIS('XB',XTEXT,XMIN,XMAX,IRFLAG)
GOTO1212
1211 CALL LNAXIS('YL',YTEXT,,,IRFLAG)
CALL LNAXIS(1XB',XTEXT,,,IRFLAG)
WRITE(5,991)ESC
1212 CALL PDATA(N,X,Y,'L','GRAY2',IMARK,ILIN,SMOOTH,SHADE,0.0)
IF(IRGS.EQ.0)GOTO1234
DO 1277 11=1,N
Y1(I1)=B*X(11)+A
1277 CONTINUE
CALL PDATA(N,X,Y1, L' ,'GRAY3',0,1,.FALSE.,.FALSE.,0.0)
TYPE 121,ESC
WRITE (5,1278)B,A,R2,CEE
1278 FORMAT(/1X,' Y= ',G12.4,'*X + ',G12.4,' :R92 = ',G12.4,
+'STD ERR = ',G12.4)
1234 TYPE 121,ESC
121 FORMAT('+',1A1,'[H 0-QUIT; 1- REPLOT; 2-SCREENDUMP',$)
ACCEPT 122, IANS
122 FORMAT(12)
IF(IANS.EQ.1)GOTO 51
IF (IANS.NE2)GOTO 2550
WRITE(5,1221)ESC
1221 FORMAT('+',1A1,'[H',80X)
CALL CPYSCR
GOTO1234
2550 STOP 'END'
9911 WRITE(5,9912)
9912 FORMAT(' ERROR IN ENTRY PLEASE RE-ENTER')
GOTO 59
END
C
C
C
SUBROUTINE STRIP (TEXT,N)
BYTE TEXT(1)
DO 20 I=N,1,-1
IF (TEXT(I).EQ.32.OR.TEXT(I).EQ.0)GOTO20
TEXT(1+1)=0
RETURN
20 CONTINUE
RETURN
END


175
The gap size distribution was measured for a set of
spatial simulations with input energy distributed hierarchi
cally, evenly and randomly. Figure 59a represents a com
bined gap size distribution measured from a set of three
different simulation runs using a hierarchical input energy
source. The gap size distribution is skewed to one set of
large gaps and a few smaller patches. With a random energy
source the results (Figure 59b) resemble the size class
distribution of the natural system (Figure 58) with more
small patches and fewer large ones. With an evenly dis
tributed input energy source and no diffusion, the system
pulses in a synchronous manner that generates a gap the size
of the simulation (100%) with each pulse. With diffusion
present, the patch size is dependent on the edge effect. If
the edge effect is canceled the result is the same; however
with a diffusive loss or gain along the edge, the patch size
is reduced from 100% due to the uncoupling of the synch
ronous pulsing at the edges.


222
3390 UYSCAL=1!:'USER PLOTTER Y SCALE FACTOR
3400 UROT=0! :'USER PLOTTER ROTATION ANGLE
3410 ARRLEN=.15
3420 ARRWID= .07
3430 ARROFF=.03
3440 IARRTY=3
3450 PDLEN=.1
3460 PULEN=.05
3470 XLAST=0!:'LAST X COORD CALCULATED
3480 YLAST=0!:'LAST Y COORD CALCULATED
3490 RETURN
3500 'CLOSE FILE THEN REOPEN IT
3510 CLOSE #1
3520 OPEN"COM1:9600,E,7,2,RS,CS,DS,CD" AS #1:'OPEN AUX PORT FOR I/O
3530 RETURN
3800 PRINT "STREAM MODE SUBROUTINE"
3810 PRINT #1,"#]J":' TURN ON STREAM MODE
3811 GOSUB 6001
3820 PRINT #1,"#]M":' TURN ON INCREMENTAL MODE
3821 GOSUB 6001
3899 RETURN
3900 PRINT SUBROUTINE FOR SETTING POINT MODE"
3910 PRINT #1,"#]I":' RESET FOR POINT MODE FOR BEGINNING SETUP
3911 GOSUB 6001
3999 RETURN
4000 PRINT "FILE HANDLING SUBROUTINE":' SUBROUTINE FOR FILE HANDLING
4004 PRINT "CURRENT DATA FILES: ":PRINT
4005 FILES "B:*.*"
4006 PRINT:PRINT:PRINT:
4010 'PRINT "FILE HANDLING"
4020 PRINT "WOULD YOU LIKE TO OPEN A NEW FILE OR APPEND TO AN EXISTING"
4030 INPUT FILE (1-NEW, 2-OLD)",FILEMODE
4040 IF FILEMODE <1 OR FILEMODE >2 THEN 4020
4050 IF FILEMODE =2 THEN 4100
4060 INPUT "WHAT IS THE NAME FOR THE FILE (1-8 CHARACTERS) "y FILENAME$
4061 IF LEN (FILENAME$)>8 THEN 4060
4062 IF INSTR(FILENAME$:") <>0 THEN PRINT "INPUT FILENAME
ONLY WITHOUT DRIVE SPECIFIER":GOTO 4000
4070 NTEMP=INSTR(FILENAME$
4075 IF NTEMP=0 THEN 4085
4078 NLEN=LEN(FILEAME$)
4080 FILENAME$=MID$(FILENAME$,1,NTEMP-1)
4085 FILENAME$="B:"+FILENAME$
4090 OPEN "O",2,FILENAME$+".DAT"
4091 OPEN "O",3,FILENAME$+"PRN"
4093 INPUT "WHAT IS DESCRIPTION OF THIS DATA SET" ,DESC$
4094 PRINT #3,CHR$ (34)+DESC$
4095 GOTO 4220
4100 'OPEN FOR APPEND
4110 INPUT "WHAT IS THE NAME OF THE EXISTING FILE ",FILENAME$
4120 IF LEN(FILENAME$)>8 THEN 4110
4130 IF INSTR(FILENAME$/":") <>0 THEN PRINT "INPUT FILENAME
ONLY WITHOUT DRIVE SPECIFIER":GOTO 4100
4140 NTEMP=INSTR(FILENAME$


Figure 32. Simulation of the parallel production-
consumption model in Figure 13. Run with available power
increasing from 50 to 300 and the initial value of the
consumer (Q4) equal to 50 (lOx base run in Figure 28).
(a) Climax species (Ql)
(b) Intermediate producer species (Q2)
(c) Weed species (Q3)
(d) Consumer species (Q4)
(e) Percent power used (J0-Jr)/J0
(f) Total biomass


120
If the input energy was less (J0=50, half of the base
run) then the pulse came at a later time (Figure 37b) and
the frequency of pulsing had a longer period. The production
was lower and the stored biomass (Q2) took longer to reach
the level that would trigger the pulse in the consumer (Q3).
When the input energy was raised to twice the level of
the base run (J0 = 200), the consumer pulsed only one time
(Figure 37c) and then remained at a low level instead of
decaying away entirely as in the base run. With a low level
of consumer, the stored biomass was not able to build up and
remained at a lower steady-state level.
The total power used for each of these runs was related
to the input power. As the input power went up, the percent
power used also went up from 93.3 at 50%, to 96.5 at 100%
and 98.3 at 200%.
The quadratic pathway between the stored biomass (Q2)
and the consumer (Q3) was responsible for the pulsing much
as the autocatalytic pathway of a Lotka-Volterra model is
responsible for its oscillating limit-cycle behavior. With
only the linear path between Q2 and Q3, the behavior was not
pulsing or oscillatory (Figure 38). The stored biomass grew
while the nutrients were used up. In this time frame (760
years), the values did not reach a steady state and 93% of
the available power was used. When simulated for 2000 time
units (Figure 38b) the percent power used dropped off to a
low steady state value. The system became nutrient limited


99
100


97
longest turnover time per unit) also rose to a steady state
value. During this time, the productivity climbed to a
local maxima, then dropped slightly, finally climbing to a
slightly higher steady state. The percent power used for
the whole run was 95.5%.
In order to test the role of each of the producers
early in the simulation, a series of runs were made with the
initial condition of one of the producer species set to zero
(Figure 29 a,b,c). With no initial climax species (Ql)
present (Figure 29a) the shrub species (Q2) became dominant
in the final steady state. The percent power used for the
run was 94.7%, slightly less than the base run configura
tion. This configuration did not support as high a level of
consumer (Q4) compared to the base model run (75.2 vs.
90.8) .
When the shrub species (Q2) was absent (Figure 29b),
the percent power used for the run and steady state values
for the consumers were similar to the base run. Without the
shrub species present to compete during the middle period,
the final climax species (Ql) peaked earlier and higher than
in the base run.
When the weed species (Q3) was initially absent (Figure
29c), the system was not self sustaining. The primary
reason was that during the early part of the simulation, the
consumer (Q4) was dependent on the weed species (Q3). With
no Q3 present, the consumer Crashed very quickly. The whole
system then crashed because the consumer feeds back in


4
oxidation-reduction reaction involving raalonic acid, brmate
and a cerium catalyst (Winfree 1973). An example of the
time and spatial development of this reaction is shown in
Figure la.
Morphological development in biological systems has
been studied and modeled by Meinhardt (1982). Patterns form
when autocatalytic growth in a system is combined with
lateral inhibition (negative spatial feedbacks). Once auto
catalytic activity starts, there must be a longer range
negative feedback (spatial inhibition of the spread of this
autocatalysis) or the whole system will pulse in a burst of
autocatalytic consumption. This sets up spatial chemical
gradients along which morphogenesis is thought to occur
(Figure lb) .
Hilborn (1979) experimented with predator-prey models
based on an aquatic ecosystem. Hilborn1s model had 100
spatial cells arranged in a linear chain with the ends
connected to form a circle. Both predators and prey were
allowed to diffuse across cell boundaries. The model was
simulated with initial conditions set so that all cells had
prey but only one cell had a predator. The model (Figure
2a) was allowed to iterate for 1000 time intervals, gen
erating the pattern seen in Figure 2b. Further experiments
showed that there was no tendency towards equilibrium in
longer runs of the model.
The spatial development of insect eyes and insect legs
has been modelled by Ransom (1981) using an autocatalytic


28
develop a distribution of trees based on all of the input
parameters. These gap models generally do not account for
any outside disturbances that generate gaps.
Various gap models (Phipps 1979, Shugart and West 1980,
Shugart, Mortlock, Hopkins, and Burgess 1980, Shugart and
Noble 1981, Doyle 1982, Doyle, Shugart, and West 1982,
Shugart 1984, and Pickett and White 1985) have been utilized
to study forested ecosystems around the world. These models
have various gap sizes ranging from 100m'"2 to 833m~2.
Spatial Systems and Models
A spatial predator-prey insect microcosm was used by
Huffaker (1958) to study two species of mites. The prey
mite fed on oranges while the predator mite fed on the prey.
In one set of experiments, the oranges were distributed in a
10x12 grid with partial barriers between the oranges and one
prey placed on each of the 120 oranges. Five days later 27
predators were dispersed on the oranges. The resulting dy
namics in populations both over time (8 months) and space
are shown in Figure 7. In other experiments with oranges in
different arrangements, the oscillatory behavior was not
seen. Huffaker concluded that the predator-prey oscillation
would only occur when there was migration from the outside
or a sufficiently complex spatial arrangement of prey and
barriers to allow localized growth of the prey followed by
consumption by the predator.


242
PROGRAM GRAPH2
C vers RGRF
C
C WRITTEN BY JOHN RICHARDSON
C
C CALL TO REGLIN ADDED 6/15/83
C
C MAXIMUM NUMBER OF DATA POINTS SET TO 250
C
C
C MODIFIED 4/11/83 FOR RGL LIBRARY
C
BYTE XTEXT(80),YTEXT(80),TITLE(80),ESC
BYTE FNAME(16),IFNAM(16)
LOGICAL IRFLAG,SMOOTH,SHADE
DIMENSION X(250),Y(250),Y1(250)
DATA TITLE/80*0/
DATA XTEXT/80*0/
DATA YTEXT/80*0/
DATA FNAME/16*0/
DATA IFNAM/16*0/
ESC=27
TYPE 10
10 FORMAT(' GRAPHING PROGRAM FOR GIGI'
&/' COMPLIMENTS OF JOHN RICHARDSON'/)
WRITE(5,105)ESC,ESC
105 FORMAT( 2X, A1, PrTM 1' A1, )
TYPE 111
111 FORMAT(' PROGRAM REQUIRES THE TT: BUFFER BE SET TO NOWRAP'//
&' SET /NOWRAP=TI:'//
&' IF THIS IS NOT DONE PLEASE EXIT PROGRAM AND CORRECT THIS'//)
TYPE 15
15 FORMAT(' IS DATA IN A DATA FILE? (1=YES, 0=NO, -1=EXIT) '$)
ACCEPT 16,IANSI
16 FORMAT(12)
IF (IANSI.LT.0)STOP'MAKE CHANGES AND RERUN'
IF (IANS1.EQ.0) GOTO 11
TYPE 161
161 FORMAT(' FILE NAME FOR DATA: '$)
ACCEPT 162, (FNAME(I),1*1,16)
162 FORMAT(16A1)
OPEN(UNIT=1,NAME=FNAME,TYPE*'OLD',FORM*'FORMATTED')
READ (1,1621)TITLE
1621 FORMAT(1x,80A1)
WRITE(5,1621)TITLE
READ(1,1622)NPAIRS
1622 FORMAT(1X,I3)
DO 1630 1=1,NPAIRS
READ(1,*)X(I),Y(I)
C1625 FORMAT(2G15 6)
1630 CONTINUE
N=NPAIRS
GOTO 51
11
TYPE 20


209
C
C
C
SUBROUTINE GGOFF
C THIS WILL SEND THE ESC NEEDED
C TO TURN OFF THE GIGI GRAPHICS MODE
BYTE ESC
ESC=27
WRITE(3/100)ESC
100 FORMAT('+', 1A1,' ')
RETURN
END
C
C
C
SUBROUTINE GGERA
C ROUTINE TO PERFORM SCREEN ERASE
WRITE (3,100)
100 FORMAT(,+',,S(E)')
RETURN
END
C
C
C
SUBROUTINE GGDMP
C ROUTINE TO PERFORM SCREEN DUMP TO LA34/LA100 PRINTER
WRITE(3,100)
100 FORMAT('+','S(H) )
RETURN
END
C
C
C
SUBROUTINE GGINIT
C ROUTINE TO INITIALIZE THE GIGI
WRITE(3,100)
100 FORMAT('+',"W(R,14,P1,NO,SO,A0)')
RETURN
END
C
C
C
SUBROUTINE GGAXIS(IX,IY,IFX,IFY)
C
C ROUTINE TO INITIALIZE THE AXIS OF THE GIGI
C WHERE IX = LOWER LEFT CORNER X VALUE
C IY = LOWER LEFT CORNER Y VALUE
C IFX= UPPER RIGHT CORNER X VALUE
C IFY= UPPER RIGHT CORNER Y VALUE
WRITE(3,100)IX,IFY,IFX,IY
100 FORMAT('+','S(A[',15,',,15,'][',15,',',15,'])')
RETURN
END
C


Figure 43. Simulation of pulse model (Figures 14 and 18)
with cells in a linear ring and a high level of diffusion
between consumers of each cell (DK=.l) and random distribu
tion of producers and consumers around ring. For each time
unit (e.g. A=0) density of producer and consumer in the
matrix is shown along with size class distribution. The
time series from A to 0 summarizes the temporal pattern of
totals in matrix.
>14000
12000
10000
000 /2
000
-4000
2000
Consumer
Iso oloeso*


o
5E-5
135


WRITE(4,4584)
4584 FORMAT(//' FINAL CONSUMER DISTRIBUTION')
WRITE(4,4585)((ICON(I,J),1*1,12),J=12,1,-1)
4585 FORMAT(12(1X,I6))
WRITE(4,4571)Q3TOT
4571 F0RMAT(//1X,'VALUES FOR CONSUMERS TOTAL CONSUMERS =',G15
WRITE(4,458)((I,Q3(I),X1(I),Y1(I),IXYZ(I)),1=1,NPROD)
458 FORMAT(1X,14,' Q3= ',F8.2,' X=',I2,' Y =',12,1X,I4)
CALL CLOSE(1)
END
C
C
C
SUBROUTINE SORT1(X,IX,N,SF)
DIMENSION X(1),IX(1)
DO 20 1=1,N
IX(I)=X(I)/SF
20 CONTINUE
DO 40 1=1,N
DO 40 J=I,N
IF(IX(J).LT.IX(I))G0T04 0
ITEMP=IX(I)
IX(I)=IX(J)
IX(J)=ITEMP
40 CONTINUE
RETURN
END


8082
81
82
83
84
841
842
85
86
9
11
91
92
121
122
12
13
C
14
15
150
232
TINT=ITINT
TINTC=ITINTC
WRITE (5,81)
CONTINUE
FORMAT(1 HOW LONG TO RUN? [ R] '$)
READ (5,82)TTIME
FORMAT(G6.0)
WRITE(5,83)
FORMAT(' WHAT IS DT [R] '$)
READ (5,84)DT
FORMAT(G10.6)
XDT=DT/TINT
X DTC= DT/TINTC
WRITE(5,841)
FORMAT(' WHAT IS NUTRIENT CONC. OF OUTER NONREACTIVE'/
& RING ( 39000 IC; 0.0 TO ? ) [R] '$)
READ (5,842)Q40IC
FORMAT(G16.5)
WRITE(5,85)
FORMAT(' WHAT IS DIFFUSION COEFFICIENT? [R] '$)
READ (5,86)DK
FORMAT(F8.5)
WRITE( 5,9)
FORMAT(' INPUT THE SEARCH LENGTH FOR PREDATOR [I] ',$)
READ (5,11)N
FORMAT(12)
WRITE(5,91)
FORMAT( 1X, FEEDING AND DOUBLING THRESHOLD [R,R] '$)
READ(5,92)PTHRSH,THRESH
FORMAT(2G8.2)
WRITE(5,121)
FORMAT(1X,'INPUT (0-SUCCESSION; 1-STEADY STATE) [I] '$)
READ(5,122)ISSUC
FORMAT(12)
WRITE(5,12)
FORMAT(' WHAT ENERGY TYPE WOULD YOU LIKE'/
+ '1: STD INPUT'/
+ '2: CONSTANT INPUT'/
+ '3: RANDOM INPUT',20X,'ENERGY TYPE [I] '$)
READ(5,13) IETYP
FORMAT(11)
IF(IETYP.EQ.1)GOTO150
WRITE(5,14)
FORMAT(' WHAT IS THE MEAN VALUE OF ENERGY '$)
READ(5,15)XMEAN
FORMAT(F5.2)
CONTINUE
GPP=0.
CNSUMP=0.
Q1TOT=0.
Q2TOT=0.
Q3TOT=0.
Q4TOT=0.
PROD=0.
!GPP COUNTER (TOTAL)
!CONSUMPTION BY CONSUMERS (TOTAL)
!AMOUNT OF PRODUCERS
¡AMOUNT OF STORAGE
¡AMOUNT OF CONSUMERS
¡AMOUNT OF NUTRIENTS
¡TOTAL PRODUCTION (GPP+CNSUMP)
! DK
! DK
! DK
! DK
! DK
! DK
! DK
! DK
! DK
! 3.0
¡3.0
¡3.0
¡3.0


117
When Q4 was below a certain threshold the system could
not be sustained regardless of the input energy. After an
initial threshold level of consumers was reached, the system
was stable, similar to that described above for Q3. Since
Q4 has a direct feedback on Ql, Q2, and Q3, the interaction
of these in the production term can determine whether or not
the system was stable. If the value of Q4 was too low then
there was little production and the system crashed.
Simulation of the Pulse model
Single Run Simulations
A simulation of the base run pulse model (Figure 14) is
shown in Figure 37a. As Q2 increased, the available carbon
or nutrient carbon tank (Q4) decreased proportionately. As
the stored biomass increased there was a threshold level at
which the consumer (Q3) began to grow rapidly and pulsed.
This pulse consumed Q2 and released the carbon back into the
available carbon pool (Q4). The threshold of pulsing was
dependent on the level of both Q2 and Q3. The level of Q3
before the pulse was, however, directly related to the level
of Q2 and the input diffusion pathway. After the pulse, the
consumer (Q3) decayed back to a low level.
The cycle repeats itself at a frequency of approximate
ly 325 years. The power used varied during the simulation
with the highest rate occurring shortly after the pulse,
when the nutrients have been concentrated in Q4 as available
carbon.


