Title: Energy systems and inertial oscillators
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Title: Energy systems and inertial oscillators
Physical Description: xiv, 242 leaves : ill. ; 28 cm.
Language: English
Creator: Zwick, Paul Dean, 1946-
Copyright Date: 1985
 Subjects
Subject: Force and energy   ( lcsh )
Bioenergetics   ( lcsh )
Oscillations   ( lcsh )
Environmental Engineering Sciences thesis Ph. D
Dissertations, Academic -- Environmental Engineering Sciences -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
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Thesis: Thesis (Ph. D.)--University of Florida, 1985.
Bibliography: Bibliography: leaves 238-241.
Additional Physical Form: Also available on World Wide Web
General Note: Typescript.
General Note: Vita.
Statement of Responsibility: by Paul Dean Zwick.
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Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000873478
notis - AEH0783
oclc - 014589026

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ENERGY SYSTEMS AND INERTIAL OSCILLATORS


by

PAUL DEAN ZWICK















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






UNIVERSITY OF FLORIDA


1985










ACKNOWLEDGEMENTS


I would like to thank my advisor and committee chairman, Dr. John

F. Alexander Jr., for his guidance and patience during the research and

writing of this dissertation. I also wish to thank Dr. Howard T. Odum

for his aid and discussion of the general concepts of energy

selectivity. Dr. Earl M. Starnes, Dr. Frank G. Nordlie, and Dr. George

R. Best must also be thanked for their fruitful discussions and input to

this research.

I also wish to thank my fellow students for listening to all the

talk about autocatalytic oscillators, and in particular for the pointed

questions regarding the operation of rhythmic guides. In addition Kathy

Levy must be thanked for finalizing the transcript for the Graduate

School.

Finally, I wish to dedicate this dissertation to my spouse Malea L.

Zwick. Malea's patience, support, humor, and self sacrifice allowed the

work to continue. Without her support and dedication this work may

never have been completed.









TABLE OF CONTENTS


Page

ACKNOWLEDGEMENTS ................................................. ii

L IST OF TABLES .................................................. vi

LIST OF FIGURES ................................................. vii

LIST OF SYMBOLS ................................................. xii

ABSTRACT ........................................................ xiii

INTRODUCTION .................................................... 1

Previous Study of Electrical and Mechanical Inertial
Mechanisms ........................................ 5
Previous Studies of Inertial Mechanisms in Environmental
Systems ........................................... 11
Internal Timing vs External Timing of Inertial
Oscillators ....................................... 14
Definition of Energy Selectivity ....................... 15


METHODS ......................................................... 17

Overall Research Procedure ............................. 18
Phase I ........................................... 18
Phase II .......................................... 18
Phase III ......................................... 19
Phase IV .......................................... 19
Phase V ........................................... 20
Energetics Diagramming ................................. 20
Identification of the System Boundary ............. 21
Identification of Energy Flows Across the
Boundary ....................................... 21
Organization of the System Components ............. 21
Connection of System Components With
External Sources ............................... 26
Diagramming the Subsystems ........................ 26
Evaluation of the Energy Circuit Model ............ 26
Translation of the Energy Circuit Model into
its Corresponding Differential Equations ....... 27
Simulation of the Model ........................... 30
Validation of the Model ........................... 30





Page

RESULTS ......................................................... 32

Concepts ............................................... 32
Electrical and Mechanical Oscillator
32
Models ......................................... 3
System components ............................
Power transfer and frequency selectivity...... 43
Autocatalytic Inertial Oscillator Model ........... 54
Autocatalytic inertial oscillator timing
functions .................................... 57
Autocatalytic inertial oscillator phasing with
an external source ........................... 64
Frequency setting and damping ................ 69
Biological Models with Autocatalytic Inertial
Oscillators ...................... ............ 74
Energy Selective Biomass Generation ............... 81
Power Transfer for an Energy Selective
Producer ....................................... 84
Competition Between Energy Selective Units ............. 90
Simulation 1: Competitive Exclusion of a
Simple Generalist by a Simple Specalist ........ 95
Simulation 2: Competition Between Identical
Producers ...................................... 101
Simulation 3: Competitive Exclusion of a
Simple Specialist by a Simple Generalist ........ 109
Simulation 4: Competition of two Producers
for a Limited Resource Within Their Niche
Acceptance Ranges but not at Either Producers'
Free Running Rhythm ............................ 115
Simulation 5: Competition of Similar
Producers with Overlapping Niches .............. 121
Simulation 6: Competition for Flow Limited
Energies Compossed of the Addition of two
Fundamental Cycles about a Mean Energy
Level .......................................... 126
Simulation 7: Competition for Flow-Limited
Energies Composed of two Fundamental
Cycles by Multiplication About a Mean
Energy Level ................................... 132
Simulation 8: Competition Between a two-
Rhythm Complex Generalist and a Single-
Rhythm Specialist for Flow-Limited Source
Composed of Two Fundamental Cycles
by Addition .................................... 138
Models for a Selected Spartfna Salt Marsh .............. 144
Original Kaituna Salt Marsh Minimodel ............. 145
Kaituna Salt Marsh Minimodel with a Solar Input
Energy Annual Rhythmic Pattern Constructed
of a Series of Diurnal Rhythms .................. 153
Kiatuna Salt Marsh Minimodel with the Addition
of an Autocatalytic Clock as an Internal
Behavior Guide for the Spartina Producer ....... 161





Page

Kiatuna Salt Marsh Minimodel with Spartina
Production Module which has an
Autocatalytic Oscillator Producing a
Niche Incapable of Utilizing the Solar
Input Energy ................................... 1
Kaituna Salt Marsh Minimodel with Competitive
Exclusion of a Producer with an Inadequate
N iche .......................................... 75
Kaituna Salt Marsh Minimodel to Include Compe-
tition Between Producers with the Ability
to Obtain Solar Energy at Different Rhythms ..... 184
Kaituna Salt Marsh Minimodel with Competition
Between an Energy Selective Producer and
a Non-Energy Selective Producer for a clow-
Limited Cycle Solar Energy Supply ...............
Kaituna Salt Marsh Minimodel with a Two-Oscllator
Producer and an Energy Source Composed of
Two Fundamental Rhythms ......................... 200

DISCUSSION ....................................................... 209

The Autocatalytic Inertial Oscillator ................... 209
External and Internal Timing of System
Oscillations ....................................... 211
Energy Selectivity Systems .............................. 211
Energy Selectivity and the Autonomous Inertial
Oscillator ...................................... 212
Energy Selectivity and the System .................. 215
Energy Selectivity and the Energy Acceptance Niche .216
Energy Selectivity and Diversity ................... 217
Future Investigations .................................... 219
The Autocatalytic Inertial Oscillator and Design
Characteristics ....................................
Energy Selectivity ................................. 220
The Effects of Energy Selectivity Within
Autocatalytic Systems ........................... 221
Energy Selectivity and Competitive
Strength in Constant Conditions ................. 221
Energy Selectivity and the Effects of
High Quality Pulsing Feedback ................... 222
The Possible Relevance of Autocatalytic Oscillators
to Physiological Clocks within Organisms ........ 222


APPENDIX COMPUTER PROGRAMS ..................... ...... ...... ... 228

BIBLIOGRAPHY ..................................................... 238

BIOGRAPHICAL SKETCH .............................................. 242
BIOGRAPHICAL SKETCH ..............................................










LIST OF TABLES


Table Page

1 Evaluation, Name, Description, and Equations of the
Energy Flows and Storages for the Electrical Band-
pass Oscillator Circuit ................................... 36

2 Evaluation, Name, Description, and Equations of the
Energy Flows and Stroages for the Mechanical Mass-
Spring Oscillator ......................................... 42

3 Evaluation, Name, Description, and Equations of the
Energy Flows and Storages in the Model of the Energy
Selective Producer ........................................ 77

4 Evaluation, Name, Description, and Equations of the
Energy Flows and Storages in the Model for Energy
Selective Competition ..................................... 93

5 Evaluation, Name, Description, and Equations of the
Energy Flows and Storages of the Original Kaituna
Salt Marsh Model .......................................... 148

6 Evaluation, Name, Description, and Equations for the
Modified Kaituna Mini-model ............................... 166

7 Evaluation, Name, Description, and Equations for the
Competition Modified Kaituna Mini-model ................... 183

8 Evaluation, Name, Description, and Equations for the
Energy Flows and Storages for Competition Between
Energy Selective and Non-energy Modules in the Kaituna
Marsh Model ............................................... 198

9 Evaluation, Name, Description, and Equations for the
Energy Flows and Storages in the Kaituna Marsh Model
with the Two Clock Complex Producer ....................... 208









LIST OF FIGURES


Figure PagE

1 Fundamental inert;al 0omponenr .......................... 3

2 Electrical Inert a31 Oscillator ........................... 7

5 Mechan cal Inert'al ;sci ll at r ........................... 10

4 Van Der Pol Oscillator ................................... 13

5 Energy C'rcu't Symbols .................................... 25

6 Diagram ShoNing tne Energy CircuiT '-'odel Employed to
E DifterentIal Equations ..........................* .......... 29

7 Tne Electr'cal Tank Circu*t Csc*lla or ....................

8 Plot of Electrical Oscillator System S3ora'es vs.
39
Time ...................... ................................

The Mechanical '.lass, Spring, Damper Osc Ilator ............

10 The Pesis+ive Power Transfer Character' s'cs for -ne
45
Bandpass Electrical Oscillator ............................

11 The Frecuency Power Character sTrcs ot tne cjnpfss
E electrical Oscillator .....................................

12 Summary ot the Power Characterist;cs 'or +he -.inp tJ ;
50
Electrical Oscillator .....................................

15 Diagram Shong thie "athemat;cal Pelar onsh'DS r Ine
Bandpass Electricjl Oscillator ............................ 53

14 Enerqy Circuit Diagram of the ^utoc.'~. I, -tc -Ic
56
C lock ................ ......................................

15 Energy Circuit Model Js:ng The 4u-'ocat'- c Icc-
59
as Dart of a ,iork a3te .....................................

1b Biological Tim'ng Nth o en' ai :omponen ................ 61

17 Biological Tim'ng Using the Source Phytnh as well as
the Potentaal 3nd Kinetn c Components of the Clock .........





Figure Page

18 Biological Timing as a Result of Clock Component
Amplitudes ................................................ 66

19 Complex Biological Behavioral Timing as a Result of
the Combination of the Previous Timing Initiators ......... 68

20 Plot of Autocatalytic Clock's Ability to Self-Adjust
to a Change in Source Rhythm .............................. 71

21 Energy Circuit Diagram of an Energy Selective
Producer .................................................. 76

22 Plot of Energy Acceptance Niche for an Energy Selective
Unit with a Intrinsic Free-Running Rhythm Equal to
One Day ................................................... 80

23 Plot of Biomass vs. Time for an Energy Selective
Producer .................................................. 83

24 Plot of Power Characteristics for an Energy Selective
Producer .................................................. 86

25 Plot of Biomass, Unit Power, and Power Transfer for
an Energy Selective Producer with an Intrinsic Free-
Running Rhythmic Period Longer Than the Period of the
Available Energy .......................................... 89

26 Energy Circuit Diagram for Energy Selective
Competition ............................................... 92

27 Simulation 1 System Relative Period and Energy-acceptance
Niche Diagrams ............................................ 98

28 Simulation Plots for Competition Between a Specialist
and a Simple Generalist ................................... 100

29 Energy Circuit Diagram for Competition Between Identical
Energy Selective Producers ................................ 103

30 Simulation 2 System Relative Period and Energy-acceptance
Niche Diagrams ............................................ 106

31 Simulation Plots for Competition Between Identical
Energy Selective Producers ................................ 108

32 Simulation 3 System Relative Period and Energy-acceptance
Niche Diagrams ............................................ 111

33 Simulation Plots for the Competitive Exclusion of a
Specialist by a Simple Generalist ......................... 114

34 Simulation 4 System Relative Period and Energy-acceptance
Niche Diagrams ............................................ 117

viii





Figure Page

35 Simulation Plots for the Competition Between Energy
Selective Producers EKisting at the Extremities of
Their Energy Acceptance Niches ............................ 120

36 Simulation 5 System Relative Period and Energy-acceptance
Niche Diagrams ............................................ 123


37 Simulation Plots for Competition of Similar Energy
Selective Oroducers with Overlapping Energy
Acceptance Niches ......................................... 125

38 Simulation 6 System Relative Period and Energy-accep-ance
Niche Diagrams ............................................ 129

39 Simulation Plots for Competiton Between Energy Select;'e
Producers for Additive Energy of Different Freauencies
from the same Energy Quality Source ....................... 131

40 Simulation 7 System Relative Period and Energy-acceptance
Niche D iagrams ............................................ 135

41 Simulation Plots for Competition Between Energy
Selective Producers for Multiplicative Ener'y of
Different Frequencies from the same Energy Quality
Source .................................................... 137

42 Simulation 8 System Relative Period and Energy-acceptance
Niche Diagrams ............................................ 140

43 Simulation Plots for Competition Between an Energy
Selective Complex Generalist and a Specialist ............. 143

44 Energy Circuit Diagram of the Original Ka;tuna Salt
Marsh Model ............................................... 147

45 The System Period Relationships for the Original Kiatuna
Marsh Mode l ............................................... 150

46 Simulation Plots for the Original raituna Marsh Model ..... 152

47 Illustration of Input Energy Change :rom a ''ean Energy
Input to a Diurnal Energy Input ........................... 156

48 The System Period Relationships for the Original Kiatjna
Marsh Model Modified to Include Diurnal Source Rhythm ..... 158

49 Simulation Plots for the Original Kaituna Marsh Model
with the Addition of Diurnal Solar Energy Input ........... 160

50 The System Period Relationships and Energy-acceptance
Niche Diagrams for the Modified Kaituna Model with
One Energy Selective Spartina Producer .................... 63

ix






Figure Page

51 Energy Circuit Diagram of the Kaituna Marsh Model
Modified to Incorporate an Energy Selective Spartina 165
Producer ..................................................
169
52 Simulation Plots for the Modified Kaituna Marsh Model .....169

53 The System Period Relationships and Energy-acceptance
Niche Diagrams for the Modified Kaituna Model with
One Energy Selective Spartina Producer
with an Inadequate Acceptance Niche for the Available
Solar Energy .............................................. 172

54 Simulation Plots for the Modified Kaituna Model with
a Spartina Producer with an Energy Acceptance Niche
Incapable of Utilizing the Input Solar Energy ............. 174

55 The System Period Relationships and Energy-acceptance
Niche Diagrams for the Modified Kaituna Model with
Two Energy Selective Producers in Competition for
Available Source Solar Energy ............................. 177

56 Kaituna Marsh Model Modified to Include an Energy
Selective Competitor for the Spartina Producer ............ 180

57 Simulation Plots for The Competition Modified Kaituna
Marsh Model Showing the Competitive Exclusion of a
Producer with an Inadequate Energy Acceptance Niche ....... 182

58 The System Period Relationships and Energy-acceptance
Niche Diagrams for the Modified Kaituna Model with
Two Energy Selective Producers in Competition for
Available Source Solar Energy ............................. 187


59 Simulation Plots for the Competition Modified Model of
the Kaituna Marsh Between Competitive Producers for
Different Energies from the same Energy Quality
Source .................................................... 190

60 The System Period Relationships and Energy-acceptance
Niche Diagrams for the Modified Kaituna Model with
One Energy Selective Producers and One Non-energy-
selective Producer in Competition for Available
Source Solar Energy ....................................... 193

61 Energy Circuit Diagram for Competition Within the
Modified Kaituna Marsh Model Between an Energy Selective
Producer and a Non-Energy Selective Producer .............. 195

62 Simulation Plots for Competition Between Energy
Selective Producer and a Non-Energy Selective
Producer .................................................. 197





Figure Page

63 The System Period Relationships and Energy-acceptance
Niche Diagrams for the Modified Kaituna Model with
One Two-rhythmed Energy Selective Producers ............... 202

64 Energy Circuit Diagram of the Kaituna Marsh Model with
a Two Osci lator Producer ................................. 205

65 Simulation Plots of a Two Oscillator Producer Within the
Kaituna Marsh Web Model .................................... 207

66 Energy Circuit Diagram for a Temperature Compensated
Autocatalytic Inertial Oscillator ......................... 226









LIST OF SYMBOLS


Symbol Definition

RI Electrical load resistance

Rs Electrical source resistance

L Inductor within an electrical system

C Capacitor within an electrical system

f Frequency at high end of electrical oscillator
h bandwidth

f Frequency at low end of electrical oscillator
I bandwidth

to Intrinsic frequency of oscillator

fr Resonant frequency of circuit, same as to

DD Total darkness 24 hours/day

LD Combination of light and darkness during the day

BW Bandwidth of oscillator

P Potential component of clock or oscillator

K Kinetic component of clock or oscillator

DT Differential time










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



ENERGY SYSTEMS AND INERTIAL OSCILLATORS

By

PAUL DEAN ZWICK

December 1985




Chairman: John F. Alexander Jr.
Major Department: Environmental Engineering Sciences


The objective of this dissertation was to analyze the effect of

inertial oscillators in energy systems. Traditionally defined

in differential equations for hardware, generalized systems diagramming

is used to extend the knowledge of inertial mechanisms to environmental

systems. Equations and diagrams are presented together, first in

definition and review, then as elements that are incorporated in

ecosystem patterns.

Starting with the simplest configurations, inertial elements were

combined into oscillators and these into ecosystem configurations. Each

model uses computer simulations to characterize response in the time and

frequency domains. A unit oscillator mechanism was included as a clock

mechanism in ecosystem prototypes. Potential for this kind of a

mechanism was considered in competition, energy selectivity (the ability


xiii






to acquire energy input in a selective manner through use of an inertial

oscillator), diversity, and survival of systems. For each model,

diagrams, equations, time simulations, and frequency responses are

given.

The results indicate that the development of an internal

oscillation, accomplished for this research by an autocatalytic inertial

oscillator, increases the efficiency of energy acquisition within a range

of acceptable frequencies, increases power transfer for the system to

other systems within the ecosystem, increases ecosystem diversity, and

provides the system with a greater competitive advantage within its

niche. The most competitive system is produced as a result of its

ability to generate multiple internal rhythms that it then employs to

better time changes in its environment. Competitive coexistance and

exclusion are produced by combinations of system configurations emolying

internal inertial oscillators.


xiv









INTRODUCTION


This research is a direct outgrowth from an understanding of the

operation of some simple oscillator systems. Knowledge of tne energy

and power input and transfer characteristics of simple oscillators lead

to the idea that many if not all systems employ a combination of

inertial components to produce internal oscillations. In many

configurations that use inertial elements, inertial oscillators are

formed. Since backforce actions are known in the behavior of biological

and social systems not involving hardware, inertial elements and

oscillators may exist in environmental systems or may be available for

ecological engineering designs of environmental systems of the future.

Therefore, this dissertation is a study of the role of inertial elements

and inertial oscillators in basic configurations of ecosystems.

Traditionally defined in differential equations for hardware,

generalized systems diagramming helps the extention of existing

knowledge on inertial mechanisms to environmental systems. In the

following sections equations and diagrams are presented together, first

in definitions and review, then as elements that are incorporated in

ecosystem patterns. The simplest inertial element and its equations are

shown in Figure 1. An important characteristic of this simple inertial

component is the integral dependency of the flow (F) through its

pathway. The flow through this inertial component is dependent upon the

value of its inductance (L) and the integral of the potential energy (S)

forcing the flow through the pathway. The energy stored in the

































Figure 1 :


Fundamental inertial component. The figure shows the
simplest inertial component. Inertial backforce is
generated by the component as a result of its induced
magnetic field. The equations describe the flow through
the inertial component and the energy stored by the
component in its magnetic field.
























S-- -Magnetic Field


L=Component inductance
in Henries


Potential
Energy
Source










magnetic field around the element is related to the inductance of the

component (L) and the square of the flow (F). It is the energy stored

in the magnetic field of the inductor that opposes any change in flow

through the element. The opposition to changes in the flow through the

element, by the fluctuating magnetic field, produces the inertial

backforce generated by the element.