Figure 9. Energy circuit language symbols (Odum 1983).


Figure 50. Simulation of pulse model (Figure 14) with
cells arranged in two dimensions (Figure 18) without
diffusion and with a constant energy source. For each time
unit (e.g. A=0) density of producer and consumer in the
matrix is shown along with size class distribution. The
time series from A to 0 summarizes the temporal pattern of
totals in matrix.
Spatial paitara
CONSUMES
CELLS
Oaaalty
cala
O'"*2
,>14000
13000
j§ 10000
fix sooo
S# 000
W 4000
fff 2000
9/m2
%lm2


239
CHAR=' B'
WRITE (3,3985)CHAR l
CALL GGPLT(0/600,10,0)
CALL GGBOX(0,600,10,700,230)
CALL GGFILL(O)
CALL GGBOX(7,600,10,700,110)
CALL S0RT1(Q2,IX,144,400.)
CALL GGPLT(7,601,IX(1)+10,1)
DO 3991 IT1=2,100
CALL GGVEC(3,600+ITl,IX(IT1)+10)
3991 CONTINUE
C 1
C
C
4500 CONTINUE
C
C>>>> PLOT CONSUMERS <<<<<<
C !
C !
ICOLOR=0 !
IF((T-PTIMEC).LT.TINTC)GOTO 5000 !
PTIMEC=T
CHAR='B' !
WRITE (3,3985)CHAR !
3985 FORMAT(' + ','T( A1) W( S',1H',A1,1H',')') !
CALL GGBOX(ICOLOR,284,4,560,244) !
CALL GGFILL(O) !
CALL GGPLT(7,0,0,1)
CALL GGVEC(7,767,0)
DO 2005 11=1,T2 1
ICOLOR=Q3(I1)/50. !
IF(ICOLOR.GT.7)ICOLOR=7 !
CHAR=ICOLOR+48 !
CALL GGPLT(ICOLOR,X1(11)*24-4+284,Y1(11)*24-44,0) !
WRITE(3,398)CHAR !
2005 CONTINUE !
CALL GGBOX(7,600,130,700,230)
CALL SORT1(Q3,IX,T2,40)
CALL GGPLT(7,601,IX(1)+130,1)
DO 2017 IT1=2,T2
CALL GGVEC(7,600+IT1,IX(IT1)+130)
2017 CONTINUE
IF(IPTR.EQ.0)GOT05 000
WRITE (3,20171)
20171 FORMAT('+S(H)')
C !
C !
C
C SEE IF ITS TIME TO QUIT
5000 CONTINUE
C !DK
C DIFFUSION IDK
C SDK
IDK
QXT=0 0


Copyright 1988
by
John R. Richardson


243
20
30
35
36
59
60
50
601
602
603
604
606
608
611
511
51
5001
5002
501
502
503
504
FORMAT(' HOW MANY PAIRS OF POINTS TO PLOT $)
ACCEPT 30,N
FORMAT(13)
IF(I.GT.500)GOTO11
TYPE 35
FORMAT(1X,'DESCRIPTION OF DATA (UP TO 80 CHARACTERS '/)
ACCEPT 36,(TITLE(K),K=1#80)
FORMAT(80A1)
DO 50 1=1,N
TYPE 60,1
FORMAT(' X AND Y VALUES FOR POINT ',13,
' SEPARATED BY COMMAS [R] ')
READ (5,*,ERR=9911)X(I),Y(I)
CONTINUE
CLOSE(UNIT=1)
WRITE(5,601)
F0RMAT(//1X,'DO YOU WANT TO SAVE DATA (1-YES, 0
READ(5,602)ISAVE
FORMAT(11)
IF (ISAVE.NE.1)GOT051
WRITE( 5,603)
FORMAT(1 WHAT IS THE FILE NAME FOR THE DATA '$
READ(5,604)IFNAM
FORMAT(16A1)
CALL ASSIGN(2,IFNAM)
WRITE(2,606)TITLE
FORMAT(1X,80A1)
WRITE(2,608)N
FORMAT(1X,13)
DO 511 1=1,N
WRITE(2,611)X(I),Y(I)
FORMAT(1X,G15.6,' ',G15.6)
CONTINUE
CLOSE(UNIT=2)
XMAX=0.
YMAX=0.
A=0.
3=0.
R2=0.
CEE=0.
TYPE 5001
FORMAT(' DO YOU WANT TO RUN REGRESSION ON DATA? (1-YES,
ACCEPT 5002,IRGS
FORMAT(11)
IF (IRGS.NE.0)CALL REGLIN(N,X,Y,A,3,R2,CEE)
TYPE 501
FORMAT(' WHAT IT THE X- AXIS DESCRIPTION')
ACCEPT 502,XTEXT
FORMAT(80A1)
CALL STRIP(XTEXT,80)
TYPE 503
FORMAT(' WHAT IS THE Y-AXIS DESCRIPTION')
ACCEPT 504,YTEXT
FORMAT(80A1)
-NO)'$)
)
0-NO)')


193
Table 3.
Coefficient values for parallel production-consumption
model in Figure 13.
K1
.003
Production coefficient for
Q1
K2
.005
Production coefficient for
Q2
K3
.007
Production coefficient for
Q3
D1
.1
Drain coefficient for Q1
D2
.2
Drain coefficient for Q2
D3
.3
Drain coefficient for Q3
K7
.006
Consumption coefficient for
Q1
K8
.015
Consumption coefficient for
Q2
K9
.040
Consumption coefficient for
Q3
D4
.08
Drain coefficient for Q4
K0
.1
Intake coefficient for Q4
FI
.01
Feedback loss coefficient for Q4




102
Q4 1
Tme
Time
100.


INPUT POWER
PERCENT POWER USED
iuu>uimsjco(e
'm


201
PROGRAM SUCIO
C SUCGGX
C VERS 1.1
C FEBRUARY 5, 1984
C
BYTE FILE(16),ESC,DES(40)
REAL M1,M2,M3,M4,M9
REAL K1, K2 K3 K4, K5 ,K6,K7,K8,K9,K0,L1,L2,L3,J,J0
DIMENSION FILNAM(6),IY(50,200)
DATA FILE/16*0/
DATA DES/40*0/
FT1(A,B)=ABS(AINT(A/B)-A/B)
D(X,Y)=(X/Y)*ALOG(X/Y)
C
C
WRITE(5,100)
100 FORMAT(1X,' SUCGGM GENERATES 6 DATAFILES',
&/' BE SURE THAT THEY DONT ALREADY EXIST',
&/' WHAT IS THE DATA FILE FOR THIS MODEL RUN ?')
READ(5,101)(FILE(I),1=1,16)
101 FORMAT(16A1)
C
C
C WRITE(5,1011)
C1011 FORMAT(' WHICH Q TO SAVE (1,2,3,4,5=% POW USED,6=BIOMSS)'$)
C READ(5,1012)IQSAV
C1012 FORMAT(13)
WRITE( 5,1013)
1013 FORMAT(' WHAT IS THE INCREMENT IN JO? [R] '$)
READ(5,1014)XINC
1014 FORMAT(G15.5)
C
C
c WRITE(5,99)
C 99 FORMAT(' HOW LONG TO RUN? ')
C READ(5,98)TIME
C 98 FORMAT(F6.0)
TIME=100. ¡X.1
C WRITE(5,981)
C981 FORMAT(' DO YOU WANT A HARDCOPY? (1-YES,0-NO)'$)
C READ (5,982)ICOPY
C982 FORMAT(12)
OPEN(UNIT=1,NAME=FILE,TYPE='OLD',FORM='UNFORMATTED')
READ(1)E1,E2,E3,E4,E5,E6,E7,E8,E9,E10,E11,E12,E13,E14,E15,
C COEFFICIENTS **********************************************
+NUM,K0,K1,K2,K3,K4,K5,K6,K7,K8,K9,D1,D2,D3,D4,L1,L2,L3,F1
+,JO,Q1INIT,Q2INIT,Q3INIT,Q4INIT
CLOSE(UNIT=1)
C INITIAL CONDITIONS********************************************
XJ0INI=J0
NSLICE=25
NCNTS=150
1 CONTINUE
C


217
C SURFACE FOR RIGHT SHIFT
200 CALL GGPLT(COLOR,X6,Y6,1)
CALL GGVEC(COLOR,X7,Y7)
CALL GGVEC(COLOR,X1,Y1)
201 CONTINUE
C
C VERTICAL AXIS TICS
C
DO 20 1=0,10
IY0=(I*25)+Y2
IF(DELTAX.LT.0)GOTO19
CALL GGPLT(COLOR,X2,IY0,1)
CALL GGVEC(COLOR,X2-6,IY0)
GOTO20
19 CALL GGPLT(COLOR,X4,IY0,1)
CALL GGVEC(COLOR,X4+6,IY0)
20 CONTINUE
C
C HORIZONTAL AXIS TICS
C
DO 25 1=0,10
IX0=I*NPNTS*IXSCLE/10+X0
CALL GGPLT(COLOR,1X0,Y0,1)
CALL GGVEC(COLOR,1X0,Y0-6)
25 CONTINUE
C
C ANGLE AXIS TICS
C
ZLINE=NLINE
DO 35 ZI=0.,ZLINE,ZLINE/10.
IF(DELTAX.LT.0)G0T027
IX0=X0-DELTAX*ZI
IY0=Y0+DELTAY*ZI
GOTO28
27 IX0=X1-DELTAX*ZI
IY0=Y1+DELTAY*ZI
28 CONTINUE
IC4=ISIGN*6
CALL GGPLT(COLOR,IX0,IY0,1)
CALL GGVEC(COLOR,IX0-IC4,IYO-6)
35 CONTINUE
C
C BACK AXIS LINE
C
C CALL GGPLT(COLOR,X2,Y2,1)
C CALL GGVEC(COLOR,X4,Y4)
C
C TOP AXIS LINE
C
CALL GGPLT(COLOR,X3,Y3,1)
CALL GGVEC(COLOR,X6,Y6)
RETURN
END


Figure 24. Simulation of three path competition model with
various pathways enabled (Figure 12). Percent power used as
a function of energy input (JO).
Linear pathway:
Autocatalytic pathway:
Quadratic pathway:
J1=K1* R
J2=K2*Q*R
J3=K3*Q*Q*R


CHAPTER 2
METHODS AND MODELS
Ecosystem concepts, configurations, and models were
represented with energy circuit language from which simula
tion programs were derived. The energy circuit language,
developed by H. T. Odum (Odum 1971, Odum and Odum 1981 and
Odum, 1983), is a symbolic language for modelling ecosystems
and their components. Elements of storages, flows, and
interactions in this symbolic language keep track of the
laws of energy conservation. The energy diagrams also show
the correct kinetic interaction between parts of the system.
The level of aggregation or disaggregation that is needed to
understand and model a system for a particular purpose can
be achieved by drawing and revising diagrams using this
energy circuit language. A diagram of most of the important
symbols with a brief description of each is presented in
Figure 9.
One of the benefits of using the energy circuit
language is that it is possible to go from a conceptual
model to the development of the differential equations
needed to simulate the model in a few steps. Each of the
pathways on the diagram represents a flow that in turn can
be represented by terms in the differential equations that
37


216
COMMON /IAREA/MASK
IXOFF=200
IF(DELTAX.LT.0)IXOFF=50
ISIGN=DELTAX/1AB S(D ELTAX)
C
C
XO=IXOFF
Y0=20
C
X1=IX0FF+IXSCLE*NPNTS
Y1=20
C
X2=IXOFF-NLINE*DELTAX
Y2=20+NLINE*DELTAY
C
X3=X2
Y3=Y2+250
C
X4=X2+NPNTS*IXSCLE
Y4=Y2
C
X5=X0
Y5=Y0+250
C
X6=X4
Y6=Y3
C
XORG=X2
YORG=Y2
C
X7=X1
Y7=Y5
C
CALL GGPLT(COLOR,X6,Y6,1)
IF(DELTAX.LT.0)GOTO15
IYTEMP=MASK(X4)
CALL GGVEC(COLOR,X6,IYTEMP)
GOTO16
15 CALL GGVEC(COLOR,X4,Y4)
CALL GGVEC(COLOR,X1,Y1)
16 CALL GGPLT(COLOR,X1,Y1,1)
CALL GGVEC(COLOR,X0,Y0)
IF (DELTAX.GT.0)CALL GGVEC(COLOR,X2,Y2)
CALL GGPLT(COLOR,X2,Y2,1)
161 IF(DELTAX.GT.0)GOTO191
IYTEMP=MASK(X2)
CALL GGPLT(COLOR,X2,IYTEMP,1)
191 CALL GGVEC(COLOR,X3,Y3)
IF(ISIGN.GT.0)GOTO200
C SURFACE FOR LEFT SHIFT
CALL GGPLT(COLOR,X3,Y3,1)
CALL GGVEC(COLOR,X5,Y5)
CALL GGVEC(COLOR,X0,Y0)
199 GOTO201


174
Table 2. Area o£ gaps digitized from photographs of Luquillo
tropical rain forest.
Picture number
1
2
5
8
Total
Number of gaps
53
43
35
74
205
Mean
0.0761
0.0517
0.0455
0.0473
0.0554
Std. Error
0.0212
0.0096
0.0043
0.0083
0.0066
Minimum
0.0093
0.0125
0.0094
0.0067
0.0067
Maximum
1.121
0.360
0.100
0.5525
1.121
Area % of total
5.38
2.97
2.12
4.66
3.785
* Means are not significantly different p=.005


214
READ(5,622)DELTAX,DELTAY,IXSCLE
622 FORMAT(314)
C
C
WRITE(5,623)
623 FORMAT(' WHAT IS CROSS HATCH INTERVAL [I] '$)
READ(5,624)NXHTC
624 FORMAT(13)
C
C
DELTAX=DELTAX*ISHFT
171 CALL GGON
CALL GGINIT
CALL GGAXIS(0,0,767,479)
CALL GGERA
C CALL GGBOX(7,0,0,767,479)
C
C SET UP DATA POINTS FOR X AXIS
C
DO 19 NPOINT=1,NCNTS
IX(NPOINT)=NPOINT*IXSCLE
19 CONTINUE
nl.ine=1
DO 11 NPOINT=1,NCNTS
IOUTY(NPOINT)=IY(IABS(NLINE-NSLICE*NRUN),NPOINT)/4
11 CONTINUE
DO 20 NLINE=1,NRUN,1
DO 10 NPOINT=1,NCNTS
IOUT(NPOINT)=IY(IABS(NLINE-NSLICE*NRUN),NPOINT)/4
10 CONTINUE
CALL GG3DX(COLOR,IX,IOUT,IOUTY,NCNTS,NLINE,DELTAX,DELTAY,NXHTC)
DO 30 NPOINT=1,NCNTS
IOUTY(NPOINT)=IOUT(NPOINT)
30 CONTINUE
20 CONTINUE
CALL AXIS(COLOR,NRUN,NCNTS,DELTAX,DELTAY,IXSCLE)
C
C
CALL GGOFF
WRITE(5,2000)ESC,FNAM
2000 FORMAT(' + ',A1,1[ H' /' ',20A1)
WRITE(5,2001)ESC
2001 FORMAT('+',A1,'[H 0-QUIT, 1-SCREENDUMP, 2-SCRDMP NO LABEL '$)
READ(5,2002)IANS
2002 FORMAT(12)
IF(IANS.EQ.0)GOTO2100
WRITE(5,2004)ESC
2004 FORMAT(,+',Al,,[H,,80X)
IF (IANS.NE.2)GOTO 20035
WRITE(5,2005)ESC
2005 FORMAT('+'A1,'[H',80X/80X)
20035 CALL GGON
WRITE(5,2003)


151
Simulation of Two Dimensional Surface Models
The simplest simulation of the two dimensional pulsing
model, with no diffusion and an evenly distributed energy
source (Figure 50), had a time series output identical to
the basic pulse model (Figure 37a). Even though the model
was disaggregated into 100 cells, each of the cells was
identical. In this run, each of the cells was synchronized
(by the initial conditions) and the pulsing was based only
on the internal frequency of the model (T=250, 50E and
T=600, 50L). There was little change in the size distribu
tion of the producers and the consumers during the
simulation.
The influence of an energy source that is hierarchical
ly distributed from the center of the matrix outward gen
erates a different pattern (Figure 51). The production was
higher in the center of the matrix than at the outer edges.
In this simulation without diffusion there was no edge
effect. The first pulse came at the center of the matrix
(highest input energy) and then moved outward to the edge in
a series of pulses. The production and consumption then
continued to oscillate. The frequency of pulsing in each
individual cell depended on the intensity of the energy
input to that cell (see also Figure 40a-f). The center
cells pulsed at a higher frequency than the outer cells due
to differences in input energy. The time series of the


34
included size class distributions taken in 1943, 1946, 1951
and 1976 (Figure 8)-. It can be seen that there is a shift
over time in the different size classes. The peak year for
the 0-8 cm class is 1946 while the peak in the 3-12 cm class
occurs in 1951 and the peak in the next three size classes
occurs in 1976. The smallest number in the lower two clas
ses also occurs in 1976. The last severe hurricane struck
Puerto Rico in 1932, and this movement through the size
classes appears to be the growth and development of an age
class of trees that grew back after the hurricane. The
hurricane in this case acts as an organizing disturbance to
reset succession of patches on a large scale.
The models simulated include the main integrative mech
anisms observed in ecosystems for coupling production and
consumption of spatially distributed units. Energy use of
these configurations was obtained from the simulations to
test the hypothesis that commonly observed organizational
designs with a successional regime that alternates pro
duction and consumption, tend to maximize system power in
the long run.