In this study system configurations with inertial mechanisms and

oscillators were analyzed and related to the design of environmental

systems. The energy circuit model simulations and plots explain and/or

define the concept of energy-selective power utilization and transfer.

First, two simple oscillator models are presented, one an electrical

resistance-inductor-capacitor circuit, and the other a mechanical mass-

spring-damper model. Second, an autocatalytic inertial oscillator is

shown. The autocatalytic inertial oscillator model provides a means of

introducing an internal control behavior into energy circuit systems

which in turn produce energy-selective behavior within models of energy

circuit producers and consumers. Next, the simulations for models of

energy-selective competition are shown. This series of models analyzes

the advantages or disadvantages of energy-selective behavior for

individual producers in competitive situations. The model simulations

also aid in developing an insight into the overall system benefits from

an individual system's energy-selective behavior. The last series of

model simulations examine the effects of including inertial oscillators

into a salt marsh located in New Zealand. The model was previously

calibrated with biomass data and simulated seasonal productivity

(Campbell, 1984). The Kaitun marsh simulations are the result of

modifications to the original Kaituna marsh model. Energy selective





5

producers are substituted into the marsh model in place of the or'*'nal

non-energy-selective producers in order to analyze the effects ot enerq/

selective behavior within a known system. Following the 7nser-'on of

energy-selective behavior into the Kaituna marsh model, the effec' of

competition is shown by the addition of competing energy-selec+'.e

producers into the modified marsh model.


Previous Study of Electrical and Mechanical Inertial Mechanisms

The ab lity to act with a behavior tnat produces a selec' .e

acquisition of input energy relies upon the ability of the system 'o

produce internal oscillations. The simple electrical ana m-chan'cal

oscillators provide an example of the concept. Electrical circuits 'nat

exhibit their own internal inertial oscillations are called Danlpass

oscillators or filters, blocking filters, tuners, receivers, or simply

oscillators. One example of a mechanical oscillating system 's an

automobile's suspension system.

Figure 2 shows the electrical bandpass oscillator, f7rst /l T

electrical diagramming and second with energy circuit Jiagramm'ng. 'he

differential equations for both diagrams are presented. 'h, "nertal

generated oscillation is the result of a transfer of energy :nternall/

between the two components of the system, the capacitor (3) anj the

inductor (L). Interestingly the period of the free-running osc llat'on

is directly related to the two components (C) and (L) and no- ro 'he

rate of flow of the electrons.

The frequency setting coefficient k3 is a constant and may be

calculated usiny two different methods. The first method is the most

direct and is solely dependent upon the inductance (L) and capacitance

(C) of the circuit (see equation 6). The second method relies upon































Figure 2:


The electrical inertial oscillator. The figure shows both
the electrical diagram and the energy circuit for this
inertial oscillator. The equations describe the rate of
change of the charge (storage Q) in the oscillator. The
frequency setting coefficient k3 is a constant for flow
through patway F3 and is calculated from the system
component values. Coefficient k3 may also be calculated
from the free-running period of the inertial oscillator.
Figure (a) is the electrical diagram. Figure (b) fs the
energy circuit diagram.












R

SS

2 c L R1

iI 2 i3





dQ/dt=1/Rs*(S-(Q/C))-(1/R*C)*Q-1/(L*C).'Q
iI=1/Rs*(S-(O/C)) =1/(L*C)0.5

i2=(1/L*C):OQ 2=1/(L*C)=2.0*-*fQ
i3=(1/R *C)-Q O=storage of charge in
caoacitor (C)
C=capacitance (Farads)
L=inductance (Henries)
f =intrinsic freauency of tne
oscillator

(a)

1/Rs*(S-(Q/C))--,

------ -- ---- I/(L C). O

/(R C)*Q )
k1 =1/1 ,


k =1/(L*C)= -

dQ/dt=1/Rs*(S-(Q/C))-1/(RI*C)*Q-1/(L*C)'Q
dQ/dt=kl*(S-(Q/C))--k2*Qk3*:O

Energy in Q=02/(2*C) Energy in K=(L/2)*(l/(L*C)*.'O)2 /(8LC)








8


being able to discern the free-running period of the circuit. Equation

6 also provides the calculations for period method to determine k3.

Which ever method is employed to calculate frequency setting the end

result is that k3 is a constant and not a variable flow coefficient.

Figure 3 shows the mechanical mass-spring-damper inertial

oscillator. The figure presents the mechanical diagram and the energy

circuit diagram with their accompanying differential equations. The

mechanical oscillator is much like the electrical oscillator and with

the exception of the system components and units has differential

equations that are much alike.

The frequency setting coefficient (k3) for the mechanical

oscillator may also be calculated using two methods, and is a constant

and not a variable.

The previous figures clearly show that the basic design for

development of an internal oscillation in simple inertial oscillating

systems is the combination of two components that act in a reciprocal

manner. These components, one potential and the other kinetic combine

to transfer energy between themselves and as a result provide their

system with an internal inertial oscillation. Furthermore, the simple

oscillator systems, such as the electrical and mechanical oscillators,

maximize their power transfer in the most efficient manner when the

period of their internal oscillations exactly match those of their

external supply. When the period of the system's internal oscillation

is longer or shorter than that of its energy source, the power transfer

and power input to the system decreases.































Figure 3:


The mechanical inertial oscillator. The figure shows the
mechanical diagram and the energy circuit model for the
mechanical inertial oscillator. The equations describe
the rate of change of the mass position (x). The
coefficient k3 is a constant for setting the period of the
system oscillation and is calculated from the system
components or from knowing the system free-running period.
Figure (a) is the mechanical diagram and (b) is the energy
circuit diagram.


















D x=change in mass position
1 m=mass
f =intrinsic frequency of
MASS 0the oscillator
ks=spring constant

D2 Spring



dx/dt=1/01*(S-X*ks)-(ks/D2)*X-(ks/m) *.


(a)


(ks/D2)


ks /m) X




k2=kks02
s,, "5
k3 "S


dx/dt=1/Dl*(S-X*ks)-(ks/D2)*X-(ks/m)0.5

dx/dt=kl*(S-k*X)-k2*X-k 3*' X


1/D *(S-X*k )-,
I J j





11

Previous Studies of Inertial Mechanisms In Environmental Systems

Alexander i?7-, s*ua'ed lhe cycles o' ore- an 'sor- :er r, : cr.

a quanti+ative theory base upon energy anal sls 'or gloo3l cyc'es.

surges, and sasters. STorms, ear'lhq ias es, ,r an "'-es, I i c s,

volcanic eruptions, wars, and forest f res oere rcdeled.e .r:e r

disorder were recognized as comDonen- parts of 3 c'cle +t at a -,.J

disorder to recycle ordered material to stmnul ate ne *'ron i -'- --

s/stem. global model as employee by le-anrer (1 ') +tc ,; I ,i

effects of the cascaaing of catas*ophic events A1th'n a 3eb s's-:-.

One particular model developedd by Lle I ander (''7- th7 E .c;n n

Surge ''odel, was employed to simulate actual d'sas-er process s.

model, produced a pulse self-started J sorder, Thereo,' r-ec.l '" r :

to disorder ana producing a new growth period.

CamDbell 19_4, 'nvest'qated -ne role ot ecosstem s'-u r jr- "

absorbing or filtering external energy inpujs to the ecos,s--m.

organization of system filter properT'es A-s shcon r

ecosystem with the ability to adapt to a ,variety o :npr-j tr- j.n -'

Turnover time for 'he structural storaces .'th'n svs*e'n l en '"

to the input energy frequencies usually determined The power _jpj'r-e

the sys-em. Furthermore, Campbell discovered that *r.e h' r fr- J

cutoff of the individual components *n an ecosystem an 'heTr jr -' jl ar

resonant peaks ere determined by the turnover '"-e '* t'- '-:jr...

However, these oscillators do no involve inert al elements.

In h's book Systems Ecolony: A'n Introduct on c.dum 1 13, =reb-nr n

number ot models that utilize poten-1 l and K'ne "cr comooners. "u:'i

also developed the energy symbol for inerf'al response--a bearneJ 3an-

(Figure 5) (Odur, 1967). The energy circuit moel shohn 'n F' jre .-as



































Figure 4:


Odum's diagram of the Van Der Pol oscillator. The model
utilizes a kinetic storage as a backforce to produce the
system osci nation. Odum 1983, pages 176-77.














13























CC JT .CL
















c: G Y



<^ ^ ,~~~- = -c e e a :"


F??CE BALA''CEZ

EV(1-X() = kX-c'7






14

drawn by Odum as a translation of the Van Der Pol oscillator. An

oscillation is developed in the system because of the presence of an

opposing internal backforce. The system does not contain a potential

storage, only a storage of velocity energy, but an oscillation is

produced by the interaction of an inertial backforce, a frictional

force, and a driving force.

Odum, 1983 further describes the control effect of pulses on

timers. Using a model of a mechanical chain which ends in a timing

pulse, he shows that timing devices require a reference pulse. The

pulse mechanism has a kinetic property in the oscillating wheel.

Feedback of this timing pulse provides control to the slower cogwheels

in the mechanical chain. The conclusion states, "If most systems

develop chains and develop high quality with pulsing delivery of the

feedbacks, a clock is automatically provided so that the systems can

utilize timing functions" (Odum, 1983, p. 284).


Internal Timing vs External Timing of Inertial Oscillators

Whether the autocatalytic oscillator is environmentally dependent,

environmentally sensitive, or completely autonomous has an effect on the

way the rhythm is produced. In fact, the major point of contention about

biological clocks centers around their utilization of or dependency upon

persistent environmental signals. The two points are best explained by

the use of two different types of man-made clocks. The internal

autonomous biological clock is likened to an alarm clock. The alarm

clock contains all the components necessary to produce internal timing.

The operation of the clock is completely autonomous, it keeps time by

its own mechanism. The external clock is best described by the sun

dial. The sun dial has an internal mechanism but the timing is the






15

result of an external signal. Therefore the sun dial acts only as a

receptor.