213
PROGRAM PLOTZ
C
C
C **6/27/83
C CHANGED AXIS ROUTINE FOR THREECORNERED ORIGIN**
C
C VERSION 1.6
C WRITTEN BY JOHN RICHARDSON
C APRIL 27, 1983
C
C SURFACE PLOTTING PROGRAM
C
DIMENSION IY(50,200),IOUT(200),IOUTY(200)
DIMENSION IX(200),MASK(800)
BYTE FNAM(20),DES(40),GON(3),GOFF(2),COLOR
BYTE BLACK,BLUE,RED,MAGENT,GREEN,CYAN,YELLOW,WHITE,ESC
INTEGER DELTAX,DELTAY
COMMON /IAREA/MASK
DATA FNAM/15*0,'.','D','A','T',0/
DATA MASK/800*0/,DELTAX/6/,DELTAY/6/
BLACK= 0
BLUE=1
RED=2
MAGENT=3
GREEN=4
CYAN=5
YELLOW=6
WHITE=7
COLOR=GREEN
IXSCLE=3
ESC=27
WRITE(5,499)ESC,ESC
499 FORMAT('+',A1,1PrTM11,A1,' ')
C !SET TERMINAL TO ANSII MODE
TYPE 500
500 FORMAT(1X,'INPUT FILE NAME ')
ACCEPT 501,(FNAM(I),1=1,15)
501 FORMAT(15A1)
OPEN(UNIT=1,NAME=FNAM,FORM='UNFORMATTED',TYPE='OLD')
READ(1),(DES(I),1=1,40),NRUN,NCNTS
+,((IY(J,K),K=1,NCNTS),J=1,NRUN)
CLOSE(UNIT=1)
TYPE 5
5 FORMAT(' REVERSE THE SLICES? (1-YES, 0-NO) ')
ACCEPT 6,NSLICE
6 FORMAT(15)
WRITE(5,61)
61 FORMAT(' SHIFT SUCCESSIVE SLICES (1 = LEFT, -1 = RIGHT) '$)
READ (5,62)ISHFT
62 FORMAT(13)
C
C
WRITE(5,621)
621 FORMAT(' WHAT ARE THE VALUES FOR DELTAX, DELTAY, IXSCLE [I] '$)


78
at any given input was highest with all three pathways pres
ent. For any combination of pathways that contained the
quadratic path (J1+J2+J3 or J2+J3 or J1+J3), power used
increased with power input to reach the same asymptote (>95%
power used). A slightly lower level was reached for path
ways dominated by the autocatalytic pathway (J2 or J2+J1).
This asymptote was approximately 90% power used with in
creasing power input. With only the linear pathway enabled,
no change occurred in percent power used with increasing
power.
A unique situation occurred when the quadratic pathway
(J3) existed alone. A low initial storage (Q) did not pro
vide enough feedback on the J3 pathway to allow growth.
Percent power used was never significant. The simulation
with only J2 and J2+J3 showed zero percent power used at low
input levels, then rose quickly at higher input power.
The size of the storage (Q) was varied to see the
effects on energy usage (Figure 22). This was achieved by
varying the depreciation coefficient (K4) in multiple run
while increasing power input in the three path model. At
high values of K4 (fast turnover times), increases in per
cent power used at steady state with increasing power were
small. With decreasing values of K4 (slower turnover
times), percent power used increased for the initial and
final values of input power.
The addition of multiple drains with different struc
tures (Figure 11) did not have as great an effect on the


I J
O Q2/Q4 48000
o ppu mo
i
Q2/Q4 48000
i l
0 PPU 100
I 1
o PPU 100
119


45
This model was simulated in BASIC (program THREEPATH in
Appendix) on a Digital Equipment Corporation (DEC) PDP 11/34
using a DEC VK-100 graphics terminal (GIGI). Measurements
were made of the percent of the input power used while
applying various levels of input power and varying the
frequency of input power. Simulation runs were also made
with one or more of the three pathways set to zero to deter
mine the impact of the various pathways on the overall
system behavior and power utilization.
In conjunction with the three path model in Figure 10,
a similar model with the same inputs but with additional
higher order drain pathways was simulated to determine the
effects on total power usage (Figure 11). In any system
that has crowding effects or high storage costs, these
drain pathways may determine how the system processes energy.
The model has a linear drain, an autocatalytic drain and a
quadratic drain.
The basic three path model was tested for the effects
of size and turnover time on the percent power used for
various power inputs by varying the drain coefficient (K4 on
Figure 10) in multiple simulation runs.
The percent power used when the three path model com
petes with individual storages with single pathways (Figure
12) was also simulated to see how the various pathways may
help or hinder a system. The competitors are individual
tanks with single pathways corresponding to the three path
ways in the three path model.


Figure 4. Basic autocatalytic model with flow-limited
energy source.
(a) Diagram with kinetic terms
dQ = Kl*JR*Q K2* JR*Q K4*Q
JR = JO / (1+K0*Q)
(b) Diagram with flow terms
RO = X0*JR*Q
R1 = Kl*JR* Q
R2 = K2*JR*Q
R3 = K3*JR*Q
R4 = K4*Q


h co c( c;:j -t' -h oj CM o
OlO-iHO-^C
HOOJIO
CO CM CTj -t< H CO CM O rt<
rfiOffiOD (C
'CB0H<0
o -oi" ooo< > ** <: :>o o
OCtOQC
-OIIII"-: >00
< > < > o *- < :> < :> : > <:> ** <: >
ooooc
>o<
**< -.imn- .
ooo<
9299#oo6#
GO#OQ-C
>000
< > > o < > < > ** c > o <: >
oooo-oiiik::
"IIIH--
** : >o-o oc:
**' >f)OO#00(X :
O><'
**< X ><">
Pm Pm Pm Pm Pm Pm Pm Pm Pm Pm
Pm I~m Pm Pm Pm Pm Pm Pm Pm Pm
.mu' " 1 :'H 1 '! 1 : "'i: 1
********************
********************
OIIIMII!"
oil" -mili" ** -t|i.o -Tjkra
-Tjk Q t* h rj tj k r j -t| k i :i -r|krj
c> o o ooc-r o cz> CO
WUO 'TO'-IUJPJWU ooimooooo 0+Z>
Ol
OJCft Q.f O) -rtO-JCfe^n > CO O 1. 4 CO n -i
o> -'tOK-** we -tot cs co o c6 -o 1^3 co?


xml version 1.0 encoding UTF-8
REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd
INGEST IEID EAF2HW2NT_LR5QN7 INGEST_TIME 2015-03-27T19:23:34Z PACKAGE AA00029747_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES


166
- 1
IIll
/

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kfrM
m u
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|| fiis
WW
1:p fl
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II :: : :
r


123
because most of the nutrients were tied up in the stored
biomass.
When the pulse model was run without feedbacks into Q4
(pathways R6 and R8 cut off) the model continued to pulse
but began to decline (Figure 39). The percent power used
dropped as the level of Q4 dropped until one final pulse and
then everything decayed to a low steady state condition.
Multiple-run Simulations
When the input power was increased, the result was most
noticeable on the stored biomass (Q2) and the consumer (Q3,
Figure 40). At low values of JO there was no pulsing within
the time frame of the simulation (760 years). As JO was
increased, the pulsing began as a result of the stored
biomass (Q2) increasing to a threshold level at which Q3
pulsed and consumed the stored biomass (Q2). As JO was
further increased, the pulsing frequency increased. At high
levels of JO the first pulse decayed and the system switched
to a steady state with Q2 being maintained at a low level
(see Figure 37c for example). The total power used (Figure
40e) increased linearly as JO increased with small fluc
tuations over time due to the pulsing of Q3. The percent
power used (Figure 4 0 f) was less than 80% for low values of
JO then rose rapidly through the pulsing and leveled off as
JO approached 250. The percent power used was reduced by
the initial consumption but returned to a maximum after the
pulse. The percent power used increased as the available


49


ACKNOWLEDGEMENTS
I would like to express my sincere appreciation to Dr.
Howard T. Odum, my committee chairman, for the insights and
inspiration he gave during the completion of this work. His
holistic views and open-mindedness provide an extremely
fertile field to develop and pursue ideas in systems
ecology. Other members of my committee (Drs. J.F.
Alexander, G.R. Best, K.C. Ewel and C.L. Montague)
provided useful feedback in class and with this
project.
The support and patience of my wife Karen has
sustained me while my two children, Matthew and James,
have provided joy and purpose for the completion of
this dissertation.
Work was done in the Department of Environmental
Engineering Sciences, University of Florida, and was
supported by graduate research funding from the
Graduate School of the University of Florida.


Figure 52. Simulation of pulse model (Figure 14) with
cells arranged in two dimensions (Figure 18). Energy
source is hierarchically distributed from center outward and
diffusion is between consumers of each cell (DK=.001). For
each time unit (e.g. A=0) density of producer and consumer
in the matrix is shown along with size class distribution.
The time series from A to 0 summarizes the temporal pattern
of totals in matrix.
Sn.tlal si,, ,1a.
distribution
1000 1
fcL
l
10 CELLS
10 CELL
40.0001
I
S\ duo.,.
10
10
I >14000
tsooo
10000
000
000
§5 4000
*** 2000
_Jo
Con.ua>.,
1*0 olo.so*


PERCENT POWER USED
CO
O


153


Figure 60. Character set for displaying spatial graphs on
GIGI computer terminal suitable for use with screen copy
onto printer. Each dot pattern is represented by the hexa
decimal code on the left edge of each plot.
(a) 80 dots
(b) 40 dots
(c) 27 dots
(d) 20 dots
(e) 16 dots
(f) 12 dots
(g) 7 dots
(h) 3 dots


161
ferent from the random ring simulations (Figures 48 and 49)
in that the input energy was randomly distributed while in
the case of the ring model the initial producer-consumer
pairs were randomly distributed. Little synchronization of
the matrix occurred because the random energy distribution
caused locally high concentrations of producers every time
there was a pulse. In the simulation of the ring with
randomly distributed producers and consumers, at a high
level of diffusion the pulse wave moved fast enough to
reset all of the producers and consumers to similar values.
With a low diffusion value, the wave traveled so slowly that
it did not get around the ring, and multiple foci of pulsing
developed.
Simulation with diffusion between the nutrient com
partments of each cell (Q4) of the model instead of to the
consumers (Q3) can be seen in Figure 54. With a random
distribution of energy and a high level of diffusion
(DK=0.1) the pulsing was almost totally uncoupled. By the
end of the run (T=750, Figure 540) there was constant pul
sing in one cell or another, and the overall level of pro
ducers as seen in the time series graph was fairly constant.
The spatial configuration of the model was also tested
with a moving consumer. This is similar to the diffusion
runs of the model but represents an active process with
discontinuous (non-uniform) movement of consumers from cell
to cell. The consumer was allowed to search for the largest
producer to consume before moving. The model was tested


CALL GGPLT(4,ITIME,IXC, 1)
IXOQ2/SFACT
CALL GGPLT(3,ITIME,IXC,1)
IXOQ3/10.0
CALL GGPLT(1,ITIME,IXC,1)
IXC=Q4/100.
CALL GGPLT(5,ITIME,IXC,1)
IXC=(J0-J)*(250./J0)
CALL GGPLT(2,ITIME,IXC,1)
C
C FIND WHICH Q TO SAVE IN ARRAY
20000 CONTINUE !GOTO(21000,22000,23000,24000,25000,26000)IQSAV
C GOTO110
21000 IY(1,NRUN,INT(T/5.)+1)=INT(Q1/2.)
C GOTO110
22000 IY(2,NRUN,INT(T/5)+1)=INT(Q2/20)
C GOTO110
23000 IY(3,NRUN,INT(T/5.)+1)=INT(Q3/2.)
C GOT0110
24000 IY(4,NRUN,INT(T/5.)+1)=INT(Q4/40.)
C GOTO 110
25000 IY(5,NRUN,INT(T/5)+1)=INT((J0-J)*5.)
C GOTO110
26000 IY(6,NRUN,INT(T/5)+1)=INT((POWUSE-80)*50)
110 CONTINUE
M1=AMAX1(Ml,Q1)
M2=AMAX1(M2,Q2)
M3=AMAX1(M3,Q3)
M4=AMAX1(M4,Q4)
IF(T.LT.TIME)GOTO 5
C
ENCODE(80,11003,YTEXT)NRUN,ALPHA(IVAL),VARS(IVAL)
+,EUSED,100*EUSED/PAVAIL
11003 FORMAT(2X,12,' VARIABLE ',A4,' = ',G15.4,' POWER USED ',
+G12.6,' PPU: ',G12.6)
IF (IPLOT.EQ.0)WRITE(4,11004)YTEXT
11004 FORMAT(1X,80A1)
IF (IPLOT.EQ.1)CALL GGTEXT(7,0,460,YTEXT,1,0)
IF (ICOPY.EQ.1)WRITE(3,11001)
11001 FORMAT('+S(H)')
IF(NRUN.LT.NSLICE)GOT050 0
IF (IPLOT.EQ.0)GOTO24999
CALL GGERA
CALL GGOFF
24999 IQSAV=1
24995 CONTINUE
CALL ASSIGN(2,1PULSAV*,6)
ENCODE(40,25001,DES)IQSAV,FILE
25001 FORMAT(1X,' TANK Q',11,' FOR DATA FILE ',3A4)
WRITE(2)(DES(I),1=1,40),NRUN,NCNTS,
+ ((IY(IQSAV,J1,K),K=1,NCNTS),J1=1,NRUN)
CLOSE(UNIT=2)
IQSAV=IQSAV+1
IF (IQSAV.LT.7)G0T024995
!Q1 GREEN
!Q2 MAGENTA
!Q3 BLUE
!Q4 CYAN
¡POWER USED RED


91
When the input power was raised to 10000 (Figure 27)
the percent power used went up for all combinations of
pathways except the linear path. The quadratic pathway was
operational at this high power level but with a significant
minimum at 2 cycles per run. Other combinations had small
minima and maxima that are hard to see at the scale of this
graph.
The response of the model to various frequencies and
power input is shown in Table 1. Simulation runs with
pathway J1+J2 had a maximum in percent power used at all
three power inputs while the combination of J2+J3 had a
minimum in percent power used at all three power inputs.
The combination of all pathways (J1+J2+J3) has a peak of
maximum percent power utilization at low power and low
frequency input. At higher power levels percent power util
ization (with all three pathways enabled) was lower with
some shifting in the frequency at which this occurs.
Simulation of Parallel Production-Consumption Model
Single Run Simulations
The parallel production model showed a successional
pattern with the initial dominant species (Q3, with the
fastest turnover) growing up, then declining as Q2 became
the dominant species and finally Q1 (with the slowest turn
over) reached a maximum and then dropped back to a slightly
lower steady state (Figure 28). The consumer (Q4, with the


56
and flows to calibrate it to a tropical rain forest ecosys
tem. The original model was run on an Electronics
Associates Incorporated model 2000 Analog/Hybrid computer.
The models presented in this dissertation were simulated on
a DEC PDP 11/34. The multiple simulations of the pulse
model were generated with a version of the program that
would run 25 simulations while varying a coefficient or
initial condition over those 25 runs and generate data files
that were then displayed with the FORTRAN program PLOTZ (See
Appendix). The source listing of the FORTRAN pulse program
is in the Appendix.
The pulse model was calibrated with tropical forest
ecosystem values for carbon flows and storages (Jordan and
Drewry 1969, Odum and Pigeon 1970, and Brown, Lugo, Silander
and Liegel 1983). The energy diagram of the model is given
in Figure 14 and the equations, coefficients and initial
conditions of the state variables are given in Appendix
Table 4.
Pulse Model With Prey-Predator Sectors
An additional higher trophic level consumer was added
to the pulsing consumer model (Figure 14) in order to test
the relationship of turnover time and hierarchical matching
of consumers (Figure 15). The extra consumer added to the
model had the same structure as the lower level pulsing
consumer (Q3), with both linear and quadratic pathways.
This model was tested by varying the turnover time of the


27
is based on the nonlinear 'Lotka-Volterra equations' and
generally does not include outside influences. The other
uses forced linear systems of differential equations and
does have inputs from outside the system. Neither of these
methods typically contains any spatial considerations and
both deal with systems near equilibrium. Systems near equi
librium tend to move toward that equilibrium and are char
acterized by spatial uniformity (Prigogine 1984 and Field
1985).
In this study, open non-equilibrium models are de
veloped that combine non-linear and -oscillatory interactions
between production and consumption with outside forcing
functions that provide resource controls. A pulsing, hier
archical model of production and consumption is used to
generalize about
succession and
reg
ression.
Spatial inter
actions generated
by this
model
are
studied
to understand
the energetic and
kinetic
basis
for
pattern
formation in
ecosyterns.
Gap Models and Patch Dynamics
Several previous studies based ecosystem models on
disturbance gaps. The JABOWA forest simulator model by
Botkin, Janak and Wallis (1972) keeps track of the birth,
growth, and death of a group of trees from seedlings on to
maturity within a certain gap size. Subroutines are used
for crowding, shading, and response to individual nutrients
and energy sources. The simulation then allows the gap to


143
m
1 11 J
1 II J
1 II J

Ul 1
Ip
|garro
J :
: S:: :
WBMMW333J
u
r j
< a

¡
L
Li? mam a
Q
D
m
i
X


Figure 54. Simulation of pulse model (Figure 14) with
cells arranged in two dimensions (Figure 18). Energy
source is randomly distributed and diffusion between
nutrient storages (Q4) of each cell is set to high level
(DK=.l). For each time unit (e.g. A=0) density of producer
and consumer in the matrix is shown along with size class
distribution. The time series from A to 0 summarizes the
temporal pattern of totals in matrix.


a
>14000
a
>380
M
12000
M
300
4
10000
280
ooo g/2
200
000
180
V.w
4000
100
2000
;;;
80
oduO
0
m Co
0


Figure 20. Energy utilization of individual components in
the three path model in Figure 10. Input power is increasing
through time.


APPENDIX
192


Figure 53. Simulation of pulse model (Figure 14) with
cells arranged in two dimensions (Figure 18). Energy
source is randomly distributed and diffusion is between
consumers of each cell (DK=.001). For each time unit (e.g.
&=0) density of producer and consumer in the matrix is shown
along with size class distribution. The time series from A
to 0 summarizes the temporal pattern of totals in matrix.
I >14000
uooo
10000
000
000
$*§ 4000
aooo
/*
__vjo L_Jo
Produaar Conauar
ilia olaaaaa alza olaaaaa


218
1 PROGRAM MEASURE3 WRITTEN BY JOHN R. RICHARDSON
2 VERSION=2!
3 VERSION 1.0 BASELINE SET 6/1/85
5 MAIN PROGRAM BEGINS AT LINE 1000
6 VERSION 1.5 6/3/85
7 CLEANED UP OLD FORTRAN CODE, ADDED DOUBLE OUTPUT FILE MODE
8 SAVE TRUE DIGITIZER VALUES FOR PLOTTER FILE OUTPUT
9 CALCULATE AREAS BASED ON SCALED DATA
10 VERSION 1.6 6/7/85
11 FIXED ERROR IF NO FILES OF B DRIVE,
12 CHANGED DATA ARRANGEMENT IN OUTPUT FILES
13 VERSION 1.7 6/24/85
14 ADDED ERROR OUTPUT ROUTINE FOR ERRORS OTHER THAN NO FILES
15 VERSION 2.0 ADDED SCALE3 SUB FOR DIFFERENT XSCALE AND YSCALE
16 7/2/85
99 *****************************************************************
100 SUBROUTINE DIGINI (LINE 3000-3490) OPENS DIGITIZER
110 AND SETS INITIAL PARAMETERS FOR PROGRAM
130 SUBROUTINE STREAM MODE (3800-3899) TURNS ON STREAM MODE
150 SUBROUTINE POINT MODE (3900-3999) TURNS ON POINT MODE
170 SUBROUTINE FILE HANDLER (4000-4220) OPENS DATA FILE FOR OUTPUT
190 SUBROUTINE SCALE2 (5000-5610) HANDLES SETTING UP USER
200 COORDINATES AND ORIGINS AND OFFSETS
220 SUBROUTINE DELAY (6000-6010) ARE TIMING ROUTINES THAT
230 MAY BE NEEDED FOR SENDING SETUP INFORMATION TO DIGITIZER
250 SUB INPUT (8000-8070) GETS DATA SENT FROM DIGITIZER
270 SUB DIGURU (9000-9070) SCALES DIGITIZER INPUT TO REAL WORLD
290 COORDINATES
300 SUBROUTINE AREAP (1660-2030) GETS INPUT POINTS FOR AN AREA
330 SUBROUTINE AREAX (2070-2580) CALCULATES THE AREA
350 SUBROUTINE PERIX (2610-2810) CALCULATE THE CLOSED AND OPEN
370 PERIMETERS FROM A SET OF GIVEN POINTS
1000 '**************************************************************
1010 '************************* MAIN PROGRAM START *****************
1020 '**************************************************************
1040 ON ERROR GOTO 20000
1060 CLS:PRINT:PRINT:PRINT:PRINT
1070 PRINT "AREA MEASUREMENT PROGRAM VERSION ";VERSION
1120 GOSUB 4000:' CALL FILE HANDLER
1130 GOSUB 3000:' CALL DIGINI
1140 GOSUB 5000:' CALL SCALE2
1260 'CONTINUE
1270 GOSUB 1660:' CALL AREAP (X,Y,AREA,NPOINT,IERR,RESOL)
1280 IF (IERR= 0) THEN GOTO 1290 ELSE PRINT "ERROR ";IERR;"
HAS OCCURRED NOT ENOUGH POINTS FOR AN AREA":GOTO 1270
1290
1340
1350
1351
1370
1380
1390
1400
1410
GOSUB 2070:' CALL AREAX(X,Y,AREA,NPOINT)
GOSUB 2610:' CALL PERIX(X,Y,PERI1,PERI2,NPOINT)
AREA=AREAIO
FOR ";NPOINT;"
PRINT THE MEASURED AREA IS [";AREAIO;"]
PRINT CLOSED PERIMETER= ";PERI2
PRINT OPEN PERIMETER= ";PERI1
PERIOUT=PERI2
PRINT BELL$;" KEEP THIS AREA OR RE-MEASURE [K or R]
POINTS"


23
previous ecosystem regenerates a facsimile of the original
ecosystem through a sequence of unidirectional stages that
reaches a steady state system called a climax. This climax
may be arrested at some point and in some cases succession
may cycle between several stages. This definition is
broader than most but is an attempt to describe the whole
process instead of the more narrow 'growth-phase1'
definition.
Regression from a climax state may occur in several
ways. In some cases it comes about as a pulse of con
sumption from within the ecosystem boundaries such as tree-
falls, landslides or disease outbreaks. It can also come
about from disturbances from larger outside events such as
hurricanes or drought. The frequency and amplitude of these
disturbances tend to be inversely correlated: larger dis
turbances occur less frequently than smaller ones. This
phenomenon is referred to as a hierarchy of disturbances
(Bennett and Chorley 1978). The interaction of these dis
turbances along with the internal fluctuations may lead to
the 'bewildering variety of patterns' to which Horn refers.
Edges
Ecosystems can generally be broken up into subsystems
that have uniform characteristics. These subsystems have
boundaries where the composition changes from one particular
type to another. The development of these edges may occur
where differing types of energy interact with ecosystem
components to generate patches and zones of transition. The


20000
21000
22000
23000
24000
25000
26000
C
C
C
C
C
C
C
C
C
C
C
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
1101
11011
25001
GOTO(21000,22000,23000,24000,25000,26000)IQSAV
G0T01101
IY(NRUN,INT(T*1.5+1))=INT(Q1/.5)
GOTO1101
IY(NRUN,INT(T*1.5+1))=INT(Q2/.5)
GOT01101
IY(NRUN,INT(T*1.5+1))=INT(Q3/.5)
GOTO1101
IY(NRUN,INT(T*1.5+1))=INT(Q4/.5)
GOTO1101
IY(NRUN,INT(T*1.5+1))=INT(PERCNT*1000)
GOT01101
IY(NRUN,INT(T*1.5+1))=INT(BIOMSS/.5)
GOT01101
SKIP PLOTTING IN THIS VERSION
GOTO 1101
IF(FT1(T,.1).GE.DT)GOTO 1101
IF(ITCNT.LT.IPLOT)GOT01101
ITCNT=0
IX=T*7
IY=Q1
CALL GGPLT(6,IX,IY,1)
IY=Q2
CALL GGPLT(1,IX,IY,1)
IY=Q3
CALL GGPLT(2, IX,IY,1)
IY=Q4
CALL GGPLT(3,IX,IY,1)
IY=Q9/DT
CALL GGPLT(4,IX,IY,1)
IY=BIOMSS
CALL GGPLT(5,IX,IY,1)
!YELLOW FOR CLIMAX SPECIES
¡BLUE FOR TRANSITIONAL SPECIES
¡RED FOR WEEDS
¡MAGENTA FOR CONSUMERS
¡GREEN FOR PRODUCTIVITY
¡CYAN FOR BIOMASS
IY=EUSED*100/JO ¡SCALE EUSED TO 0-100
CALL GGPLT(7, IX,IY+350,1) ¡PLOT POWER USED WHITE
IY=(DRAIN/DT)* 100/JO ¡SCALE DRAIN TO 0-100
CALL GGPLT(5,IX,IY+350,1) ¡CYAN FOR DRAINS
IY=DIVERS*50
CALL GGPLT(2, IX,IY+350,1)
IF(T.LT.TIME)GOTO 5
WRITE(5,11011)JO,Q1,Q2,Q3,Q4,PERCNT*100.,BIOMSS
FORMAT(1X,7(1X,F10.4))
IF(NRUN.LT.NSLICE)GOT02
CALL ASSIGN(2,'SUCMANY',6)
ENCODE(40,25001,DES)IQSAV,FILE
FORMAT(1X,' TANK Q',11,' FOR DATA FILE ',16A1)
WRITE(2)(DES(I),1=1,40),NSLICE,NCNTS,
((IY(JCNT,KCNT),KCNT=1,NCNTS),JCNT=1,NSLICE)
CLOSE(UNIT=2)
WRITE(5,1061)IQSAV


CHAPTER 4
DISCUSSION
Many of the characteristics of ecosystem function were
generated by the simulations in this dissertation. Energy
increased with growth. Net production alternated with pul
sing net consumption. Hierarchical patterns in space re
sulted from oscillations in time. Edge effects developed.
There were similarities with succession observed in nature.
Many characteristics of ecosystems were generated by mini
models that had autocatalysis, recycling, parallel pathways
of different order, spatial intercell exchanges and hier
archical distribution of time constants. In other words,
simple models emulated many features of more complex
ecosystems.
The spatial model in this dissertation differed from
many previous spatial ecosystem models that used individual
species growing and interacting together (Botkin, Janak and
Wallis 1972, Phipps 1979, Doyle 1982). This model was a
unit ecosystem model that combined all of the species into
compartmentalized production, consumption and nutrient
storages. This simplified the model but kept many of the
ecosystem characteristics.
178


PERCENT POWER USED
< : ^ i n ^ n i


Time 2000
125


32
on a combination of terrain relief, hydrology, and cor
related output from the simulation model.
Another approach to spatial modelling is to divide the
area into individual cells with a representative model in
each cell with some interaction terms among the individual
cells. This is the approach Costanza (1979) used in model
ling the economic development of South Florida.
Simulations with individual models for each cell have
certain advantages, because the interaction of neighboring
cells influences the outcome. A serious disadvantage where
the number of cells is large is the immense amount of
computer time required for the simulations. By making the
cell size larger this can be avoided, but loss of spatial
detail occurs as the cell size increases. The sub-cell
distribution modelling technique used by Pearlstine et al.
(1985) has just the opposite advantages and disadvantages.
The time requirements for simulation do not necessarily
increase as the area of cells is increased, but individual
intercell interactions are lost.
Plan of Study
Objectives
This study of energy use and pattern formation with
production consumption models has several parts:
First, the energetics of different pathway config
urations were tested using a series of minimodels. These
models were manipulated to determine the energy use of


Total Area per Class (m*m) Number of Gaps
26
Gap Distribution
Barro Colorado
Size Class (m^m)
Gap Distribution
Size Class (m^m)


200
Table 8. Percent power used as a function of different
energy input sources and diffusion rates. Edge effect model
with different levels of consumers (Q3) on outside (buffer)
edge of spatial matrix.
Diffusion rate
.001
.01
.1
Value of Q3
Energy
Percent Power Used
on outside
Distribution
edge
Hierarchical
96.6
96.5
96.1
O
o
Even
96.5
96.5
95.9
Random
96.6
96.5
95.9
Hierarchical
96.6
96.5
96.7
50.
Even
96.6
96.7
96.6
Random
96.6
96.7
96.6
Hierarchical
96.6
96.7
96.9
100.
Even
96.6
96.8
96.8
Random
96.6
96.9
96.9


Figura 17. Cell geometries considered for spatial models.
(a) Square matrix with each cell having 4 side
and 4 corner neighbors. Active 10x10 matrix
embedded in a 12x12 matrix. This one was
chosen for the spatial simulations.
(b) Hexagonal matrix with each cell having 6 side
neighbors.


60
Any ecosystem can be divided into edge and non-edge
(center) parts. The amount of edge in an ecosystem is
a function of the size and number of the individual patches
within it. For a given area, as the number of subunits
increases the percent of the subunits on the edge decreases
(see Figure 16) .
A 10x10 matrix was used in the spatial simulations,
giving 36% of the total in edge cells and 64% in non-edge
cells. This size model was chosen to reduce the edge and
yet be small enough to simulate in a reasonable time. Com
puter runs for this model lasted approximately 3 hours on a
PDP 11/34. A model with a center to edge ratio of 10:1
would need approximately 20 times as many cells. In order
to test the effects of edges on the model, a single layer of
cells was added around the outside edges of the 10x10
matrix, giving it a 12x12 total area (Figure 17). The outer
layer was not acted as a buffer to approximate conditions of
an edgeless system.
Arrangements of cells
In simulating a spatial model, many arrangements of
cells can be used. The simplest form used was a linear array
with cells arranged in a linear ring. For two dimensional
models the cell geometry chosen was a square. This was done
for several reasons:


Figure 42. Multi-run simulation of pulse model (Figure
14) with variation is turnover time of pulsing consumer.
(K12 varied from .01 to .5).
(a) Production unit (Ql).
(b) Stored biomass (Q2).
(c) Pulse consumer (Q3).
(d) Nutrient storage (Q4).
(e) Power used (JO-Jr)
(f) Percent power used 100*(JO-Jr)/(JO)


BIOGRAPHICAL SKETCH
John R. Richardson was born in Monett, Missouri, on
April 30, 1945. He attended school in Monett and graduated
from Monett High School in 1963. After four years at the
Missouri School of Mines and Metallurgy in Rolla, Missouri,
he graduated with a B.S. degree in chemistry. He attended
the University of Missouri in Columbia for two years study
ing biochemistry. In 1970 he began working at the Molecular
Virology Institute in St. Louis, Missouri.
In 1973 he started at the University of Florida and
graduated with a M.S. degree in environmental engineering
sciences. After working for several years with the Missouri
Department of Natural Resources, he returned to the Univer
sity of Florida to pursue a Ph. D. in 1973. He graduated
with a Doctor of Philosophy degree in 1988.
254


Figure 8. Size class distribution over time of plot of
trees in tropical forest at El Verde (Crow 1980).


67
The final variation was a model with production com
partmentalized as before in individual cells but with free
roaming consumers, not constrained by cell boundaries. One
consumer was allowed to consume and move about the matrix
according to a set of constraints. When the consumer grew
above a preset size, it was split into two equal halves and
each half was allowed to consume, move and split again. An
upper limit of 100 was placed on the total number of con
sumers that could be generated during the run (the total in
the 10x10 matrix of the previous model versions). This
model also had three different energy inputs and diffusion
of nutrients (Q4).
Format for Spatial Display Graphs
Data from the spatial pulsing model were displayed
using the format shown in Figure 18. The spatial distribu
tions of the producers and consumers were shown at various
times during the run (usually 50 years apart). The values
of producers and consumers in individual cells were repre
sented by the density of dots in the cell. The producer
density increment was 2000 g/m~2 with a range of 0-16,000
g/m-'2 while the consumer was represented by an increment of
50 g/m"2 and a range of 0-400 g/m~2.
Measurement of Hierarchies at El Verde Site
In order to compare hierarchical relationships that
were generated in the model with those occurring in the


189
synchronous organization of individual consumers (100 total)
over the entire matrix of cells mimiced the effects of a
large consumer with the same territory. The percent power
used for each of these simulations was the same (Table 5 and
6 in Appendix).
Coupling of the consumers from cell to cell by dif
fusion organized the consumer action over the whole matrix
depending on the strength of that coupling. Low levels of
diffusive coupling generated local areas of organization by
the consumers (Figure 53) while strong coupling organized
the disturbance over the entire matrix, (not shown but
similar to Figure 50).
Active coupling between subunits by consumers was simu
lated using a moving consumer model (Figures 55 and 56). In
this case, very different patterns were formed with a smal
ler number of consumers. The action of organizing the
entire landscape (10x10 matrix) was achieved with fewer
consumers. The energy use was not significantly different
from the other spatial simulations (Table 7 in Appendix).
The efficiency of active coupling may be higher than passive
(diffusion) coupling.
Organization at a higher level tends to have a larger
effect in generating patterns. Some of this may due to a
type of 'memory' generated in the landscape by the
disturbance-succession sequence generated by these pulsing
production consumption models. As the system pulses, small
differences between individual cells generate further dis-


15
flow into the structure. Mechanisms must develop that build
structure to capture the most energy possible. These feed
back structures then have a prior energy use embodied in
them (emergy, after embodied energy, of a structure has been
defined as the total amount of energy used in developing
these structures (Odum 1983 and 1986)). This dissertation
looks at some of the possible kinetic pathways that feed
back to process energy and the energetics of these pathways.
Pathway Configuration
A simple model demonstrates several ways in which use
ful power can be increased (Figure 4, see description of
symbols in Figure 9). This model is a single storage with
autocatalytic production drawing on a flow-limited energy
source (an energy source with constraints on the pathway,
limiting the amount of energy that can be delivered).
The efficiency of a pathway can be increased if less
energy is fed back to gain more energy. For a simple auto
catalytic system (Figure 4a and 4b) this can be done by
either using less energy to gain the same inflow (changing
the value of K2 in the model) or by increasing the inflow
for the same feedback (increasing K1 while concurrently
decreasing K3). Because there are thermodynamic limits on
any process, it may not be possible to improve designs to
increase energy flows beyond thermodynamic limits.
The first law of thermodynamics requires the conserva
tion of energy. This implies the following constraint on
the production process of the model (Figure 4).


Figure 51. Simulation of pulse model (Figure 14) with
cells arranged in two dimensions (Figure 18). Energy
source hierarchically is distributed from center outward and
no diffusion between cells. For each time unit (e.g. A=0)
density of producer and consumer in the matrix is shown
along with size class distribution. The time series from A
to 0 summarizes the temporal pattern of totals in matrix.
000
4000
2000
>960
900
260
200 */*
160
100


CO
un


Time Time
Percent Power
Used
oooo, aa. aaipoo.