In this study the autocatalytic inertial oscillator is an

autonomous internal oscillator with a free-running rhythm. The

oscillator produces rhythmic activity that will persist for some time

without any environmental cues. The development of energy-selective

behavior may require that a system possess the ability to sense changes

in its environment. With this in mind, the autocatalytic inertial

oscillator senses its environment and its activity responds to changes

in that environment. The importance of possessing an internal

oscillator that has its own free-running rhythm and is capable of

sensing and changing with its environment is studied in this research.

The ability to acquire energy in a selective manner may depend amongg

other characteristics) upon a system's ability to generate an autonomous

internal rhythm which senses changes in its environment and responding

to those changes.


Definition of Energy Selectivity

The "principle of energy selectivity" could be stated as follows.

When a system is subjected to a cyclic energy input from its environment

the system will best insure its ability to obtain energy by developing

an autonomous internal rhythmic behavioral guide that is capable of

sensing its environment. The closeness of the system's internal rhythm

to that of its primary environmental cue determines the system's energy

acceptance ability, and combined with other environmental, internal and

system factors its ultimate survivability.

Energy-selectivity may play an important role in the ability of a

system to maximize its input energy and power. Since the ability to




















aquire energy and power has a direct bearing on a system's abIity to

transfer power to the next larger system energy-selective behavior would

also play an important role in the power transfer among systems. If a

system can act in an energy-selective manner that activity may help the

system define its niche within the next larger system. One hypothesis

for this research is that energy-selective behavior provides a system

with a means of maximizing its input power, and as a result a means of

maximizing its power transfer to the next component or larger system.

Within biological systems, the ability to act in an energy selective

manner provides the system with a means for allocating input energy

among the various subsystems. The ability to allocate input energy may

allow the formation of a greater variety of niches, thereby increases

system diversity and maximizing system power.

Energy-selectivity behavior is an ability possessed by a system as

the result of its generation of internal rhythmic activity which allows

the system to define a range of energies it will accept, all other

energies are rejected.










METHODS


Starting with the simplest configurations, inertial elements were

combined into oscillators and these into various ecosystem

configurations. For each model computer simulations were used to

characterize response in the time and frequency domains. A unit

oscillator mechanism was included as a clock mechanism in ecosystem

prototypes. Potential for this kind of a mechanism was considered in

competition, energy selectivity, diversity, and survival of systems.

For each model, diagrams, equations, time simulations and frequency

responses are given.
The following models were developed or utilized to investigate

energy selectivity in biological systems:

1. a model of an electrical/mechanical oscillator;

2. a model for an autocatalytic inertial oscillator;

3. a model for an energy-selective producer;

4. models for competition between energy-selective
producers;

5. a model of the Kaituna Salt Marsh -
a) the original model was simulated to check the normal
model operational characteristics,
b) the model was then modified to account for the
diurnal solar input,
c) the model was modified to incorporate an energy
selective producer,
d) the model was modified to investigate energy
selective competition.









Overall Research Procedure


Five general sections were employed to ascertain the effects of

energy-selectivity on the power transfer of systems with cyclical

external forcing functions.


Phase I

The first phase of this research was devoted to understanding the

conceptual similarities in simple oscillator systems. Special emphasis

was placed toward understanding what characteristics of these simple

oscillators could be used to simulate the system rhythmic activity. Two

electrical/mechanical models were tested. The electrical and

mechanical models were simulated repeatedly (with any necessary

modifications to the models) until the simulation results compared with

known system responses. The two original choices were (a) a series

approach to the production of an electrical oscillation and (b) a


parallel tank circuit or bandpass approach to the production of an

electrical oscillation. Ultimately, the tank circuit oscillator

provided the the best opportunity'for expansion into a model for the

simulation of an autocatalytic inertial rhythm.


Phase II

Phase two was used to produce a phase sensing, time interpreting

autocatalytic inertial oscillator (a system rhythm generator). The

development of such a unit was deemed necessary because the simple

addition of a linear oscillator to a producer or consumer unit within a

model would provide only a rhythmic linkage and not phase adjustment or

age-sequence timimg. Many models for the autocatalytic inertial








oscillator were tested. However, the final oscillator model (presented

in the results section) is capable of sensing source phase shifts and

adjusting to those shifts. The autocatalytic inertial oscillator will

allow for the adjustment of its internal rhythm and oscillator activity

range (the time the oscillator will function in constant conditions);

and can produce a variety of source sensing, internal, and age related

timimg functions.


Phase III

The third phase of this analysis was used to simulate varia-ions

of a model for a simple energy-selective producer. The model

incorporated a autocatalytic inertial oscillator into a producer un't

thus allowing the unit to develop an energy acceptance niche within

which it could or would accept energy from one or more cyclic eneriv

sources. The unit and load power for the producer was calculated during

the simulation. The model design was based on the assumption thal the


producer's load (i.e. its output flow) was the unit's connection to a

more complicated system.


Phase IV

Phase four was employed to simulate a number of variations for a

competition model. These models simulated the competition between two

energy-selective producers for the use of an external cyclic energy

source. The source rhythm and the internal behavioral rhythms of the

competitors were varied in successive computer simulations to analyze

the effect of energy-selectivity on competition and input energy niche

development. Source energy supply was also designed to investigate the

effects of various combinations of source fundamental rhythms, i.e. the








addition or multiplication of two source rhythms to form one variable

energy supply.


Phase V

Phase five was employed to simulate the existing Kaituna salt marsh

mini-model developed by Odum, Knox, and Campbell, 1983. The model was

chosen because it provided a proven mini-model with cyclic external

forcing functions. The model had already been evaluated and was of

reasonable proportions to enable the testing of energy-selectivity

without excessive complexity. Furthermore, the model provided a

reasonable framework for the analysis of energy-selective behavior

within a webbed system.


Energetics Diagramming


The methodology employed to design or construct the models

presented in this dissertation was developed by Alexander et al.,

(1980). The procedure is as follows:
Step 1. Construct a system boundary,

Step 2. Identify the energy flows across the system
boundary,

Step 3. Organize the system's components,

Step 4. Connect the system components to the external
sources,

Step 5. Diagram the subsystems,

Step 6. Evaluate of assign values to the energy model
diagram,

Step 7. Translate the energy diagram into its corresponding
differential equations,

Step 8. Simulate the model, and

Step 9. Validate the model's response or check response to
see if it meets expectations.









Identification of the System Boundary

Systems that are open exist in a hierarchy. To differentiate

between the storage, flows, and sources for one system from that of

another--a boundary must be defined. Boundary definition for the models

employed in this research was based upon restricting complexity in model

design. The concept of power utilization and transfer is complex enough

without complicating the simulations with large cumbersome models. The

mini-model approach to analysis requires the use of a simple model to

illustrate a particular concept.


Identification of Energy Flows Across the-Boundary

Alexander et al., (1980) have developed a method for positioning

energy sources around a system boundary. First, all pertinent external

sources, with flows across the boundary, are identified. The source

with the lowest energy quality is placed in the lower left corner of the

diagram outside of the boundary. Next, the remaining sources are then

placed around the boundary in a clockwise direction by increasing energy

quality.


Organization of the System Components

Energy language, as developed by H.T. Odum (1976), is made up of a

set of symbols, shown in Figure 5. These symbols are employed to

construct energy models, and are representative of component functions

which repeatedly occur in all systems.

Figure 5a is a tank or storage and is used to identify system

state variables. All storage have a particular turnover time and

capacity to store energy. The turnover time of a tank varies depending

upon the particular pathway flow controlling the turnover. Examples of









turnover time include input turnover and output turnover. Turnover

time can be found by dividing the tank storage by an input or output

pathway flow. A tank with a storage of 1000 Cal and an output pathway

flow of 50 Cal/hr would have a turnover time of 20 hours. The capacity

of a tank is fixed and can best be described by a city water tank. The

capacity for any tank 7s determined by its design, and for discussion

let the capacity of a hypothetical city water tank be 5000 gallons.

The turnover of this tank if it were half full (2500 gallons) and

draining at 250 gal/hr would again be 10 hours, but the capacity would

remain 5000 gallons.

Figure 5b is the symbol for an external source. This type of a

source provides a constant potential which drives the flow along its

associated pathway. The greater the system requirement the larger the

flow.

Figure 5c is a energy sink and represents total energy degradation

of a system. The energy entering the system sink is no longer usable by

the system.

Figure 5d is a flow-limited source. A flow-limited source

provides flow energy only. One example of a flow limited source is the

solar energy provided to the earth. While this energy may increase or

decrease at any given moment, the total energy supplied can never exceed

the value of J, and the amount of energy available for use at any given

time is Jr. The only option available to a system dependent upon this

type of source (if it wants to get more energy) is to become more

adapted at capturing the energy as it passes. This source differs from

the previously described source because it requires the system to work

harder or to become more efficient in order to obtain energy. The






23

source in Figure 5b is capable of supplying as large a flow as the

system requires, and as a result can lead to explosive growth if not

limited by some other means.

Figure 5e is a bearded tank and stores energy as a result of an

acceleration, i.e. kinetic energy. The energy stored in this type of a

tank is due to a change in pathway flow. The beard on the tank 4as

derived by Odum as a result of the combination of two symbols--one from

energetic and the other from electronics. The two symbols were the

energy tank in Figure 5a and the electrical symbol for an inductor (see

Figure la, in the previous section).

Figure 5f is a transformation or work gate. The interaction of

two or more state variables produce a flow of energy (k*A*B). The

interaction process is defined by the mathematical symbol or equation

placed inside the symbol.

Figures 5g is a energy flow sensor. The flow of energy in pathway

F1 is sensed and utilized to produce or generate some other function

within the system. Photoperiodism relies on a sensor to determine tie

amount of light available.

Figure 5h is a group symbol for a production unit. The producer

mixes low quality energy with high quality energy to produce an

intermediate quality energy.

Figure 5i is another group symbol, the consumer. The consumer

symbol defines a unit that is self-contained and feeds control energy

back to other parts of the system.

System components are arranged within an energy circuit diagram by

energy quality. Inside the system boundary the lowest quality component

is located in the upper left corner of the diagram. Components with

































Figure 5:


Energy circuit symbols. Symbol (a) storage tank or state
variable, (b) constant potential source, (c) energy sink,
(d) flow limited source, (e) bearded tank which stores
kinetic energy and is capable of producing intertial
backforce, (f) work gate or energy transformation, (g)
flow sensor, (h) producer group symbol, (i) consumer group
symbol (odum, 1976 and 1983).
















0b
ra'



Ib)
1


/ '



bq






26

higher quality are then located in a descending order toward the lower

left corner of the diagram.


Connection of System Components With External Sources

System components are connected to an outside source when it is

determined that there exists a flow of energy from the source to the

component. Solar energy to plants (producers) would constitute a

pathway connection from a source of external solar energy to a producer

within the system boundary.


Diagramming the Subsystems

Each group symbol represents a subsystem within the model, and is

constructed in the same manner as the overall model. Each subsystem is

developed by combining variations of the remaining symbols to produce

subsystem functions. Mathematical notations are placed inside of the

interaction symbols used in the subsystems and in the overall system to

identify their specific operational characteristics.

Location of energy symbols within the subsystems is determined by

their respective energy quality. The lowest quality storage is located

in the upper left corner within subsystem, while the highest quality

storage is located in the lower right corner within the group symbol.

Interaction or transformation symbols are generally located to the right

of the tank or tanks that it supplies.


Evaluation of the Energy Circuit Model

The first and second laws of thermodynamics, conservation and

degradation of energy, are employed to evaluate energy circuit models.

Since energy can be neither created nor destroyed, energy interactions

must balance. Therefore, the energy flows entering an interaction









symbol must be equal to the energy flows leaving that symbol. The

second law requires that all storage and interaction have an energy

pathway to the system sink. These flows represent storage depreciation

and work process losses.


Translation of the Energy Circuit Model into its Corresponding
Differential Equations

Energy circuit models can be expressed in nonlinear differential

equations. For example, the model in Figure 6 contains a producer,

with an autocatalytic inertial oscillator controlling the rate of the

producer's biomass production. The model may be translated as follows:


Jr = J/(1+kO*l)


dP/dt = k5*Jr*/P/-k6*Jr/P/-k4*P-k3* fP


dB/dt = k1*Jr*l-k2*B

Variables (P) (K) and (3) are called "state variables" because they

determine the state of the system. Constants kl,k2,k3,...k7 do not

change during simulation, and are proportionality constants of flow

which determine the percent flow in their respective pathways. Flows

entering a state variable are positive (kl*Jr*l) while flows leaving a

state variable are negative (-k2*B). The interaction in Figure 6 is

multiplicative and is indicated as such by the "X" located in the center

of the symbol.

Balancing the flows around an interaction, as required by the first

law of thermodynamics, is accomplished by ensuring that each pathway

equation contains the same number and type of storage and/or external

sources. For example, each pathway in Figure 6 that touches the






























Figure 6:


Model for Translation of diagram into equations. The
energy circuit model is translated into the following
equations:


Jr = J/(1+kO*l)

dP/dt = k1*Jr*/P/-k6*Jr*/P/-k4*P-k3*/P

dB/dt = kl*Jr*l-k2*B

NOTE: The "/P/" is the absolute value of storage P














29








































1* r'
f =3






30

interaction symbol has storage "Jr" and "I" in their respective

equations, (see previous differential equations).


Simulation of the Model

Energetics models may be simulated with an analog or digital

computer. Although the analog computer is capable of instantaneous

simulation, the necessary scaling and physical setup required for a

model requires a lengthy effort. Therefore, simulation of the models

for this research was accomplished with a digital computer using the

Continuous System Modelling Program (CSMP). The models were simulated

on the Northeast Regional Data Center's main frame computers located ;n

the Space Sciences building on the University of Florida campus,

Gainesville.

The CSMP language employs "call functions" or "routines" for

integration of differential equations. The program allows the selection

of the integration time interval (DT) and length of time for the

simulation. Example equations for the rate of change of the system

state variables have already been presented. The integration of that

rate of change produces the levels for the tanks in the model at any

time during the simulation.


Validation of the Model

Validation of complex systems models constructed from sets of

nonlinear differential equations cannot be accomplished by statistical

methods. Therefore, other methods of validation must be employed. Most

often models are constructed from nonlinear equations that attempt to

reproduce the characteristics of the system. This approach is not

without its faults. Namely, a variety of models can be produced to














reconstruct a given set of data or conditions. Construction of models

with no underlying hypothesis or theory often produces models which only

attempt to replicate the data and as a result significantly reduce the

credibility of the model.

To avoid that problem the models constructed for this analysis have

supporting hypotheses as the basis for their operation and design. The

model for the autocatalytic inertial oscillator is based upon the theory

that system rhythmic activity is a function of a nonlinear oscillators

constructed of potential and kinetic components. Testing of the

oscillator model was accomplished by analysis of existing data about the

operational phenomena for system rhythmic activity. The model of the

autocatalytic inertial oscillator was constructed based upon theory and

not to replicate a set or sets of known data.

Models for the biological producers were constructed based upon the

established energy theories and modelling design algorithms developed by

H. T. Odum, 1976 and 1983. The analysis of energy-selective behavior

was then tested via the simulation of these hypothetical producers.

Finally energy-selective subsystems were added to an operational,

evaluated, and previously simulated models of a salt marsh ecosystem.

The ultimate test for these subsystems was a comparison of the model

established operational characteristics vs. its new operational

characteristics with the addition of energy-selective behavior.










RESULTS


Results and analyses are presented in four sections: concepts,

models for biological units, models for competition between biological

units, and models for a the Kaituna salt marsh, located in Nlew Zealand.

Each individual section presents the modelss, accompanying simulation

plots, and applicable discussion or explanation of the results.


Concepts


The general concept for energy-selectivity, as a result of an

intrinsic-autocatalytic-rhythmic-function, results from the extension

of concepts from the electrical and mechanical inertial oscillators.

Once the general principles for these simple systems had been

ascertained the development of a model for the autocatalytic inertial

oscillator was undertaken.


Electrical and Mechanical Oscillator Models

System components for the electrical tank circuit oscillator are

shown in Figure 7a. The inertial oscillator contains a capacitor,

inductor, and a source and load resistance, Rs and RI respectively. The

corresponding energy circuit diagram is shown in Figure 7b. While the

electrical diagram indicates component connectivity, the system storage

information for charge or potential energy in the capacitor (C),

magnetic flux or kinetic energy around the inductor (L), and circuit

total energy are not shown. The inadequacy of this type of diagram

32


































Fi figure 7:


The electrical tank circuit oscil lator. ;Jiagrjm J: in
electrical diagram of this bandpass type of electrr -I
oscil lator, b) an energy circuit diagram for inr. tan-
oscil lator. '.ote the indication of the storage _-t energy
in the energy language diagram, an important isp-c?( t
this circuit which is not shown in the electrical j' ,ram.
























dQ/dt=1/Rs*(S-(Q/C))-(1/R C)*Q-1/(L*C).


i3=(1/R1*C)*0


S=1/(L*C)0.5
2=1/(LC)=2.0*-*f0

Q=storage of charge in
caoacitor (C)
C=capacitance (Farads)
L=induc:ance (Henries)
f =intrinsic frequency of the
oscillator


I/R *(S-(Q/C))--,
s


I/(R1*C)*Q


- I/ ( -L ).- O

1Rs
k-1/(LRCC)

kl=1/ Rs


k3=I/(L' C )= -


dQ/dt=1/Rs*(S-(Q/C))-I/(RI*C)*Q-1/(L*C).0

dO/dt=kl*(S-(O/C))-k2*0-k 3*.

Energy in Q=Q2/(2*C) Energy in K=(L/2)*(1,/(L*C)*.'O)2 = /(3L*C)


il=1/Rs*(S-(O/C))






35

arises from the lack of descriptiveness, i.e. the storage information is

left to the reader's imagination. On the other hand, the energy circuit

diagram clearly indicates component connectivity as well as the storage

of energies within the system. Charge potential is stored in tank (Q)

and flux kinetic energy is stored in tank (K). Total circuit energy is

represented by tank (E). The beard on tank (K) indicates a back pressure

effect on the flow of electrons within its supporting pathway F3 and

results from magnetic field interaction with pathway electron flow.

Table 1 shows component values and flows, storage initial

conditions, description, and equations for the electrical tank circuit

oscillator. Table 1 also presents the equations for pathway

coefficients, and coefficient values for the same oscillator. Since

the model is designed to calculate its own pathway coefficients the

equations shown in Table 1 are the equations within the model that

calculate the flow coefficients. The initial conditions for tanks (0),

(K), and (E) are assumed to be zero. However, if tank (Q) has an

initial charge then the initial conditions for tanks (K) and (E) are

calculated by the model. The two equations necessary to accomplish this

are shown in the table.

One Interesting characteristic of these models is the integral

dependency of flow F3 on tank (Q). This phenomenon results from the

fact that tank (Q) provides the potential force that drives current

through pathway F3, while tank (K) opposes any change in the flow of

current through pathway F3. The result produces an integral dependent

flow in pathway 3. The build up of magnetic flux around the inductor

acts as a storage of kinetic energy, and the transfer of energy between

tanks (Q) and (K) produces the system's internal oscillation.

















Table 1: Evaluation, Name, Description, and Equations for the Energy Flows and
Storages in the Electrical Bandpass Oscillator Circuit


( oiponen t/ Stor (IPe/ low


Value/ liI tI t al (ontlit ion


3.0


11)0111
10(100



2.5e1?



(1.0
1 .01 -1
I (I I



0 II
(I. I)


Volts


Ohms
Farads
Ilenries





Mhos
tilos



Amllls
AmIps
Amlo'.
Amps


htime or Descriptive Coin.ent


I le( ritual potential d
I Oal ICes is tdtae
Source resistance
I le( tri( al cIdapac i tance
l Ile tri al ijinducltaice
P'erllt ia I eonerfly storaile
l inet lii ener(qy storaile
Iotad enPtl y ol osi i lator
I ef f it ient f low
(Iot f i ent I low
I'er otl sett inil coe l I i lent
I low
I low?
I low3


I Iuatl ion


(1?)/(0.)
(l/2) (I ?)

QIK
I/50

1/I
? .tl n f

I, ? ( I


Note: Table for Figure 7.