3
statistical methods. These 'unusual events' can be impor
tant in understanding how a system works (Weatherhead 1986).
Frequency analysis has been used for some time to study
periodic behavior of systems (Platt and Denman 1975, and
Emanuel, West and Shugart 1978). Fourier transformations
decompose the output or behavior of a system into an addi
tive series of sinusoidal processes. The variance is parti
tioned into a set of frequencies that when combined gives
the output being measured. Aperiodic behavior or systems
with known nonlinear components may also be studied with
these techniques, but the results are often not useful.
Some nonlinear systems with behavior described as 'chaotic'
have frequency domain variance as noisy as the time domain
variance (Abraham and Shaw 1984a, 1984b).
Pattern Formation
Patterns in natural systems range from the smallest
molecular patterns of motion to the placement of the stars
and galaxies in the universe. One of the most intriguing
aspects of pattern formation is the similarity of patterns
at differing time scales and sizes. From a systems point of
view this would lead one to suspect that the processes are
similar at each scale.
Chemically reacting systems give rise to various types
of patterns (Bray 1921, Nicolis and Prigogine 1969, Winfree
1973, Haken 1977a, 1977b). The Belousov-Zhabotinski reac
tion, which makes fascinating patterns, is a simple


Figure 58. Size distribution of Cecropia gaps in tropical
rainforest at El Verde.
(a) Distribution of gaps in all five photographs.
(b) Distribution of gaps in each individual photograph.


Figure 22 Simulation of three path model in Figure 10.
Percent power used as a function of energy input and size of
drain coefficient (K4 varied from .02 to 2.0).


50
For any system to survive over the long term, it must
fit into a regime of disturbances or catastrophic events
from sources outside its own boundaries. The system must be
tuned to the frequencies of those systems that influence it
in order to maximize power and survive. The three path
model was simulated with various frequencies of power input
to see how the various pathways process power at different
frequencies and amplitudes.
Parallel production-consumption minimodel
A model with producers in parallel was used to study
the effects of competition among producers (Figure 13). The
model had three producers, all having the same structure,
with one aggregate consumer that was consuming all three and
feeding back as a multiplier on the production function of
each. It is a basic predator-prey model with competition
among the different producers, along with feedback control
and energy constraints in the form of a flow limited source.
Instead of having combinations of pathways that can vary,
this model had combinations of producers that could vary.
The producers had different turnover times and coef
ficients so that Ql, Q2, and Q3 represented climax, mid-
successional (shrub) and early successional (weed) species.
The coefficient of consumption (the percent of each producer
the consumer eats per unit time) for each producer was
different. The weed species had a higher value than the
shrub species, which was higher than the climax species


194
Table 4
Steady state values, coefficients and flows for pulse model.
J0
100.
Sunlight normalized to 100
JR
4.0817993
Available sunlight at ground level
Q1
1000.
Labile storage (Primary producer)
Q2
10000.
Stored Biomass
Q3
50.
Pulse consumer
Q4
30000.
Nutrients (Available carbon)
K1
.00000417
R1
510.63309
K2
.5
R2
500.
K3
.05
R3
50.
K4
.45
R4
450.
K5
.00005
R5
.5
K6
.00045
R6
4.5
K7
.0000002
R7
5.
K8
.0000018
R8
45.
K9
.000002
R9
50.
K10
.00000417
R10
510.63309
Kll
.0005
Rll
5.
K12
.05
R12
2.5
K13
7.833E-7
(Jordan and Drewry 1969, Odum and Pigeon 1970, and Brown
Lugo, Silander and Liegel 1983)


Initial Conditions and Total Energy Use 112
Simulation of Pulse Model 117
Single Run Simulations 117
Multiple Run Simulations 123
Simulation of Pulse Model with Prey-Predator Sectors... 133
Simulation of the Ring Model 141
Simulation of Two Dimensional Surface Models 151
Rain Forest Gaps and Hierarchies 164
Size Class Distributions 164
Gap Size Measurements 171
Comparison to Models 171
CHAPTER 4 DISCUSSION 173
Maximum Power Considerations 179
Power and Feedback With Paths of Higher Order 179
Effect of Hierarchies on Performance 181
Power Used as a Function of Input Power 132
Threshold for Stable Feedbacks and Pulsing 132
Implications for Succession 134
Role of Individual Units 184
Succession and Pulsing 185
Spatial Pattern formation 186
Synchronous vs. Asynchronous Systems 136
Coupling of Spatial Units by Diffusion Processes.... 187
Organization by Higher Level Consumers 183
Power Use and Edge Effects .190
General Principles .....190
APPENDIX 192
BIBLIOGRAPHY 247
BIOGRAPHICAL SKETCH 254
v


59
highest level consumer (Q5) and measuring the percent power
used and the level of the other storages in the system.
Spatial Models
The models previously discussed were time domain models
with no spatial effects. However, because ecosystems de
velop through time and space and spatial variations can be
at least as important as variations in time, spatial models
were developed and simulated to test hypotheses concerning
spatial development of ecosystems such as energy processing
and pattern formation and hierarchical control of pattern
formation.
The basic spatial model was a collection of subunits,
each one a pulsing consumer model (Figure 14). These sub
units were organized in a spatial format. When this simple
model was simulated in a spatial format, size effects, edge
effects and the consumer range of influence can become
important. Intercell interactions between individual produ
cers, consumers, nutrients, and energy sources may be im
portant in energy utilization and pattern formation.
Effects of edges in the spatial model were of interest
in pattern formation and energy use. Special boundary con
ditions were defined for the model cells along the edge.
These boundary cells were manipulated in the simulation
model in order to study the effects of edges on energy use
and pattern formation. The boundary cells were also manip
ulated to minimize the effect of edges in certain runs of
the model.


Figure 5. Basic multiple path model. Three input
pathways represent different feedback regimes:
linear (Jl), autocatalytic (J2), and quadratic (J3) .


zzzz
o rr


Figure 43. Multi-run simulation of pulse model (Figure
14) with variation in quadratic pathway (K9 varied from
0.5E-6 to 0.53E-5 with K7 and K8 varied proportionately).
(a) Production unit (Ql) .
(b) Stored biomass (Q2).
(c) Pulse consumer (Q3) .
(d) Nutrient storage (Q4).
(a) Power used (JO-Jr)
(f) Percent power used 100*(JO-Jr)/(JO)


Figure 16. Number of edge and center cells as a function of
total number of cells in a given square area.


PERCENT POWER USED
111
1. 1.5 2. 2.5 3. 3.5 4. 4.5 5. 5.5 S.
INITIAL VALUES, Q4


31
In high altitude balsam fir forests in the northeastern
United States, waves of tree loss and regeneration are
thought to be formed by an interaction of the prevailing
wind with the larger mature trees that are exposed along the
gap-wave (Sprugel and Bormann 1981, and Sprugel 1984). The
wind in this case acts to organize the disturbance cycle
that occurs normally in this type of forest into a spatial
wave pattern instead of randomly occurring patches.
The 'ohi'a dieback phenomenon in the rain forests of
Hawaii (Mueller-Dombois 1980) is a case of localized loss of
trees in the forest not due to disease or insect pest. It
was postulated that the effects were due to local soil
moisture loss arising from some climate instability. Repro
duction of the 'ohi'a was adequate enough to regenerate the
forest after the dieback, thus providing a way for this
shade intolerant species to become the primary canopy
species without further succession. Climatic variability
was thus used to an adaptive advantage.
Spatial modelling of ecosystems can be done in several
different ways. By using a model based on the FORET simula
tion model (Shugart and West 1977) and spatially distribu
ting the output of the model according to flooding condi
tions and hydroperiod, Pearlstine, McKellar and Kitchens
(1985) suggested possible species changes due to changes in
the hydroperiod caused by a river diversion in South
Carolina. In this case the number of individual subcell
models was kept small and the spatial distribution was based


r
Figure 14.
Pulse model of
tropical forest
ecosystem model.
Individua'
L rate
equations:
Rate eqiiati
ons
for state
variables:
R1 =
K1*Q1*
Q4* JR
dQl =
R1
- R2
R2 =
K2*Q1
dQ2 =
R3
- R9 Rll
R3 =
K3*Q1
dQ3 =
R7
+ R5 R12
R4 =
K4*Q1
dQ4 =
R12
+ R8 + RS
- RIO + R4
R5 =
K5*Q2
JR =
JO/
(1 + K13*Q1
*Q4)
R6 = K6*Q2
R7 = K7*Q2*Q3*Q3
R8 = K8*Q2*Q3*Q3
R9 = K9*Q2*Q3*Q3
RIO = K10*Q1*Q4*JR
Rll = K11*Q2
R12 = K12*Q3
R13 = K13*Q1*Q4*JR


I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
//V / fyjLlWL-'
Howard T. Odum, Chairman
Graduate Research Professor of
Environmental Engineering
Sciences
X certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Assistant Research Professor
of Environmental Engineering
Sciences


227
c
c
c
£********************************************************
CALL ASSIGN (4,'TTO:')
IF (IPLOT.EQ.0)G0T0499
CALL GGON
CALL GGINIT
CALL GGAXIS(0,0,767,479)
499 Q1IC=Q1
Q2IC=Q2
Q3IC=Q3
Q4IC=Q4
NSLICE=25
NCNTS=150
NRUN=0
500 CONTINUE
C J0=XJ0INI+NRUN*4.
IF(NRUN.EQ.0)GOTO501
VARS(IVAL)=VARS(IVAL)+XINC !INCREMENT VALUE WE ARE VARYING
501 NRUN=NRUN+1
J0=XJ0INI
J=J0/(1+K13*Q1*Q4) ¡GIVE JR (J) INITIAL VALUE
T=0.
q**********************************************************
C SET K VALUES TO TRACK FOR MULTIRUN MODEL
C
K3=. 1 *K2
K4=.9*K2
C
K7=.1*K9
K8=.9*K9
C
K5=1*K11
K6=.9*K11
q***********************************************************
Q1=Q1IC
Q2=Q2IC
Q3=Q3IC
Q4=Q4IC
EUSED=0.0
M1=0.
M2=0.
M3=0.
M4=0.
R1=0
R2=0.
R3=0.
R4=0.
R5=0.
R6=0.
R7=0.
RS=0 .
R9=0.


198
Table 6. Percent power used for various runs of DSP1C
spatial model having only one consumer equally distributed
across the entire production matrix.
Successional state Steady state
High initial condition for consumer Q3 (5000).
Hierarchical 96.5 96.5
Even 96.5 96.6
Random 96.4 96.4
Low initial condition for consumer Q3 (50).
Hierarchical 96.5 96.4
Even 96.5 96.4
Random 96.4 96.3


252
Richardson, J. R., and H. T. Odum. 1981. Power and a
pulsing production model. In w. J. Mitsch, R. W.
Bosserman, and J. M. Klopatek (eds). Energy and
Ecological Modelling. Elsevier. Amsterdam, pp. 641-
647.
Rosen, R. 1970. Dynamical System Theory in Biology. Wiley-
Interscience. New York. pp. 302.
Runkle, J. R. 1985. Disturbance regimes in temperate
forests. In S. T. A. Pickett and P. S. White (eds).
The Ecology of Natural Disturbance and Patch Dynamics.
Academic Press. Orlando, pp. 17-33.
Rushing, W. N. 1970. A quantitative description of
vegetation at El Verde sites. In H. T. Odum and R. F.
Pigeon (eds). A Tropical Rain Forest. NTIS.
Washington, D. C. pp. B-169 to B-238.
Schaffer, W. M., and M. Kot. 1985. Do strange attractors
govern ecological systems? Bioscience 35:342-350.
Sergin, V. Y. 1979. Numerical modeling of the glaciers-
ocean-atmosphere global system. Journal of Geophysical
Research 84:3191-3204.
Sergin, V. Y. 1980. Origin and mechanism of large-scale
climate oscillations. Science 209:1477-1482.
Sergin, V. Y., and S. Y. Sergin. 1979. Fluctuations of
climate and glaciation of the earth. Soviet Geography
19:99-136.
Shugart, H. H. 1984. A Theory of Forest Dynamics.
Springer-Verlag. New York. pp. 278.
Shugart, H. H., A. T. Mortlock, M. S. Hopkins, and I. P.
Burgess. 1980. A computer simulation model of
ecological succession in Australian sub-tropical rain
forest. ORNL/TM-7029. Oak Ridge National Laboratory.
Oak Ridge, Tenn. pp. 48.
Shugart, H. H., and I. R. Noble. 1981. A computer model of
succession and fire response of the high altitude
Eucalyptus forest of the Brindabella Range, Australian
Capital Territory. Aust. J. Ecol. 6:149-164.
Shugart, H. H., and D. C. West. 1977. Development of an
Applachian deciduous forest succession model and its
application to assessment of the impact of the chestnut
blight. J. Environ. Management 5:161-179.
Shugart, H. H., and D. C. West. 1980. Forest succession
models. Bioscience 30:308-313.


Figure 10. Three pathway model used to test effects of
various energy inputs on kinetic mechanisms.
Linear input: J1=K1*R
Autocatalytic input: J2=K2*Q*R
Quadratic input: J3=K3*Q*Q*R
dQ=J1+J2+J3-K4*Q
R=J0-J1-K0*R*Q-K5*R*Q*Q


2T f. 40000


lt
:1; '::i'.:';;
: : i : si :
o
¡a Irjjji-'ii
* IsTlik
a
* ::i'3s;"
m
t 1:
fjbtfe
LUI
ill

!:
ii#
ULUCI
. : : -
m
.vnW Ttcjis
Ejs^to)-jsh w¡
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.
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LIU
yj
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nn
LIU
c
uu


Figure 25. Simulation of the three path model in Figure
10. Percent power used as a function of frequency of the
input power (J0=500).
Linear pathway:
Autocatalytic pathway:
Quadratic pathway:
J1=K1*R
J2=K2*Q*R
J3=K3*Q*Q*R


223
4150
4160
4170
4200
4202
4204
4206
4210
4211
4220
4250
5000
5030
5040
5050
5060
5070
5080
5100
5110
5120
5130
5140
5150
5160
5170
5180
5190
5200
5210
5220
5240
5242
5244
5246
5247
5248
5249
5250
5252
5260
5270
5280
5290
5300
5310
5320
5330
5340
5350
5351
5352
5353
5360
IF NTEMP=0 THEN 4200
NLEN=LEN(FILENAME?)
FILENAME$=MID$(FILENAME$,1,NTEMP-1)
FILENAME$*"B:"+FILENAME$
OPEN "I",3,FILENAME$+".PRN"
LINE INPUT #3,DUM$:PRINT:PRINT DUM$:PRINT:PRINT
CLOSE #3
OPEN"A",2/FILENAME$+".DAT"
OPEN"A",3,FILENAME$+".PRN"
'INPUT "BASIN NUMBER = ",WET1
RETURN
'PRINT"SUBROUTINE SCALE3" : SUBROUTINE SCALE3
GOSUB 3900: GO SET POINT MODE FIRST !
PRINT ****** DIGITIZER THREE-POINT SCALING ******"
PRINT :PRINT:PRINT :BEEP:GOSUB 6001
PRINT DIGITIZE THE ORIGIN OF THE GRAPH >>":BEEP
GOSUB 8000:XORG=XIN:YORG=YIN:' CALL DIGDRU(XORG,YORG,IBTN)
IF(VAL(CODE$)=10)GOTO 5530
PRINT DIGITIZE ANY OTHER KNOWN"
PRINT POINT ON THE SAME HORIZONTAL (X-AXIS) LINE>>"
GOSUB 8000:XHZ=XIN:YDUMM=YIN:' CALL DIGDRU(XHZ,YHZ,IBTN)
IF(VAL(CODE$)= 10)THEN RETURN
XSCALE=1!
XOFF=0!
YSCALE=1!
YOFF=0!
ANGLE=0!
XROUND=0!
YROUND=0!
XD=FNRDIST(XORG,YORG XHZ,YDUMM)
IF(XD = 0!)THEN RETURN:'GOSUB 3800:RETURN:'STREAM MODE
'PRINT TWO-POINT X DISTANCE IN DIGITIZER REAL UNITS: ";XD
PRINT "DIGITIZE A THIRD KNOWN POINT ON THE VERTICAL (Y-AXIS) LINE"
GOSUB 8000:XDUMM=XIN:YHZ=YIN
ANG1=FNANGLER(XORG,YORG,XHZ,YDUMM)
ANG2=FNANGLER(XORG,YORG,XDUMM,YHZ)
YANG=1.570796-(ANG2-ANG1)
YD=FNRDIST(XORG,YORG,XDUMM,YHZ)*COS(YANG)
IF YD=0 THEN RETURN
PRINT "X-DISTANCE= ";XD?" Y-DISTANCE= ";YD
PRINT ENTER USER COORDINATES"
PRINT OF THE FIRST POINT (REAL) [0.0,0.0]: "7
INPUT X1U,Y1U
'5 FORMAT(2F10.0)
XDEF=XD+X1U:YDEF=YD+Y1U
' WRITE(LUNO,6)XDEF
PRINT ENTER USER X COORDINATE"
PRINT OF THE SECOND POINT (REAL) ";XDEF
INPUT X2U
IF(X2U = 01)THEN X2U=XDEF
PRINT ENTER USER Y COORDINATE"
PRINT OF THE THIRD POINT (REAL) ";YDEF
INPUT Y3U
XU-X2U-X1U


156
simulation had sharp peaks due to the different frequencies
of pulsing of the independent cells. The size distributions
of the producers and consumers were based on the input
energy and are grouped accordingly. Without diffusion, the
pattern formed was entirely dependent on the hierarchical
pattern of the input energy.
The addition of diffusion between the consumers of each
cell for the previous model smoothed out the time series for
the consumers and producers (Figure 52). A low level of
diffusion (DK=0.001) enabled the first pulsing cells (lo
cated at the center of the matrix) to affect the neighboring
cells, thus spreading the pulse wave out over the matrix.
In this simulation the size distribution of the producers
and consumers tended to smooth out over time. The edge
effects were minimized in this simulation by allowing the
outer non-reactive ring of consumer cells to float at a
value that was the average of the total consumers in the
matrix.
Diffusion between the consumers at a low level had a
much greater effect in this two dimensional version of the
model than in the one dimensional ring version of the model.
When the two dimensional version was run with a random
energy source and a low diffusion coefficient (Figure 53,
DK = 0.001) the effect was similar to that seen in Figure 52.
In this case, local foci of high productivity (caused by
locally high values of input energy) led to pulses that
spread over the entire matrix. This simulation was dif-


181
and thus provided energy to the system under all input
regimes. A system with a combination of pathways then shows
greater stability under fluctuating regimes and maximizes
power with increasing energy inputs.
Multiple pathway models have been used to describe a
variety of systems. A disaster model using multiple path
ways (linear and autocatalytic) has been used to describe
earthquakes and floods (Alexander 1978). Models of chemical
reacting systems have often used multiple pathway models to
describe the kinetics of the reactions ("Brusselator",
Nicolis and Prigogine 1977 and "Oregonator", Field and Noyes
1974). "Chaotic systems" are often modeled with multiple
non-linear pathways (Abraham and Shaw 1984b).
Effect of Hierarchies on Performance
Hierarchical subunits of a system generally have in
creasing turn-over times with increasing trophic levels
(Allen and Starr 1982, Urban, O'Neill and Shugart 1987).
The addition of an extra consumer (adding a level to the
hierarchy) of the pulse model (Figure 15) must have the
appropriate turnover time to survive. If the turn-over time
was too short, not enough energy was available to that level
of the hierarchy to sustain it and the added level did not
survive (Figure 45a). If the turnover time was too long,
the rate of power use dropped and the whole system collapsed
(Figure 45c). The appropriate size consumer modified the
output behavior of the model (pulsing with a longer period),
but the system was stable and utilized slightly more power.