1. ll's in(.. t )






37

Figure 8 is a plot of the electrical inertial oscillator's

storage vs time. The system is driven by a source potential which also

varies with time. The important conceptual characteristics shown by the

plot include:

1) the oscillating nature of (0) and (K) with 'me,

2) the 90 degree or /4 radian lag of (K) to (0),

3) the effect of heavy damping on the system storage as tht
source supply is interrupted, and

4) the constant amplitude of (0) and (K) as long as source
supply is available.

Figure 9a shows the mechanical inertial oscillator. '-ree

components make up the system; they are the mass, spring, and dampers.

Within this system potential energy is stored by the spring, while

kinetic energy is stored by the position of the mass. The corresponding

energy circuit diagram is shown in Figure 9b. As with the electrical

inertial oscillator, the energy diagram more clearly indicates the

storage for potential energy (X) as shown by the displacement of 'ne

system's mass, kinetic energy (K), and total energy (E). Table 2 shows

the storage and flow initial conditions, component values, a descrip 've

comment, and the equations for the mass-spring oscillator. -n

intriguing characteristic of these two inertial oscillator systems 's

seen by a comparison of their pathway equations. Note the identical

equations for system pathway coefficients, with the exception ot

coefficient units the system equations are exactly the same. Simply

stated, the system characteristics are identical. System behavior is

identical, and since the same behavioral characteristics are present in

many simple inertial oscillators the behavior is repeated in other

systems and may be conceptually repeated in more complex systems. One






























Figure 8:


Plot of the electrical oscillator system storage vs.
time. Both the storage of potential and kinetic energies
oscillate at a frequency determined by the pathway flow
coefficient k4. The determination of k4 is always
(2.0*T*f ) where:


w=3.1415


fl=frequency of intrinsic oscillation.










39










+30 -


+15 f ^ \ ) l

1 Wi I




-15v


-3C
0 5 13

Time (Seconds)

































Figure 9:


The mechanical mass, spring, damper oscillator. Diagram
a) the mechanical diagram indicating system components and
positional relationship, b) the energy circuit diagram
indicating system components, energy storage, and
component positional relationship.

















0 x=change in mass Dosition
1 'm=mass
f =intrinsic frequency of
MAS the oscillator
MASS
ks=spring constant

02 Spring


dx/dt=1/Do*(S-X*ks)-(ks/D2)*X-(ks/m)05*.5


(a)


ks/m) .

kl=1/Dq

k2=ks/D2
k3=(k/mk )0


dx/dt=1/Dl*(S-X*ks)-(ks/D2)*X-(ks/m)0 5.'

dx/dt=kl*(S-ks*X)-k X-k3*.X


1/D,*(S-X-k).


(k /D2)*:

















Table 2: Evaluation, Name, Descriotion, and Equations for the Energy Flows and
Storages in the Mechanical Mass-Spring Oscillator Circuit



Comillonent/Stordae/f low Value/Initial C tondtion N.m t, oI llcscriptive (Conwlent I quat ion


3. 'Sin('.. )


forcee
allampin() constant I
llampinI connstant 2
Ma ss
Spri nq constantt
Potential enPerriy storanle
Kinetic enerily storaqle
Iotal energy rio oscillator
Coeffic ient flow I
Coefficlent flow 2
I'erioil settlir g coeff i ienlt
I low I
I low 2
I low I


(x"2)/(2'k?)
(M/I )' (I ''2)
PtK

I/l l
1 /011


1I'(S Lk x)

2^x
k, x


Note: Table for Figure 9.


Source

Il
"2
M
S
I'


3.0
In.l)
1011il


1 .11 -4
II. 0)
0. 11
0.0
1 .0( -
). (01 1









important concept to learn from these inertial oscillators is that

behavior or identical function can result from the combination of a

variety of different components.

Power transfer and frequency selectivity are important concepts

fundamental to understanding the operational characteristics of

inertial oscillators. Power transfer (the power available to the

load from the source) for both the electrical and mechanical inertial

oscillators is governed by a set of identical behavioral

characteristics. Figures 10a through 10f show the important power

transfer characteristics of the electrical inertial oscillator. Figures

10d to 10f indicate that for the same source potential the system power

is distributed differently with decreased loading. The load power, the

power that is transferred from the source to the load, is maximized when

the source resistance equals the load resistance (Figure 10e). When

balance occurs, source power (the power lost within the source as a

necessity for the transfer of energy from the source to the load) equals

load power, producing a 50% transfer of power form source to load.

Balance in this context describes the ratio of the load resistance to

the source resistance, a perfectly balanced system occurring when the

source and load resistances are equal. Figures 10a through 10c indicate

that the potential energy stored within the system increases as the

loading is decreased. With less energy required by the load the system

is capable of storing more energy.

Figures 11a through 11f also indicate that the power available to

the load is both frequency and balance dependent. Figures 11d through

Ilf show that the power transferred from the source to the load is

maximized when the energy delivered is of the same frequency as the






























Figure 10: Resistive power transfer characteristics for the bandpass
electrical oscillator. The greatest power transfer occurs
when the resistance of the source equals the resistance of
the load. Diagram (a) oscillator potential and kinetic
storage for larger source resistance, (b) oscillator
potential and kinetic storage for equal source and load
resistance, (c) oscillator potential and kinetic storage
for larger load resistance, (d) source and load power for
larger source resistance, (e) source and load power for
equal source and load resistances, (f) source and load
power for larger load resistance.




















* .Source



-Load


-:ME :SEC;NCS)

(d)


Source


IS "Loa

::.E SECONDS)

(e)


Source <-ad
15 3

':ME SECCND3S

(f)


T:ME SECONDSS)

(a)


T:,ME SECONDSS)

(b)


-30
0


-:.E (SECONDS)

(c)





























Figure 11:


Frequency
osci lator
frequency
intrinsic
osci I lator
frequency
oscillator
oscillator
frequency
and load
oscillator
frequency


power characteristics for the bandpass
SThe greatest power transfer occurs when the
of the source equals the frequency of the
frequency of the oscillator. Diagram (a)
potential and kinetic storage for source
greater than the oscillator frequency, (b)
storage for source frequency equal to the
frequency, (c) oscillator storage for source
lower than the oscillator frequency, (d) source
power for source frequency greater than
frequency, (e) source and load power for source
equal to oscillator frequency, (f) source and


load power for source frequency lower
frequency.


than oscillator



























0



30


TIME SECONDSS)


30











-30 3'
T30 00
T:ME (SECONDS)


.30


_ 3
0 !5 0
TIME (SECONDS)


-:"!E 'SECCNOS)


,--Source


":ME 'SECCIOS)

(e)


':ME ;SE::D3S;









inertial oscillator's internal frequency. When the source frequency is

faster or slower than the intrinsic frequency of the inertial oscillator

the oscillator operates at the source frequency. Comparison of

Figures 11a through 11c indicate that as the source frequency decreases,

i.e. increasing period the power transferred to the load reaches a

maximum when the source frequency equals the oscillator intrinsic

frequency. figuress 12a and 12b summarize the results of the previous

plots of load power with changing load resistance (RI) and with changing

source frequency (fs), respectively. The greatest power transfer for

changing load resistance (RI) occurs when the source resistance (Rs) and

load resistance (RI) are equal. The greatest power transfer for changes

in source frequency (fs) occurs when the source frequency equals the

inertial oscillator's intrinsic frequency (fo). When the two

frequencies are equal the system is said to be in resonance, and that

particular frequency is called the resonant frequency (fr). Moreover, at

resonance the circuit operates most efficiently with the least source

power lost for the most load power delivered (Figure 12b). Ultimately,

the maximum power transfer from source to load occurs when the system is

at resonance and is In balance. The maximum storage of energy within

the system occurs when the load resistance is large (the least loading)

and when the period of the source energy is long.

The same power transfer characteristics apply to the mechanical

inertial oscillator, however, the components are different. Within the

mechanical oscillator maximum power transfer occurs when the damping of

the source equals the damping of the load, and when the system

oscillation equals the source oscillation. Regardless of the system

(electrical or mechanical) the behavior for both systems is the same.


































Figure 12:


Summary of power characteristics for the bandpass
oscillator. Plot (a) source and load power with changing
load resistance, (b) source and load power with changing
source frequency. The greatest power transfer occurs when
source and load resistance and frequency are equal.















60







2 30



0




2
1 10 102 13

Load Resistance (Ohms)



60







o 30 -







0.
0.01 0.1 1.0 10.0 12C.0


Freeuency (Hertz)









The preceding paragraphs and figures have presented The concepts

for power transfer and energy selectivity with a limited reliance uoon

mathematics. However, tne electrical inertial oscillator lends itselff

readily to a mathematical presentation of these concepts. figuree '3

again shows the electrical and energy circuitt diagrams for the 'an-

oscillator (often called a bandpass oscillator Decause of its property

of passing only a range of frequencies to the loaa). Equations

through 11 show the mathematical relationships for the system, in botn

languages. The range of frequencies that are passed to the load is

often limited, for practical use, to a spec't7c subset of frequencies

called the circuit bandwidth. The bandwidth depends solely upon the

load resistance (RI), the inductor's (L) value, and a constant equal to

2.0* Useful power transfer occurs for frequencies Tnat deliver

potentials greater than or equal to 70.7' of the source polent;al

(Ekeland, 1981). The frequencies that deliver these potentials are

within the bandwidth of the inertial oscillator circuiT. The high and

low cutoff frequencies for the bandwidth are calculated from tne values

of the system components. Equations 1 and 8 describe the calculations

necessary to determine the bandwidth of the circuit shown in Figure 18.

The accompanying equations 4 and 11 indicate the relationship of circuit

components necessary to calculate resonant frequency. Desonant

frequency for an inertial oscillator defines the center of its bandw'dth

because it is the frequency that transfers the most energy and power.

The concept of maximum power transfer as a result of energy

selectivity is established for the electrical and mechanical

oscillators. The previous paragraphs and figures describe the general

characteristics of this phenomenon. The following sections describe the
































Figure 13:


The diagram shows the mathematical relationships
for the bandwidth of the electrical oscillator. Diagram
a) the electrical circuit and corresponding equations,
b) the energy circuit and corresponding equations. The
equations describe the same bandwidth because the two
diagrams are mathematically identical. The advantage of
the energy diagram is the clarity of presentation for
components and storage not as clearly indicated by the
electrical diagram.














Max


0.707


fl f

Frequency


R
Ss

C L S


-_ Potential inetic
Storage (P)-' Storage
Note: Consider R part of source (S) -

BW=R1/(2.0*-*L) k,=1/(R1*C)
f =fr-(BW/2) k3=1/(L*C) =
fh=fr+(BW/2) k3/k2=R1/L
fr=(1/2)*-*(L*C)0.5 BW=k3/(2*-*k2) R1/2-*L)

fh =fr(BW/2)
f h= f rk+(BW/2)


f =k05/(2 -)
r3








extension of these concepts into the development of a autocatalytic

oscillator, and through the use of the biological oscillator present the

concept of energy selectivity for biological systems.


Autocatalytic Inertial Oscillator Model

Electrical and mechanical inertial oscillators exhibit some of the

important characteristics required for biological rhythmic behavior,

however they do not self-adjust nor do they exhibit growth with time.

They do possess an internal timing between their potential (P) and

kinetic (K) components, and exhibit time relationships between their

internal components and an external source. While these qualities are

necessary for biological oscillators they fall far short of the

complexity generated by the addition of nonlinear autocatalytic behavior

within the inertial oscillator itself.

The autocatalytic inertial oscillator incorporates a non-linear

internal feedback within the oscillator, and as a result, produces a

phase adjusting oscillator capable of sensing environmental sources and

remaining tuned to those sources. Timing functions within the

autocatalytic inertial oscillator also grow with time and might

therefore be utilized to initiate action with Increased age.

Figure 14 is an energy circuit diagram of the autocatalytic

inertial oscillator developed for this study. Equations 1 and 2

present the mathematical relationship of the inertial oscillator system

components. The / / indicates the use of the absolute value, and is

necessary because the energy flows within the inertial oscillator are

capable of completely reversing their directional flow. Environmental

sources normally have some constant energy about which the available

energy varies. However, the inertial oscillator senses rhythmic change































Figure 14:


The autocatalytic clock. Intrinsic rhythm is made self-
adjusting by the addition of internal fe-dback.
Furthermore, the growth with time of the potential and
kinetic energy storage might provide an explanation for
clock-initiated age functions i.e. reproductive behavior.
Timing initiators also include nocturnal activity,
seasonal timed activity, and environmentally sensed
initiation of particular processes i.e. flowering and
fruiting.
















































dP/dt=k *Jr*/P/-k4*Jr*/P/-k2*P-k3*.P

Jr=J/(l+kO*/P/)

S=A*sin(. t)

k3=Frequency setting coefficient =

=2*-*f.
0o i
f=oscillator's intrinsic frequency






57

and as a result the / / is required to ensure that flow directions

within the inertial oscillator remain in proper phase relationship.

Autocatalytic inertial oscillator timing functions include five

useful timing initiators. First, the summation of potential and kinetic

energies within the inertial oscillator can be utilized as a product in

an energy transformation or work gate function. Figure 15 is an energy

diagram depicting an autocatalytic inertial oscillator as part of a

energy transformation or work function. The system is energy-selective

as a result of producer's inertial oscillator activity, and this

particular concept is presented by simulation plots in the next section.

The model is provided to familiarize the reader with the location of

components in an energy-selective production module. Source energy to

the system is flow limited, and the system has a recycle of (B) through

(N). Flow F2 indicates that the recycle of nutrients requires some

energy loss from the pathway. Figure 16 shows the concept for the

second timing initiator. The timing of the potential and kinetic tanks

(P) and/or (K) might be utilized as time indicators or process function

initiators. For example, the hunting initiator for nocturnal predators

or the tidal feeding initiator for clams, oysters, and/or mussels. For

the third timer, the timing combination of tanks (P) and (K) combine

with an ability to sense a source rhythm to begin or initiate process

functions. Examples of this characteristic could be the flowering of

plants or the thickening of animal furs during winter. Figure 17 is a

model depicting the concept of timing with a source and internal

inertial oscillator combination as the timer. Fourth, the amplitude of

tanks (P) and/or (K) are employed to initiate some age process or

function. One example, for this type of inertial oscillator activity


































Figure 15:


Autocatalytic clock as part of a workgate. The system
uses energy selectivity to acquire input energy and to
generate biomass. The diagram and following plots are
presented only to familiarize the reader with the use of
the autocatalytic clock.


















59


































Figure 16: Timing initiator for potential component of the
autocatalytic clock. This particular timing initiator
might be employed by a biological unit to initiate
nocturnal activity.
















61




















Clock





































Figure 17: Biological timimg using the source rhythm as vell as both
the potential and kinetic components of the autocatalytic
clock.












63
















Cloc
rJ -- ^ 0 ^


Jr






64

might be the initiation of a particular reproductive behavior which

results from the autocatalytic growth of tank (P) and/or (K). When one

or both of the tanks reach a predetermined level, the autocatalytic

inertial oscillator could initiate the production of hormones that in

turn produce timed reproductive behavior. Figure 18 is a model for

the age process initiator of the autocatalytic inertial oscillator.

The fifth timer results from the combination of some or all of the

previous functions and could be utilized to simulate complex forms of

behavior. The model shown in Figure 19 uses all the previous timing

functions to produce a complex timing behavior capable of timing both

diurnal as well as annual rhythms within the environment and produces

autocatalytic inertial oscillator behavior that can take advantage of

that knowledge.

Autocatalytic inertial oscillator phasing with an external source

plays an important role in the behavior of an individual system within a

given environment. Therefore, the phase relationship of the potential

(P) and kinetic (K) of the inertial oscillator in relation to the source

is important. If an individual biological system is to remain in tune

with its environment it must adapt to changes in source phasing.

Phase-shift experiments are employed to test the self-phasing of

biological systems. These experiments substitute laboratory phase-

shifted signals for normal environmental rhythms and examine the effects

of the new false signal on the system. As an example, Webb (1950)

studied a group of fiddler crabs (Uca) to analyze phase shifting in the

crab's color-rhythms. Webb substituted darkness during normal light,

and light during normal darkness producing a 12-hour phase change in

the crab's color-rhythm. The crabs subjected to the phase-shifted-








































ur 1 : io I c a lrmn -I s -s.I -
J c3 3 Iy t C cloCk r ythms '
'r F.eproatuct i ? RlT r L -i
SR- a ;e of the h 'loc icI L nIt.


















66























Cloc




J

r


































Figure 19: Biological timing as a combination of the four pre\- ous
timing initiators. This type of timing produces complex
behavior by timing the environment as well as tne
individual timing characteristics of the unit's
autocatalytc clock.


















68

























SClock






J
r









signal altered their intrinsic rhythms to self-align with the new

(albeit false) environmental signal. Therefore, any model of a

biological rhythm generator must be able to self-phase with its

environmental.

Figure 20 is a plot showing the autocatalytic inertial oscillator's

ability to self-phase when presented with a phase-shifted external

signal. The simulation begins with a source supply that is sine-

phased. The autocatalytic inertial oscillator is in tune with the

source, and storage (P) and (K) are increasing and in normal phase

relation. Between time 2.0 and 4.5 the source rhythm is eliminated.

The inertial oscillator oscillation continues with the same phasing,

however, the levels in tanks (P) and (K) decrease. This decrease is the

result of depreciation losses via pathway F2, and is required by the

second law of thermodynamics. Beginning at time 4.5 the source is

reinstated with an cosine-phased oscillation i.e. a 90 degree or 7/4

radian shift from the original source rhythm. Oscillator potential (P)

and kinetic (K) components experience a significant perturbation within

the next two cycles, after which time the autocatalytic inertial

oscillator self-tunes to the source's new phasing. Within the next

seven cycles the inertial oscillator completely re-establishes an in-

phase relationship with the source.

Frequency setting and damping of internal rhythms also play an

important role in an system's ability to maintain rhythmic behavior or

to efficiently utilize its internal rhythmic activity in a competitive

situation. The frequency setting coefficient k3 (in the model shown in
2
Figure 14) is always equal to w In radian units is 2.0*7 *fo,

where (fo) is the intrinsic frequency of the inertial oscillator.

































Figure 20: Plot indicating the ability of the autocatalytic clock to
self-adjust to changes in phase of an environmental
source. The damping during source elimination results
from a second law loss of energy via pathway F3, and is a
requirement for all systems--including internal clocks.
Full recovery as a result of the phase shift introduced
into the source required seven oscillations or periods.









































































0 7
Ti.ME CAYSS)


150 -







100 -
a












0 -








Coefficient k3 is extremely important to the operation of the

autocatalytic inertial oscillator because it controls the period of the

autocatalytic inertial oscillator's internal oscillation. The exact

method by which biological oscillators set their rhythms is still not

clearly understood, however the ability to identify these periods of

biological rhythms does exist. The periods for intrinsic rhythms are

known for many plants and animals. Within humans internal free running

biological rhythms exhibit a period range from 19 hours to 25.8 hours

and were identified from cave and bunker studies (Palmer,1976). The

following passage is an example of how biological system intrinsic

rhythms may be identified.

Over a decade ago, a young speleologist became
obsessed with the idea that it was scientifically
important for him to live alone, underground, sans
clocks, for two months. He chose an inhospitable
cave in the trench Alps where he lived at the 375-
foot level. Here the temperature hovered at 32
degrees (F), the relative humidity remained
unchanged at 100%, and the darkness was complete
save for a small battery-powered light in his tent.
Each time he awoke, ate, or prepared to retire he
called over a field telephone to a surface camp,
where his lonely words and the times of his calls
were recorded. He claimed tnat the inexorable cold
and dampness reduced his body temperature to less
than 97 degrees (F), and he was constantly
threatened by avalanches and cave-ins --- still, he
held out for the sake of science and whiled away his
time writing a best-selling novel about his
subterranean adventures, anxiety, and building
libido.

Throughout his underground stay, he tried
mentally to keep track of the passage of time on the
surface. When the men in the surface camp informed
him on September 14 that his experiment was over, he
thought it was only August 20. His subjective
judgement of the passage of time had been
exceedingly sluggish; mentally, he had lost 25 days!
A major problem in his estimation of the passage of
time was the fact that he commonly imagined that he
was taking a short siesta after his midday meal,
where in reality he had been awake for 16+ hours and






73

his "nap" lasted approximately 8 hours. However,
his living clock, as evaluated by the times of his
retiring and awaking phone calls to the surface, had
ignored his mental confusion and guided his body
functions all the while, measuring off periods of
activity and sleep that total just longer than a
day: 24 hours and 30 minutes on average (Palmer,1976
p. 125).