Figure 59. Size distribution of gaps in tropical rainforest
pulsing model simulation (Figures 14 and 18) at time =760.
(a)
Size
class
distribution
from
three separate
runs
with
hierarchical
energy
distribution.
(b)
Size
class
distribution
f rom
three separate
run
with r
andom energy
distri
bution.
model
model


9
model. By allowing cells in the model to divide and migrate
within given constraints, the model developed patterns simi
lar to those in real insects. The model allowed simple
random cell division with movement constrained to a hex
agonal direction away from the center of the cell division.
Sergin (1978, 1979, 1980) studied the oscillatory be
havior of long term climate variations using models that
combine linear and nonlinear interactions of the heat cap
acities of the oceans and polar ice sheets. The period of
the climatological events in these models is on the order of
10,000 to 100,000 years. The model of global temperatures
varies in its behavior from steady state to oscillations
based on small changes in areal coverage of continental ice
sheets.
Pattern formation based on digital, rule based systems
has been used to model biological systems. Examples such as
cellular automata (Turing 1952 and Wolfram 1984) and a 'game
of life' (Gardner 1970 and Poundstone 1985) generate complex
spatial patterns from simple rules. The 'game of life' is
generated on an N x N matrix where
1. Every active cell with two or three neighboring
cells survives to the next generation.
2. Each active cell with four or more neighbors
'dies' from overpopulation. Every active cell
with one or no neighbors 'dies' from isolation.
3. Each empty cell adjacent to three 'live' neighbors
gives birth to a new cell.


CHAPTER 3
RESULTS
Simulation of Three Path Model
Individual Pathway Tests
The amount of energy flowing through each of the path
ways in the three path model (Figure 10) depends on the
total energy input to the model. As input power (JO) was
increased (Figure 19) steady state flows for each of the
pathways changed. Each pathway predominates at certain
times. The linear path had the largest power flow when
input power was low, while the quadratic pathway had the
highest flow at higher power inputs.
When input power was increased through time (Figure
20), there was no steady state, but, like Figure 19 when
power increased, the energy flow shifted from the linear
pathway to the autocatalytic and finally to the quadratic
path. The fraction of energy remaining (Jr/JO) also de
creased over time. As input power increased, a greater
fraction of the input power was utilized.
The model was run with different pathway combinations
(Figure 21) and with various power inputs. Each curve on
the graph represents a steady state value for various com
binations of pathways present in the simulation. Power used
71


Figura 12. Three pathway model with individual competing
units having single input pathways similar to combined
model. Coefficients in Appendix.
Combination tank:
Linear input:
Autocatalytic input:
Quadratic input:
dQ=Jl+J2+J3-K4*Q
Single tanks:
Linear input:
dQl=JlX-K4*Ql
Autocatalytic input:
dQ2=J2X-K4*Q2
J1=K1*JR
J2=K2*Q*JR
J3=K3*Q*Q*JR
J1X=K1'*JR
J2X=K21*Q2*JR
Quadratic input: J3X=K3'*Q3*Q3*JR
dQ3=J3X-K4*Q3
JR=J0-J1-K2'*J2-K3'*J3-J1X-K2'*J2X-K3'*J3X


Prey: 0-5 nil density (white); 6-25 low density (light stipple); 26-75 medium density (horizontal lines); 76 or
over, high density (solid black). Predator; 1-8 (one white circle).
OCCIDEN TALI S


Figure 45. Simulation of pulse model with prey-predator
sectors (Figure 15).
(a)Simulation with turn-over time of higher level
pulsing consumer (Q5) set equal to lower level
pulsing consumer (Q3).
(b)Simulation with turn-over time of higher level
pulsing consumer (Q5) set to ten times longer than
the turn-over time of lower level pulsing consumer
(Q3) .
(c)Simulation with turn-over time of higher level
pulsing consumer (Q5) set to one hundred times
longer than the turn-over time of the lower level
puling consumer (Q3).


83
model as multiple inflow pathways. The percent power used
was lowest when all combinations of drain pathways were
enabled (Figure 23). Percent power used increased with
increasing input power. The highest value for percent power
used was achieved when only the original linear drain was
present. Any combination with the linear drain used less
power at low power inputs than the nonlinear pathways alone
or in combination. The higher order drains enabled the
system to draw more power at low levels than when combined
with linear pathways. This effect was opposite from that
with input pathways at very low power where the nonlinear
pathways did not function well (see Figure 21).
The effects of adding competition pathways to the model
(Figure 12) can be seen in Figure 24. In this case, each of
the competing pathways (single tanks Ql, Q2, and Q3 with
individual pathways) were left on throughout the simula
tions. Here again the various pathways were disabled and
simulations run with varying power inputs. The results were
similar in some ways to those in Figure 21 where at high
power inputs the percent power used approached one of two
asymptotes. The greatest percentage of power utilization
occurred when all pathways were enabled and the lowest power
utilization occurred when only J2 or J3 were enabled. The
addition of the extra competing storages increased the per
cent power used in each of the pathway combinations compared
to Figure 21. These extra pathways were always there to use


199
Table 7. Percent power used for DSP100 model as a function
of search length and input energy type.
Search length
(cells)
Successional
state
Steady
state
Hierarchical distribution
1
96.4
(17) (a)
96.5
(31)
2
96.6
(28)
95.6
(39)
3
96.6
(36)
96.5
(39)
4
95.6
(36)
96.6
(43)
5
96.5
(33) (b)
96.5
(45)
Even distribution
1
95.7
(15)
crash
(13)
2
96.4
(30)
95.5
(30)
3
96.4
(38)
96.5
(34)
4
96.4
(38)
96.5
(42)
5
96.5
(32)
96.5
(40)
Random distribution
1
96.4
(20)
95.7
(32)
2
96.6
(34)
96.6
(38).
3
96.5
(40)
96.5
(44)
4
96.6
(42)
96.5
(45)
5
96.5
(41)
96.6
(45)
(n) indicates number of consumers at end of simulation.
(a) see Figure 55
(b) see Figure 56


179
Maximum Power Considerations
The class of models studied here duplicate real systems
by reinforcing pathways that process more power. The feed
backs simulate useful power processing. These models link
kinetics and energetics in ways observed in nature.
Power and Feedback With Paths of Higher Order
Systems that generate higher order pathways to cap
ture varying energy flows may offer a competitive advantage.
The maximum power implication is that as systems develop
feedbacks (higher order pathways) they can extract more
energy from the source. Lotka (1922) stated that as long as
there was untapped available energy, systems were capable of
growth when rates of flow increased through the system.
Odum (1982 and 1983) added that as systems mature they feed
back energy which amplifies other pathways and maximizes
power. The multiple pathway configuration shown in the
three path model provides a possible mechanism for this to
occur. In the three path model simulations (Figures 19 -
27) the linear pathway had a fixed efficiency while the
autocatalytic and quadratic pathways had variable effi
ciencies (see Figure 20 and 21) depending on the input
power.
The development of multiple pathways in a system is
incurred at some energy cost to the system. The energy
costs associated with developing and maintaining the non
linear pathways must be competitive to survive. For systems


Figure 38. Simulation of pulse model (Figure 14) without
a quadratic pathway (K7, K8r K9 = 0.0).
(a) Simulation for 760 years.
(b) Simulation for 2000 years.
Legend:
PPU = Percent power used
Q1 = Production unit
Q2 = Stored biomass
Q3 = Pulse consumer
Q4 = Nutrient storage


PERCENT
88
a
L
0)
D
£C
Ul
3
o
0.
90.
. : 1
1 : 1 = 1 : i 1 ; l 1
ALL
! 1 : .
8$.

J1+J3
70.
J1 + J2
J2+J3
-
6.
^

50.
J2
40.
"
30.
J1

20.

10.

J3

0.
1
' j 1 I ; I
= 1 ~
0. 5. 10. 15 20 25. 30 35. 40 45 50.
INPUT FREQUENCY

24
presence of many spatially distributed patches may be due to
the production-consumption pulsing of components in the
ecosystem.
Hierarchies and Patches
The frequency of disturbance based on internal cycles
has been shown to be from 200-500 years in a variety of
ecosystems (Emanuel, West and Shugart 1978, Runkle 1985).
Distribution of disturbances over time varies from fairly
constant low amplitude disturbances to long-period, high
amplitude disturbances. The successional changes due to
disturbances may be related to the size and scale of the
disturbance (Peet and Christensen 1980, Peet 1981).
Brokaw (1982a, 1982b, and 1985a) found a hierarchical
distribution in gap sizes in a tropical rain forest at Barro
Colorado Island (Figure 6a). The area per size class is
plotted vs. the size class (Figure 6b). This relationship
may be important in determining patch dynamics. Brown
(1980) suggested that size class distributions may be
related to the emergy per size class (the emergy per size
class is also related to the area per class). Brokaw calcu
lated the turnover rate for the forest, based on the gap
formation, to be from 85 to 128 years depending on the
minimum size of the lowest class used.
Models
The simulation models used to study ecosystem behavior
generally fall into two classes (Shugart 1984). One of these


SPATIAL PATTERNS AND MAXIMUM POWER IN ECOSYSTEMS
3Y
JOHN R. RICHARDSON
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE
UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS OF THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA


244
CALL STRIP(YTEXT,80)
TYPE 70
70 FORMAT(' WANT TO INPUT MINIMUMS AND MAXIMUMS (1-yes, 0-no)'$)
ACCEPT 72,IMIN
72 FORMAT(11)
IF (IMIN.EQ.0)GOTO85
TYPE 74
74 FORMAT(1X,'WHAT ARE XMIN AND XMAX [R] '$)
ACCEPT *,XMIN,XMAX
TYPE 76
76 FORMAT(1X,'WHAT ARE YMIN AND YMAX [R] '$)
ACCEPT *,YMIN,YMAX
85 CONTINUE
TYPE 89
89 FORMAT(1X,' LINE TYPE (0-9) '$)
ACCEPT 891,ILIN
891 FORMAT(I1)
8910 TYPE 892
892 FORMAT(1X,'VALUE FOR DATA MARKER (0-9, -1 TO SEE LIST) '$)
ACCEPT 893,IMARX
IF(IMARK.GE0)G0T089 3 0
WRITE(5,8921)
8921 FORMAT(/' 0- POINT'
&
/'
1- SQUARE'
&
/'
2- OCTAGEN'
&
/
3- TRIANGLE'
St
/'
4- CROSS'
&
/
5- X'
&
/
6- Y'
&
/'
7- DIAMOND'
&
/'
8- ARROWHEAD
&
/'
9- HOURGLASS
&
/'
10-POINT IN -
G0T08910
8930 CONTINUE
893 FORMAT(15)
IF (IMIN.EQ.1)IROUND=0
IF (IMIN.EQ.1)G0T09910
TYPE 90
90 FORMAT(' ROUND MAX AND MIN VALUES? (1-YES, 0-NO) '$)
ACCEPT 99,1 ROUND
99 FORMAT(11)
IF(IROUND.GT.1)GOTO51
IF(IROUND.LT.0)GOTO51
9910 I RFLAG=.FALSE
IF(IROUND.EQ.1)IRFLAG=.TRUE.
WRITE (5,9901)
9901 FORMAT(' CURVEFIT THE DATA LINE (1-YES; 0-NO)'$)
READ(5,9902)ISM
9902 FORMAT(11)
SMOOTH=.FALSE.
IF(ISM.EQ.1)SMOOTH=.TRUE.
WRITE(5,991)ESC
991 FORMAT('+'1A1,'[H*)
!SEND CURSOR HOME


o o
215
2003 FORMAT(' S(H)')
CALL GGOFF
2100 END
C
C
C
C
SUBROUTINE GG3DX(COLOR,IX,IY,IYX,NPNTS,N,DELTAX,DELTAY,NXHTC)
BYTE COLOR
DIMENSION IX(1),IY(1),MASK(1),IYX(1)
COMMON /IAREA/MASK
INTEGER DELTAY,DELTAX
IXOFF=200
IF(DELTAX.LT.O)IXOFF=50
IF(N.NE.1)GOTO10
C
C SET UP MASK FOR FIRST SLICE
DO 5 1=1,NPNTS
MASK(IX(I)+(IXOFF-N*DELTAX))=IY(I)+N*DELTAY
5 CONTINUE
C
C
10 CONTINUE
DO 20 1=1,NPNTS
IXOUT=IX(I)+(IXOFF-N*DELTAX)
IYOUT=IY(I)+N*DELTAY+20
IF(IYOUT.GEMASK(IXOUT))GOTO50
GOTO 20
50 MASK(IXOUT)=IYOUT
CALL GGPLT(COLOR,IXOUT,IYOUT,1)
IF(N.LE.1)GOTO20
IF(I.EQ.1)GOTO19
IF(IMOD(I,NXHTC).NE.0)GOTO20
19 IX2=IXOUT+DELTAX
IY2=IYX(I)+(N-1)*DELTAY+20
IF(DELTAX.LT.0)GOT0190
IF(IY2.LT.MASK(1X2))GOTO20
190 CALL GGVEC(COLOR,1X2,IY2)
20 CONTINUE
GOTO 33
IF(N.EQ.1)G0T033
M1=MASK(IXOFF-(N-1)*DELTAX)
C DO 33 1=1 ,DELTAX
C MASK(IXOFF(N*DELTAX)+I)=M1
33 CONTINUE
RETURN
END
C
C
SUBROUTINE AXIS(COLOR,NLINE,NPNTS,DELTAX,DELTAY,IXSCLE)
BYTE COLOR
INTEGER DELTAX,DELTAY,MASK(1)
INTEGER X0,Y0,X1,Y1,X2,Y2,X3,Y3,X4,Y4,X5,Y5,X6,Y6,XORG,YORG
INTEGER X7,Y7


107
50 o
Tim
Tim
100


17 Cell geometries possible for spatial models 64
13 Format of spatial model display graphs 69
19 Steady state power utilization of units in
the three path model (Figure 10) as a
function of input power (JO) 73
20 Energy utilization of individual components
in the three path model in Figure 10 75
21 Steady state energy flows on various pathways
and combinations of pathways in the three
path model (Figure 10) as a function of input
power (JO) 77
22 Simulation of three path model in Figure 10 80
23 Simulation of three path model with multiple
drain pathways in Figure 11. Percent power
used as a function of energy input (JO) 82
24 Simulation of three path competition model
with various pathways enabled (Figure 12) 35
25 Simulation of the three path model in Figure
10. Percent power used as a function of
frequency of the input power (J0=500) 88
26 Simulation of the three path model in Figure
10. Percent power used as a function of
frequency of the input power (J0=2000) 90
27 Simulation of the three path model in Figure
10. Percent power used as a function of
frequency of the input power (J0=10000) 93
28 Simulation of the parallel production-consumption
model in Figure 13. Model base run 96
29 Simulation of the parallel production-consumption
model in Figure 13 99
30 Simulation of the parallel production-consumption
model in Figure 13. Multiple simulations of
the model with available power increasing
from 50 to 300 102
31 Simulation of the parallel production-consumption
model in Figure 13. Multiple simulations of
the model with percent power used for entire
run vs input power 10 4
vi i


100
the production function of all of the producers in the
system.
When the model was simulated with no initial consumer
(Q4) it crashed even faster (not shown) than in Figure 29c
because of the feedbacks in the model from the consumer to
the producers.
Multiple Run Simulations
The behavior of the parallel production model with
varying input power is seen in Figure 30a-f. In this set of
runs the base model was run for 100 time units. For each
successive run, the input power (JO) was increased, varying
from 50 to 300. As the energy input increased, the peaks of
the producers were higher (Q1-Q3), with Q3 (the weed
species) showing the most change in amplitude (Figure 30c).
The climax species (Ql) peaked sooner as the input power in
creased .
The simulation of succession to a climax was thus
speeded up by increasing the energy input at lower levels,
but at higher levels the increase in energy had little
effect on the transition to dominance of the climax species.
The effect of increasing energy input was also seen in
the level of the consumer (Q4 in Figure 30d) With in
creasing power, the consumer was maintained at a propor
tionately higher steady state.
For this set of simulations, as the input power in
creased, the percent power used increased asymptotically
(Figure 31). There was a diminishing return on the input


18
K0*Jr*Q + K2*Jr*Q = Kl*Jr*Q + K3*Jr*Q (1)
Substitute R (flow) terras as abbreviations for terras
in equation (1):
R0 + R2 = R1 + R3 (2)
Inputs of energy of any process must equal the outputs.
Efficiency is defined as:
Efficiency = (Output'of useful power)/Inputs
or in terms of our equation:
Efficiency = R1/(R0+R2) (3)
where R3 is waste heat generated in the process (re
quired by the second law of thermodynamics). Because R3
cannot be zero, there is a natural upper limit to the effi
ciency of any process.
another method to increase energy flow from a flow
limited source is to have multiple pathways capture avail
able energy, each effective at a different energy level
(Figure 5). Multiple pathways (J1,J2,J3) use stored energy
to build structures to capture available energy. A linear,
donor-controlled pathway (J1) requires little structure and
employs no feedback in order to capture energy, but has
severe limitations (its efficiency cannot change) due to the
dependency on the energy source. An autocatalytic pathway
(J2) feeds back embodied energy (structure built by the
system) to draw in more energy. The quadratic pathway (J3)
is a co-operative phenomenon in which the structure of the
system is interacting with itself to feed back embodied
energy to draw in more power. A system that develops such


Figure 21. Steady state energy flows on various pathways
and combinations of pathways in the three path model (Figur
10) as a function of input power (JO).
Linear pathway:
Autocatalytic pathway:
Quadratic pathway:
J1=K1*R
J2=K2*Q*R
J3=K3*Q*Q*R


53
(Odum 1969) A list of coefficients is given in Appendix
Table 3.
Several variations of this basic model were written in
FORTRAN and BASIC computer languages and simulated on both a
PDP 11/34 and on a Heathkit H8. The source listing for the
standard parallel production-consumption model (SUC10) is
presented in the Appendix.
Pulse Model
A general pulsing ecosystem model (Figure 14) was de
signed to test various hypotheses about energy flows and
pulsing, hierarchical organization, and spatial development
of ecosystems. Some of the structure of the model was
derived after the tests of the thredpath model and the
parallel production-consumption model. The model had many
characteristics of ecosystems such as:
1. Flow limited resources (representing solar based
energy resources).
2. Nutrient storage within the boundaries of the model.
3. Units of production, consumption and storage.
4. Feedback of consumers on production through nutrient
recycle.
5. Consumption at low maintenance rates and at high
pulsing rates.
6. Production through a fast turnover storage into a
long turnover biomass storage.
The basic pulsing ecosystem model was tested for dif
ferent flow rates, initial storages and energy inputs. From
this, a baseline understanding of the dynamic behavior of
the model and energy processing capabilities (as percent
power used) was developed.
The pulse model (Figure 14) was similar to the one in
Richardson and Odum (1981) with some changes in coefficients


170
Diameter Distribution
Height Distribution
Radiation Site
Wmmm
LZn Y7\ I
10 12 14 16 18 20 22 24 26 26


141
Simulation of the Ring Model
The linear array ring model was simulated with high
diffusion (DK=.l) between consumers in adjacent cells
(Figure 46). Initially the concentration of producers and
consumers around the ring was constant except for a single
consumer at a high level (Q3 (2,2) =100.; lower left hand
corner of consumer matrix). At T=50 years (Figure 46A), the
consumers had pulsed in both directions around the ring and
completely encircled the ring by T=100 (46B). At T=150
(46C), the production was beginning to spread around the
ring from the lower left corner and continued through T=
200, 250, 300, 350 (46D-H). The consumers again began to
grow (T= 350, 46H) and spread around the ring again. This
was followed by another wave of production and consumption
(T= 500-750, 46J-0).
In runs with lower diffusion (0.01) between consumers,
the pulse wave traveled slow enough that the wave only moved
part way around the entire ring before the internal pulse
frequency allowed the remainder of the consumers to pulse,
thus stopping the wave. With an even lower diffusion coef
ficient of 0.001, the wave moved 3 cells before stopping.
With a diffusion coefficient of 0.01, the wave moved 10
cells before being stopped by the natural internal pulse
frequency.
A different pattern developed when the producers and
consumers in the model were distributed in a random pattern
around the ring (the individual cell concentration of pro-


PERCENT POWER USED PERCENT POWER USED
104
M,
9.
39.
79.
69.
59.
49.
39.
29.
19.
9.
9 39 69 99. 129. 159. 189 219 249 279 399.
INPUT POWER (J9)
-1-1. I
[ ; I : I
-1
I ; I 1
INPUT POWER


Figure 35. Simulation of the parallel production-
consumption model in Figure 13. The initial value of weed
species (Q3) was varied from 0 to .5 and input power was
held constant (J0=100, base run value).
(a) Climax species (Ql)
(b) Intermediate producer species (Q2)
(c) Weed species (Q3)
(d) Consumer species (Q4)
(e) Percent power used (J0-Jr)/J0
(f) Total biomass


211
C 1X1 = Integer value of X opposite
C IY1 = Integer value of Y opposite
C
C IF THE FILL IS TURNED ON THE BOX WILL BE FILLED AUTOMATICALLY
C
BYTE COLOR
CALL GGPLT(COLOR,IX,IY,1)
CALL GGVEC(COLOR,1X1,IY)
CALL GGVEC(COLOR,IX1,IY1)
CALL GGVEC(COLOR,IX,IY1)
CALL GGVEC(COLOR,IX,IY)
RETURN
END
C
C
C
SUBROUTINE GGCIRC(COLOR,IX,IY,IRAD)
C SUBROUTINE TO DRAW A CIRCLE AT POINT IX,IY WITH A RADIUS OF IRAD
C COLOR = BYTE variable 0-7 for color
C IX = Integer value of X
C IY = Integer value of Y
C IRAD = Integer radius of circle
C
C IF THE FILL IS TURNED ON THE CIRCLE WILL BE FILLED AUTOMATICALLY
C
BYTE COLOR
CALL GGPLT(COLOR,IX,IY,0)
WRITE(3,100)COLOR,IX,IY+IRAD
100 FORMAT('+ W(I',11,')C[',14,',',14,']')
CALL GGPLT(COLOR,IX,IY,1) !LEAVE CURSOR AT CENTER
RETURN
END
C
C
c
SUBROUTINE GGTEXT(COLOR,IX,IY,TEXT,ISIZE,ITILT)
C SUBROUTINE TO WRITE TEXT AT IX,IY ON SCREEN
C Writes text at ix,iy with size and rotation of
C characters given
C COLOR = BYTE variable 0-7 for color
C IX = Integer value of X
C IY = Integer value of Y
C TEXT = BYTE array containing 80 char or less
C ISIZE = Integer value for text size 0-8
C IROT = Integer value of degress of rotation for
C line of text (multiple of 45)
C
C THIS SUBROUTINE CALLS LENGTH TO DETERMINE
C THE LENGTH OF THE STRING
C
BYTE COLOR,TEXT(1)
N=0
CALL GGPLT(COLOR,IX,IY,0)
CALL LENGTH(TEXT,N)


Figure 46. Simulation of pulse model (Figures 14 and 18)
with cells in a linear ring and diffusion between consumers
of each cell in ring (DK=.l). For each time unit (e.g. A=0)
density of producer and consumer in the matrix is shown
along with size class distribution. The time series from A
to 0 summarizes the temporal pattern of totals in matrix.
Initial conditions of consumers were set to near zero except
for one "seed" consumer at lower left corner of matrix which
was set to 100.
Spatial pallara
X- \ ..
CONSUMER
CELLS
>14000
M 12000
13 ,ooo
jSsj sooo
W. S000
$8 4000
**? 2000
g/a.2
I >380
300
2S0
200
ISO
:>*' too
jff 80
Consumar
alta olaaaas
s/-a


133
This implies there is a 'window' of size for the consumer to
pulse.
Changing the rate constant (K9) of the quadratic path
way caused the pulsing consumer to change frequency, in
creasing the frequency of pulsing with an increasing coef
ficient value (Figure 43). There was a point in this set of
simulations where the pulsing ceases but in this case the
size of the consumer remains small. When the quadratic
pathway became dominant at low consumer levels, the system
did not pulse but completely consumed the stored biomass
storage (Q2).
When simulated without the quadratic pathway and chang
ing the coefficient of the linear pathway (Kll), the model
did not pulse, the consumer (Q3) remained at a low level and
the stored biomass (Q2) built up (Figure 44, compare to
single run figure 38). As the linear pathway increased,
there was a slight increase in the consumer (Q3) with less
of a build-up in the stored biomass (Q2). In all cases,
through time the power use and percent power used dropped
off.
Simulation of Pulse Model with Prey-Predator Sectors
Simulation of the pulse model with an additional prey-
predator sector (Figure 15) investigated how turnover time
is related to hierarchical consumers (Figure 45). With a
drain coefficient on Q5 the same as or larger than that of
the normal pulsing consumer (K17=0.05 or 0.5), the effect


STEADY STATE PATH FLOUI
S+3


168


202
C BIG OUTER LOOP
C
DO 1062 IQSAV-1,6
C
C
NRUN=0
2 CONTINUE
J0=XJ0INI+NRUN*XINC
NRUN=NRUN+1
T=0
PERCNT=0
PAVAIL=0
PUSED=0
DUSED=0
Q9=0
BIOMSS=0
M1=0
M2=0
M3=0
M4=0
M9=0
P=0
BMAX=0
Q1=Q1INIT
Q2=Q2INIT
Q3=Q3INIT
Q4=Q4INIT
Q1SIZE=30.
Q2SIZE=5.
Q3SIZE=1.
Q4SIZE=20.
C SET OUTPUT VECTOR AND FLAG **********************************
DT=. 1
C WRITE(5,108)DT
C 108 FORMAT(' TIME INTERVAL DT= ',F5.3)
C ISTEP=1/DT !Dr'S PER T
C IPLOT=0 !PLOTTING INTERVAL IN DT'S
C ITCNT= 0 ¡ITERATION COUNTER
C WRITE(5,1081)
C1081 FORMAT(' WHAT IS PLOTTING INTERVAL PER TIME UNIT [I] '$)
C READ(5/1082)IPLOT
C1082 FORMAT(12)
C WRITE(5,1083)
C1083 FORMAT(1X,' INPUT VALUES FOR SIZES Q1-Q4 [R] '$)
C READ (5,1084)Q1SIZE,Q2SIZE,Q3SIZE,Q4SIZE
C1084 FORMAT(4F8.3)
C WRITE GIGI STARTUP INFORMATION
C CALL GGON
C CALL GGINIT
C CALL GGERA
C CALL GGAXIS(0,0,767,479)
C CALL GGBOX(7,0,0,767,479)
C CALL GGBOX7,0,0,767,350)
C


188
The level of the hierarchy in which inter-cell coupling
takes place plays an important part in the development of
spatial patterns. When this coupling took place at the
level of nutrient exchange from cell to cell (Figure 54),
the effect was hardly detectable, even at high diffusion
levels.
Spatial patterns generated are not totally dependent on
the natural energy inputs but organize using those natural
energy regimes. Spatial diversity thus depends on the the
landscape energy pattern, the interactions between the sub
units, the hierarchy level of the interaction and the
existing pattern of vegetation.
Most of the models in this dissertation used only
diffusive coupling between spatial subunits of the model.
Many systems have more complex interactions between subunits
than this simple linear coupling. The active transport
systems of biological systems are good examples of the more
complex coupling that can occur in living systems. The
moving consumer model represents a more complex coupling
between individual cell units.
Organization by Higher Level Consumers
The role of the consumer in these models was very im
portant in organizing pattern formation. When the spatial
model was simulated with one consumer spread evenly over the
matrix, the result was exactly the same as when the model
was simulated with all cells uncoupled and a single consumer
in each cell of the matrix (Figure 50). In this case, the


182
The highest level of the hierarchy in this model determined
the frequency and scale of pulsing. Therefore there are
optimum turnover times for maximum performance.
Conversely, as input power increases, a higher level of
consumers may be supported. This was seen in the parallel
production-consumption model (Figure 32d) and the pulse
model (Figure 40c).
Power Used as a Function of Input Power
The general trend for all of the models tested here was
that as the input power increased, the percent of input
power that is utilized increased. This occurred in the
three path model (Figures 21,- 22, 23), the parallel
production-consumption model (Figures 30e, 32e, and 36), the
pulse model (Figure 40f) and the spatial models. This
appears to be a function of the non-linear pathways that
feed energy back to increase the efficiency with increasing
available energy. Individual simulations of these models
with only linear pathways did not show this behavior.
Threshold for Stable Feedbacks and Pulsing
The pulse model exhibited a double threshold phenom
enon. At low power inputs the model did not pulse and at
high power inputs the model did not pulse (Figures 37 and
40). In the middle power range, the model pulsed and the
pulse frequency was a function of the input power. Local
ized maxima of power utilization may occur in the pulsing
range due to synchronization of inputs with natural internal


21
higher order feedback pathways may exhibit a greater rate of
use of available energy.
This added quadratic pathway is available to utilize
any energy left after the efficiency is raised to the upper
limit for the autocatalytic pathway. This is a mechanism
that can draw in energy that would normally be unavailable
to the system. The quadratic pathway may have a high cost
to develop and maintain this pathway but it enhances overall
use of that extra energy by the whole system. This may give
a competitive edge in some circumstances over systems with
out higher order pathways, particularly when available
energy may be fluctuating. Available power will be in
creased by switching from one pathway to the other depending
on the energy source. Some pathways are more efficient at
low energy levels while others are more efficient at high
energy levels, thus allowing such systems to efficiently
utilize fluctuating power sources.
Pulsing and Patterns in Ecosystems
Succession and Disturbance
Any climax state is eventually interrupted by disturb
ances that generate patches in which succession is re
initiated. The gaps in a forest may be generated by local
outbreaks of consumers within the forest, tree mortality, or
outside disturbances such as fires, hurricanes, volcanic
activity, and landslides (Runkle 1985). The role of the
landslide as a gap-forming mechanism has been described in


226
WRITE(5,1011)FILE
1011 FORMAT(1X,3A4)
C WRITE( 5,1016)
C1016 FORMAT(' DO YOU WANT HARDCOPY (1-YES, O-NO) '$)
C READ(5,1017)ICOPY
ICOPY=0
C1017 FORMAT(12)
C WRITE (5,1018)
C1018 FORMAT(1 WHICH Q TO SAVE (1,2,3,4,5=JR,6=%POW USED)'$)
C READ (5,1019)IQSAV
IQSAV=1
C1019 FORMAT(13)
C WRITE(5,1020)
C1020 FORMAT(1 DO YOU WANT TO PLOT THE GRAPHS (1-YES, 0-NO)'$)
C READ( 5,102DIPLOT
IPLOT=0
C1021 FORMAT(11)
C WRITE(5,99)
C99 FORMAT(' HOW LONG TO RUN? ')
C READ(5,98)TIME
C98 FORMAT(G6.0)
CALL ASSIGN(1,FILE)
READ(1)E1,E2,E3,E4,E5,E6,E7,E8,E9,E10,E11,E12,E13,E14,E15,
C COEFFICIENTS **********************************************
+NUM,K1,K2,K3,K4,K5,K6,K7,K8,K9,K10,K11,K12,K13
+,XJ0INI,Q1,Q2,Q3,Q4
C INITIAL CONDITIONS*****************************************
CLOSE (UNIT=1)
Q1IC=Q1
Q2IOQ2
Q3IC=Q3
Q4IC=Q4
C********************************************************
C
C
WRITE(5,1012)K1,K10,K2,K11,K3,K12,K4,K13,K5,XJ0INI,K6,Q1,
+K7,Q2,K8,Q3,K9,Q4
1012 FORMAT(1X,'1-K1 1,G12.6,' 10-K10',G12.6/
+ 1X,
'2-K2 '4
,G12 6,'
11-K11',G12.6/
+ 1X,
'3-K3 4
, G12.6 '
12-K12',G12.6/
+ 1X,
'4-K4 '4
,G12.6,'
13-K13',G12.6/
+ 1X,
'5-K5 i
,G12.6,'
14-XJ0INI',G12.6/
+ 1X,
'6-K6 '4
,G12.6,'
15-Q1IC ,G12.6/
+ 1X,
'7-K7 '4
G12.6,'
16-Q2IC ,G12.6/
+ 1X,
'8-K8 '4
,G 12.6, '
17-Q3IC',G12.6/
+ 1X,
'9-K9 '4
,G12.6 '
18-Q4IC,G12.6/
+ 1X,
' INPUT VARIABLE
NUMBER TO VARY => '$)
READ ( 5,10 13) IVAL
1013 FORMAT(12)
WRITE (5,1014)ALPHA(IVAL),VARS(IVAL)
1014 FORMAT(' VARIABLE ',A4,' = ',G12.6/
+ HOW MUCH TO INCREMENT? '$)
READ(5,1015)XINC
1015 FORMAT(G15.6)


560 PRINT
570 IF Q$<>" Y" THEN 640
580 PRINT FNP$(1,T*2,X1+275),FNP$(2,T*2,X2+275)
,FNP$(3,T*2,X3+275),FNP$(4,T*2,5+J9/1)
590 PRINT FNP$(5,T*2/JO/100+5)
600 PRINT FNP$(7,T*2,Q/1000+385)
610 PRINT FNP$( 4,600,270) ;"T(S1) JR=" ;J9?"'"
620 PRINT FNP$(6,T*2,170+(100*J9/J0))
630 Q7=Q7+Q REM TOTAL Q TO GET AVERAGE
640 T=T+T9 IF T<360 THEN 370
650 PRINT FNP$(4,600,270);"T(S1)' "?"
660 J5=P1/J7*100
670 PRINT FNP$(7,0,270);"T(S1)'POW USED=";P1
;"POW AVAIL=";J7;"PERCENT USED=";J5;"AVE Q=";Q7/T
680 INPUT X
690 PRINT CHR$(27)+""
700 PRINT P1/J7;"FRACTION OF TOTAL POWER USED"
710 END


44
pathway (J2), and a quadratic pathway (J3). The tank has a
linear drain.
The model represents a system that can change its use
of three functional pathways to get energy. The linear
pathway represents the energy flow that a system can re
ceive without any feedback in this pathway, only pathway
resistance to the flow. Because it is a donor controlled
pathway, the system has no control on the flow. Diffusion
pathways are an example of this type of energy flow. The
linear pathway is very efficient because it takes almost
nothing to receive the energy.
The autocatalytic pathway has a feedback from the sys
tem storage for interacting with an energy source to facili
tate the capture of more energy. If energy is available to
support the storage this pathway may lead to a competitive
advantage over the linear pathway. The efficiency of the
autocatalytic pathway depends on the energy source, the
storage and the pathway coefficient. A pathway of this type
has the capability of capturing more available energy.
The quadratic pathway has a self-stimulating feedback
(see equation on Figure 10) from the storage to capture
available energy. Examples of cooperative feeding that may
fit this model are common in ecosystems such as pack hunting
by some carnivores, cell and organ system interactions and
the cooperative work by humans in developed nations.