The above passage illustrates the intrinsic rhythm setting and

damping properties of internal biological rhythm generators (living

clocks as described by Palmer). The frequency of biological oscillators

is determined under constant conditions, and many experiments are

directed solely at determining these rhythms. Interestingly, one way to

find the intrinsic rhythm of an electrical or mechanical inertial

oscillator is to remove the oscillator from its energy supply and time

the period of its intrinsic oscillation.

The duration of oscillator activity, the amount of time an

autocatalytic inertial oscillator rhythm remains active when isolated

from its energy cue by placing it into constant conditions is also

important. The ability of a biological system to adapt to interuptions

in energy supply is critical. With heavy damping an autocatalytic

inertial oscillator loses its rhythm (in constant conditions) within a

short period of time. However, with light damping the autocatalytic

Inertial oscillator could maintain a rhythmic pattern for long periods

of time--as in the above passage for two months or longer. Damping

determines the longevity of the autocatalytic inertial oscillator's

activity when environmental conditions become constant. One example of

an interruption in an environmental source supply is a drought in a

normally wet area.

With these characteristics in mind, the simulation of biological

rhythmic activity by an autocatalytic inertial oscillator requires that






74

the internal autocatalytic inertial oscillator characteristics be

adjustable. Coefficients k3 and k2 satisfy these requirements. Pathway

coefficient k3, sets the intrinsic rhythm, while coefficient k2 adjusts

tne dampirj. Damping of the autocatalytic inertial oscillator's free-

running rhythm is the direct result of the depreciation losses within

the inertial oscillator, and in the model is accounted for by pathway

c2.

Biological Models with Autocatalytic Inertial Oscillators


The previous section presented results for the energy circuit

autocatalytic inertial oscillator. Development of this oscillator is

an important initial step in the analysis of energy selectivity and its

effect on the power maximization and power transfer within biological

systems. The following model, plots, and figures illustrate the effect

of energy selectivity on power transfer and biomass generation for a

producer with an internal autocatalytic inertial oscillator. The model

is intended to illustrate the effects of energy selectivity on a

hypothetical biological system.

Figure 21 is an energy diagram for an energy-selective producer.

Table 3 presents the storage initial conditions, the pathway

coefficients, a descriptive comment, and the equations for the model in

Figure 21. The production of biomass (pathway F1) is regulated by the

activity of the producer's internal autocatalytic inertial oscillator.

Since the autocatalytic inertial oscillator is energy selective, the

work gate or energy transformation function is also energy-selective.

Pathway F2 is the autocatalytic inertial oscillator's sensor and

provides information to the inertial oscillator about the rhythmic

changes that occur in its environment. Pathways F6 and F7 are


































Figure 21 :


Energy diagram for an energy-selective producer. The
clock produces an internal rhythm required for the
production of unit biomass. Equations 1 through 4
describe the component relationships in mathematical
Language.












































Jr=J/(l+kO)
dB/dt=k Jrl-k2B

dP/dt=k5 r/P/-k6 r/P/-k4P-k 3P
















Table 3: Evaluation, Name, Description, and Equations for the Energy Flows and
Storages in the Model of the Energy Selective Producer


'tor laqe/ Iow/t (mnlboolent


niomass
Cloct



0
I'

S?
1
I


I)


Va lue

1.01 o(>


I 0o I
0. I


3 Of *4
3 11 15


1.01 "1


3 01 14






5 ) 3


I 01 4


lIan w or Il'stription (


fdl ii ulllmi. O f pro)ll 'dce
(a l I rlll ', t r I In cI l I
Cal L'oti t )al riterl, ly in i Ih" k
(al I inet t energy in c lo)
(al/d Solar enierqy to prodil.er
I alld HIlllllMSS p odul t I)oll
" ail/d Solar enerqy sensor
Lal/d Oo(llass export
( al/4l Input (of osenlsor to (im t
Cal/d Autocatalytc fuedbaclt
of cl I I oteIt t a I to
sensor
Slal/ld lIe'ret iaton of cllol
pot i t a 11
( a /d ()eprl t lat lon of ( Iot t
p)ot ntial w1 th inertial
bacd force
Potent lal ( apat ii, y to i Ili
I'er I l set I ill) (t oel I Ic itlnt


I ql a t I oll





(l'l)";)/(?'l )
I I I I


I *lr





SI

.1



I 'I'




2 I '


Note: Table for Figure 21.





78

depreciation pathways. A linear depreciation is provided for by pathway

F6 and represents the normal depreciation losses within the inert;al

oscillator. These losses include physical losses such as cellular

breakdown. Pathway F7 represents the depreciation losses experienced

by the autocatalytic inertial oscillator however these loss are opposed

by a kinetic inertial backpressure. Kinetic inertial oackpressure is

felt as a byproduct of the flow in pathway F7.

The total energy of the autocatalytic inertial oscillator ;s the

sum of the inertial oscillator's potential and kinetic energies, and is

employed as the work gate regulator. When the source rhythm is

outside of the producer's range of energy acceptance the producer

rejects the available energy and production of biomass decreases.

An energy-selective producer increases its biomass or net

production within a defined range of energies, i.e. a biological energy

acceptance niche. The term bandwidth is used to describe the energy

acceptance range for the electrical and mechanical inertial oscillators.

The analogy for bandwidth in ecological systems is NICHE. Figure 22a

shows an energy acceptance niche for a producer with a free-running

internal biological inertial rhythm having a 24 hour period of

oscillation. The width of an energy-acceptance niche could be employed

to determine whether the producer is a specialist or simple generalist.

The models in Figure 22b show the relative frequency relationships for

the simulations employed to produce the energy-acceptance niche in the

Figure 22a. The source rhythms are depicted in the circular source

symbols while the producers' intrinsic rhythms are depicted in the

bullet symbols used for energy circuit producers. The figure clearly

indicates that the most biomass is generated when the source and

producer rhythms are equal.




























Figure 22: Plot Indicating the niche width of an energy selective
producer with an intrinsic rhythm of one day. The niche
might be called an energy acceptance niche since it
defines the range of frequencies or rhythms within which
the producer accepts energy input. This figure was
produced by plotting the steady state biomass for five
simulations of the model in figure 21. For each
simulation the internal rhythm of the producer was held
constant at 1 period per day, while the source rhythm was
varied during the simulations. The source periods ranged
from 0.01 to 100 oscillations per day. The mini-diagrams
in (b) are only to aid In distinguishing between the
relative period changes. Since the change in source
period varied by order of several magnitudes plotting
these rhythms for presentation was impractical.








80






















I I


creauencv Oscllat'ons/lay

(a)








81

Energy Selective Biomass Generation

This section shows the results of the analysis for the energy-

selective generation of biomass. The producer simulations show that

energy transfer for the generation of producer biomass occurs at the

frequencies within the energy-acceptance niche of the producer.

Frequencies outside of the producer's energy-acceptance niche generate

no biomass.

The plots in Figure 23a show the effect of energy-selectivity on a

producer's biomass for source energies less than, equal to, and greater

than the producer's intrinsic rhythm. The plots clearly show the

producer's capability for accepting or rejecting rhythmic energies.

Biomass attains its greatest level when the producer's free-running

autocatalytic inertial oscillation equals the frequency of the available

energy (plot line a). When the available source energy oscillates with a

rhythm outside of the producer's acceptance range the producer rejects

the energy (plot lines b and c). Figure 23b shows the producer's

internal autocatalytic inertial oscillator generates a rhythmic behavior

within the producer that is either in or out of step with the source

rhythm, thereby selectively accepting or rejecting the opportunity to

utilize available energy. The three mini-models show the relative

period relationships for the source and intrinsic producer rhythms.

Whether the former or the latter is true, the producer by its

autocatalytic oscillator-regulated activity defines its acceptance niche

within which it accepts energy. Interestingly, biomass generation about

the producer's fundamental rhythm appears relatively symetrical.
































Figure 23:


Simulation plots and relative frequency diagram for the
energy-selective producer in Figure 21. Plot line (a)
shows the growth in producer biomass for source energy
inside the producer's energy-acceptance niche. Plot lines
(b) and (c) show the decrease in biomass for source
energies at frequencies outside of the producer's energy-
acceptance niche. The greatest biomass generation occurs
for supply energy at the free-running rhythm of the
producer's intrinsic rhythm.








83
















o


x
b c

0




0 1 2

Time (Years)

In


(b)






84

Power Transfer for an Energy Selective Producer

Figures 24a and 24b show the producer's unit power and power

transfer for the three plots in Figure 23. The maximum power

transferred by the producer to the remainder of the system (via F3)

occurs when the rhythms of the producer and the system are matched. For

all source rhythms outside the producer's energy-acceptance niche power

and energy are rejected and power transfer is minimized. Producers that

develop acceptance niches too far removed from their environmental

signals do not possess the ability to pass on an appreciable amount of

power from an external source to the remainder of the system. A

producer capable of attracting more energy and power and transferring

more power to the next component in a system is also contributing more

toward the power maximization of the system as a whole. This concept

plays an important role in the "Maximization of Power" as introduced by

Lotka, 1922 and later refined by Odum, 1976.

Essentially, the principle of maximum power, describes a system

structure that gets the most power for available energy input while at

the same time utilizing that input power most efficiently. Efficient

power transfer, therefore, becomes an important characteristic for

stable system behavior. Were systems to reward selfish producers or

consumers the system would be relegating itself to receiving less energy

from components lower on the food chain (than would be expected by the

Principle of Maximum Power), and as a consequence would support a more

limited food chain.

One way for a system to maximizes its power input while insuring

stability over a wide range of inputs is to develop producers and

consumers that create or occupy specific spaces or niches. A system



































24: Plot of power transfer for the energy-selective
producer. Power transfer to the load (the rest of the
system) via pathway F3 reaches a maximum when the internal
rhythm of the producer equals the external oscillation of
the source.


Figure




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