PERCENT POWER USED
CO
U1


240
DO 5002 1=2,11
I DK
DO 5002 J=2,11
! DK
DO 5001 IT*11,1+1
! DK
DO 5001 JT-J*1,J+1
! DK
QXT=QXT+DK*(Q4(IT,JT)-Q4(I,J))*DT
SDK
5001
CONTINUE
! DK
Q4T(I,J)=Q4(I,J)+QXT
! DK
QXT=0.0
! DK
5002
CONTINUE
! DK
DO 5003 1=2,11
!DK
DO 5003 J=2,11
SDK
Q4(I,J)=Q4T(I,J)
SDK
5003
CONTINUE
T=T+DT
IF(T.LT.TTIME)GOTO300
SDK
C>>>>>> END OF MAIN LOOP <<<<<<
C
CALL GGOFF
DO 439 1=1,12
DO 439 J= 1,12
ICNT(I,J)=ICNT(I,J)*DT
439 CONTINUE
CALL ASSIGN(4,'SURF4')
WRITE(4,440)VERS,TITLE,BUF1,BUF2
440 FORMAT('1','SURFACE MODEL VERSION NO. ',F6.2,1X,10A1,
& 1X,9A1,1X,8A1)
WRITE(4,454)ETOT,ETYPE(IETYP),PROD,TOTPOW,TOTPOW/(TTIME*100.),
& GPP,CNSUMP,N,ISSUC,DK
454 FORMAT(1X,' INPUT ENERGY TOTAL= ',F10.2,' ENERGY TYPE ',A4/
& 1X,' TOTAL PRODUCTION =',G15.6/
& 1X,' TOTAL POWER USED =',G15.5,' AVE POWER/CELL = ',G15.6/
& 1X,' GPP= ',G15.6,' TOTAL CONSUMPTION* *,G15.6/
& 1X,' SEARCH LENGTH =',I3,' STARTING CONDITION = ',12/
& 1.X,' DIFFUSION COEFFICIENT = 'F7.5)
WRITE(4,455)TTIME,DT,Q4TOT/1000,Q4TOUT/1000.,
& (Q4TOUT+Q4TOT)/10 0 0.
455 FORMAT(1X,' FOR ',F10.0,'ITERATIONS DT= ',F7.3/
& 1X,' TOTAL NUTRIENTS (KG) = ',F10.2/
& 11X,'Q4 OUTER TOTAL (KG) = 'F10.2/
& 1 1X, 'TOTAL INNER AND OUTER (KG) = ',F10.2/
& 1X,10X,' NUTRIENT MATRIX Q4(I,J)'/)
WRITE(4,456)((Q4(I,J)/1000.,1=1,12),J=12,1,-1)
456 FORMAT(1X,12F7.2)
WRITE(4,457)PTHRSH,THRESH,Q2TOT/1000.
457 FORMAT(//1X,'VALUES FORQ2(I,J) PRODUCERS'/
& 1X,'PRODUCER THRESHOLD FOR CONSUMER MOVING* ',F10.2/
& 1X,'CONSUMER THRESHOLD FOR DIVIDING INTO = ',F10.2/
& 1X,' TOTAL PRODUCERS (KG) = *,F10.2)
WRITE(4,4561)((Q2(I,J)/1000.,1=2,11),J=11,2,-1)
4561 FORMAT(8X,10F7.2)
WRITE(4,4581)
4581 FORMAT(//' CONSUMER VISITATION MATRIX '/)
WRITE(4,4582)((ICNT(I,J),1=1,12),J=12,1,-1)
4582 FORMAT(12(1X,I6))


105
power as the effect was greater at low power than it was at
higher levels of power.
When the input power was varied as in the previous
example (50 to 300) but the initial condition of the con
sumer was started at a higher level (Q4INIT=50, lOx base run
value) the results were similar to the previous run but
damped (Figure 32a-f). The shift in time of the peak of the
climax species (Ql) was less than before and the amplitudes
of the initial peaks of Q2 and Q3 were less. Percent power
used per time increment also was higher in the earlier
stages of this run compared to the previous run (compare
Figure 32e with 30e). With higher initial levels of the
consumer, the model generated more power earlier through the
feedback of the consumer on the producers.
When the input power was held constant (J0=100) and the
initial condition of the consumer (Q4) varied, the model
displayed two different behaviors (Figure 33a-f). With few
consumers initially, the system crashed, unable to proceed
through the normal growth sequence. When the initial quant
ity of the consumers (Q4) was above a critical level, the
system grew and went through a normal growth sequence. A
sharp transition occurred in the percent power used as
Q4INIT was increased (Figure 34).
Because the consumer (Q4) was feeding back as a multi
plier to the producers, some minimum critical value must
exist for the consumer population to stabilize this model.


Figure 29 Simulation of the parallel production-consumption
model in Figure 13. See Figure 28 for legend and ordinate
scale.
(a) Simulation run with initial value of climax
species (Ql) set equal to zero.
(b) Simulation run with initial value of interme
diate species (Q2) set equal to zero.
(c) Simulation run with initial value of weed
species (Q3) set equal to zero.


36
Distribution
of trees over time


PE RCENT POWER USED
cs>


197
Table 5. Percent power used as a function of input energy
sources and diffusion in different ecosystem levels.
Successional
Steady state
(a) No Diffusion
Energy Distribution
Percent
Power Used
Hierarchical
96.5
96.6
Even
96.5
96.6
Random
96.6
96.6
(b) Diffusion
between nutrient (Q4) tanks
(Successional initial conditions)
Diffusion rate
.001 .01
.1
Energy Distribution
Percent Power
Used
Hierarchical
96.5 96.5
95.5
Even
96.5 96.5
96.5
Random
96.6 96.6
95.6
(c) Diffusion
between consumer (Q3) tanks
(Successional initial conditions)
Diffusion rate
.001 .01
.1
Energy Distribution
Percent Power
Used
Hierarchical
96.6 96.6
96.6
Even
96.5 96.5
96.5
Random
95.6 96.5
96.5
(a) Model DSP1. See Figure 50 for example run.
(b) Model DSP1. See Figure 54 for example run.
(c) Model DSP1QZ See Figure 53 for example run.


52


14
maximum power output, a general restatement of Lotka's
original idea of maximum energy flow but with an important
distinction. Odum (1971, 1982, 1983a, and 1983b) further
clarifies maximum power as useful power where 'use' is
feedback of the product of energy use to amplify other
pathways.
In describing cycles of life, death and regeneration,
Calow (1978) has found that although Lotka's principle
holds, there seem to be no a priori grounds for placing
restrictions on how this use of energy should be achieved.
He further stated that selection would have shifted in the
course of time from one of maximizing speed to maximizing
efficiency. This is a restatement of the strategy of eco
system development utilizing r and K growth (Odum 1969) .
Jantsch (1980) suggests than maximum engagement in
matter (i.e., energy storage) and maximum process intensity
(i.e., entropy production) are criteria for ecosystem
stability. Non-equilibrium structures thus come about by
fluctuations in the mechanisms which result in modifications
of the kinetic behavior of these structures.
Design for Maximum Power
The important question here is how do systems build
structure in order to maximize utilization of available
power. Odum's theory (1971 and 1983) is that by feeding
back energy (derived from structure that is being built)
reinforcement occurs that increases efficiencies and energy


SUBROUTINE REGLIN(N,X,Y,A,B,R2,CEE)
BASED ON PROGRAM IN 'COMMON BASIC PROGRAMS'
LON POOLE AND MARY BORCHERS P. 145
DIMENSION X( 1),Y( 1)
REAL J,K,L,M
J=0.0
K=0 0
L=0.0
M=0 0
A=0.0
B=0.0
R2=0.0
CEE=00
DO 100 1=1,N
J=J+X(I)
K=K+Y( I)
L=L+X(I)*X(I)
M=M+Y(I)*Y(I)
R2=R2+X(I)*Y(I)
CONTINUE
XN=N
B=(XN*R2-K*J)/(XN*L-J*J)
A=(K-B*J)/XN
J=B*(R2-J*K/XN)
M=M-(K**2)/XN
K=MJ
R2=J/M
CEE=SQRT(K/(XN-2))
RETURN
END


180
with small storages (i.e. fast turnover times), the quad
ratic pathway can be non-functional (Figure 21, pathway J3).
Because non-linear pathway flows are a function of both the
energy source and the storage, there are conditions when the
pathway has a threshold for operation (Figure 21, pathway J2
and J2+J3 and Figure 27 pathway J3). Low energy systems may
not have enough energy available to allow development of
these higher order pathways.
Human systems may be a good example of how these path
ways may operate. Nomadic, subsistence societies can be
considered as basically linear systems that utilize avail
able resources with few or no feedbacks. By developing
autocatalytic feedbacks, primitive societies move up to
developing societies building structures to process more
energy (farming, mining, transportation and manufacturing).
As growth continues, systems develop within society that
have higher order quadratic feedbacks to facilitate proces
sing energy (communications, banking and finance, and infor
mation systems). Because the higher order pathways are
dependent on storages and energy flows, the structures may
not be stable with reduced energy.
For a system pathway to utilize fluctuating energy
flows, it must have enough structure to sustain the system
when the non-linear pathways are not functioning (at lower
energy levels). While the nonlinear pathways were dependent
on the frequency and amplitude of input energy (Figures 25,
26 and 27) the linear pathway had no frequency dependency


228
R10=0
R11=0.
R12=0
EUSED=0 0
PAVAIL=0.0
C
C WRITE(5,1081)
C1081 FORMAT(' SCALE FACTOR FOR Q2200. OR 1000. [R] )
C READ(5,1082)SFACT
C1082 FORMAT(G7.2)
C WRITE( 5,108)
C108 FORMAT(' WHAT IS THE TIME INTERVAL DT [R] ')
C READ(5,109)DT
C109 FORMAT(G5.3)
TIME=750.
DT=. 1
SFACT=100.
NTIME=TIME
XDT=DT/10
C START OF LOOP ***********************************************
IF (IPLOT.EQ.0)GOTO5
CALL GGERA
CALL GGBOX(7,0,0,767,479)
CALL GGTEXT(7,626,475,XTEXT,1,0)
CALL GGBOX(7,620,452,767,479)
<2 PRINT INITIAL CONDITIONS*************************************
5 T=T+DT
C RATE EQUATIONS***********************************************
J=J0/(1+K13*Q1*Q4)
POWUSE=10 0.*(JO-J)/JO
PAVAIL=PAVAIL+J0
EUSED=EUSED+JO-J
R1=DT*K1*Q1*Q4*J
R2=DT*K2*Q1
R3=DT*K3*Q1
R4=DT*K4*Q1
R5=DT*K5*Q2
R6=DT*K6*Q2
R11=DT*K11*Q2
R7=DT*K7*Q2*Q3*Q3
R8=DT*K8*Q2*Q3*Q3
R9=DT*K9*Q2*Q3*Q3
R10=DT*K10*Q1*Q4*J
R12=DT*K12*Q3
C LEVEL EQUATIONS *********************************************
1091 CONTINUE
Q1=Q1+R1-R2
Q2=Q2+R3-R9-R11
Q3=Q3+R5+R7-R12
Q4=Q4+R4+R6+R12+R8-R10
IF(FT1(T,1.)GTDT)GOTO 110
IF(IPLOT.EQ.0)G0T02 0000
ITIME=T
IXC=Q1/10


BIBLIOGRAPHY
Abraham, R. H., and C. D. Shaw. 1984a. Dynamics-The
Geometry of Behavior. Part 1: Periodic Behavior.
Aerial Press. Santa Cruz, Calif. pp. 220.
Abraham, R. H., and C. D. Shaw. 1984b. Dynamics-The
Geometry of Behavior. Part 2: Chaotic Behavior. Aerial
Press. Santa Cruz, Calif, pp. 139.
Alexander, J. F. 1978. Energy basis of disasters and the
cycles of order and disorder. Ph.D. dissertation.
University of Florida. Gainesville, Fla.
Allen, T. F. H., and T. 3. Starr. 1982. Hierarchy.
Perspectives for Ecological Complexity. University of
Chicago Press. Chicago, pp. 310.
Anderson, D. J. 1986. Ecological sucession. In J. Kikkawa
and D. J. Anderson (eds) Community Ecology, Pattern
and Process. 31ackwell Scientific Publication.
Melbourne, Australia.
Bennett, R. J., and R. J. Chorley. 1978. Environmental
Systems. Methuen. London, pp. 624.
Botkin, D. B., J. F. Janak, and J. R. Wallis. 1972. Some
ecological consequences of a computer model of forest
growth. J. Ecology 60:349-872.
Bray, W. C. 1921. A periodic reaction in homogeneous
solution and its relation to catalysis. Journal of
American Chemical Society 43:1262-1267.
Brokaw, N. V. L. 1982a. The definition of treefall gap and
its effect on measures of forest dynamics. Biotropica
14:158-160.
Brokaw, N. V. L. 1982b. Treefalls: frequency, timing and
consequences. In E. G. Leigh Jr., A. S. Rand, and D. M.
Windsor (eds). The Ecology of a Tropical Forest.
Smithsonian Institution Press. Washington, pp. 101-
108.
Brokaw, N. V. L. 1985a. Gap-phase regeneration in a
tropical forest. Ecology 66:682-687.
247


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233
147
C
151
C
C
51
511
551
552
52
521
C
C
C
C
ETOT=0.
TOTPOW=0.0
T=0.
T1= 1
T2=T1
¡TOTAL ENERGY INPUT (SUM OF MATRIX)
¡MEASURE TOTAL POWER USED: SUM OF EUSED
¡
¡NUMBER OF CONSUMERS
Q1IC=1000. ¡1C CONDITION FOR Q1
Q4IC=39000.
Q2IC=1000.
IF(ISSUC.EQ.O)GOTO147
Q4IC=30000. !IC CONDITION FOR NUTRIENT TANK
Q2IC=10000. !IC FOR PRODUCER STORAGE
CONTINUE
Q3IC=50
THRESH=500. ¡DOUBLING THRESHOLD FOR CONSUMER
IF (IOFLAG.EQ.O)GOT05 21
CALL GGON
IF (IPTR.EQ.0)GOTO151
WRITE(3,20171)
CALL GGINIT
CALL GGAXIS(0,0,767,479)
CALL GGERA
CALL GGBOX(7,0,0,767,479)
WRITE(3,51)
¡3.0
¡3.0
¡3.0
I 3.0
CLEAR MACROS AND DEFINE ONE TO DRAW BOXES
FORMAT( +', @. @:A P[+0,+0]W(S1)V[,+24]V[+24,]V[,-24]V[-24,]
+ W( SO) @ ?')
WRITE (3,511)
FORMAT ( +' @ : B T(A1) P[+0 ,+0] V[,+24] V[+24 ] V[,-24] V[-24 ]
+ W(SO)T(A0) <3;')
WRITE(3,551)
FORMAT('+L(A1)'/
+'+LM 7"FFFFFFFFFFFFFFFFFFFF;'/
+'+L"6"AA55AA55AA55AA55AA55;'/
+'+L"5"92492492492492492492;'/
+'+L"3"84210842103421084210;')
WRITE(3,552)
FORMAT('+L"4"88442211884422118844;'/
+,+L"2"42009100240091004200; '/
+'+L"1"20000840021000042000;'/
+ '+L"0"00002000000200002000; '/
+'+L"B"00000000000000000000;')
DO 52 1=0,7
CALL GGPLT(1,735,(1+1)*24-16,0)
CHAR=I+48
WRITE(3,398)CHAR
CALL GGBOX(7,725,0,767,248)
CALL GGBOX(7,0,0,767,248)
CALL GGBOX(7,575,0,725,248)
CONTINUE


219
1420 INPUT ANS$
1440 IF (ANS$ ="R")GOTO 1260
1480 PRINT #2,XT(1), YT(1),PENDUM;AREA;PERIOUT;NPOINT
1481 PRINT #3,AREA;PERIOUT
1490 PRINT #2/XT( 1) ,YT(1)/PENUP
1500 FOR 1=2 TO NPOINT
1510 PRINT #2,XT(I), YT(I),PENDWN
1520 NEXT I:'200 CONTINUE
1525 IF WET5=99 THEN 1590
1530 PRINT #2,XT(1),YT(1),PENDWN
1590 PRINT M DO YOU WANT TO INPUT ANOTHER AREA (Y OR N)
1600 INPUT ANS$
1620 IF (ANS$ = "Y") GOTO 1260
1630 CLOSE:PRINT "TYPE SYSTEM TO EXIT FROM BASIC OR RUN TO RERUN":END
1640 ****************************************************************
1641 ************************** END OF MAIN PROGRAM *****************
1642 '***************************************************************
1660 PRINT "SUBROUTINE AREAP":IERR=0
SUBROUTINE AREAP(X,Y,AREA,NPOINT,IERR,RESOL)
1700 IERR=0
1710 NPOINT=0
1720 AREA=0
1740 PRINT ENTER FIRST POINT BY PRESSING THE '1* KEY "
1745 PRINT ENTER REMAINING POINTS BY PRESSING ANY KEY BUT '2'"
1750 PRINT THEN QUIT ENTERING POINTS BY PRESSING '2' "
1760 GOSUB 9000:' CALL DIGURU (XIN,YIN/CODE)
1770 XOLD=XIN
1780 YOLD=YIN
1790 IF (CODE$ <> "1") GOTO 1760
1800 NPOINT=1
1810 X(NPOINT)=XOLD
1811 XT(NPOINT)=XTRUE
1820 Y(NPOINT)=YOLD
1821 YT(NPOINT)=YTRUE
1830 IF (CODE$ = "2") GOTO 1950
1840 GOSUB 9000:'200 CALL DIGURU (XIN,YIN,CODE)
1850 IF (CODE? = "2" ) GOTO 1950
1870 NPOINT=NPOINT+1
1880 X(NPOINT)=XIN
1881 XT(NPOINT)=XTRUE
1890 Y(NPOINT)=YIN
1891 YT(NPOINT)=YTRUE
1900 XOLD=XIN
1910 YOLD=YIN
1930 PRINT XIN,YIN,"CODE =";CODE$:BEEP
1940 GOTO 1840
1950 IF (NPOINT < 3) THEN IERR =1: '300
1970 IF (IERR <> 0)THEN RETURN
2020 IERR= 0
2025 'GOSUB 3900
2030 RETURN
2070 PRINT SUBROUTINE AREAX( X, Y,AREAIO,NPOINT)"
2130 PRINT **** DIGITIZER AREA CALCULATION ****"
2140 NUMPNT=0:'200