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Maintenance of Intraguild Predation in Jumping Spiders

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Maintenance of Intraguild Predation in Jumping Spiders
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2008

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Approximation ( jstor )
Ecological modeling ( jstor )
Ecology ( jstor )
Foraging ( jstor )
Modeling ( jstor )
Predation ( jstor )
Predators ( jstor )
Productivity ( jstor )
Spatial models ( jstor )
Spiders ( jstor )

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University of Florida
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7/24/2006

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MAINTENANCE OF INTRAGUILD PREDATION IN JUMPING SPIDERS


By

TOSHINORI OKUYAMA

















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


2006

































Copyright 2006

by

Toshinori Okuyama















ACKNOWLEDGMENTS

A number of people helped me to complete this project, which makes this

dissertation so much more meaningful to me. I don't list them here, but I hope

my appreciation is well understood. Still I must acknowledge my committee (Ben

Bolker, Jane Brockmann, Craig Osenberg, Jim Hobert, Bob Holt, and Steve

Phelps) for their valuable criticism and encouragement throughout, which I value

very much.

I am extremely grateful about the Teaching Assistantship and Research

Assistantship opportunities as well as the CLAS fellowship for their support and

experiences. Complex Systems Summer School of the Santa Fe Institute also

provided an extremely enjo'-J .-'L environment in which I was able to initiate a part

of this dissertation.

Lastly, interactions I have had with Ben Bolker have been my most valuable

experience here at UF. Ben improved not only my project but also my way of

approaching ecological problems. If I accomplished anything worthwhile in the

future, it is because I was fortunate enough to work with him.















TABLE OF CONTENTS


page


ACKNOW LEDGMENTS .............................

LIST OF TABLES .. .. .. ... .. .. .. .. ... .. .. .. ... .. ..

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

A B ST R A CT . . . . . . . . .

CHAPTER

1 GENERAL INTRODUCTION .......................

1.1 Intraguild Predation ... .. .. .. ... .. .. .. .. ... ..
1.2 Jum ping Spiders . . . . . . .
1.3 Spatial Structure .. . . .. . ...... ..
1.4 Anim al Behavior ... .. .. .. .. ... .. .. .. ... .. ..
1.5 Adaptive Behavior Under Spatially Structured Environments ...
1.6 Dynamics of Jumping Spider Activity .................
1.7 Synthesis .................. ............


2 INTRAGUILD PREDATION WITH SPATIALLY
INTERACTIONS . ............

2.1 Introduction .. ... .. .. .. .. ...
2.2 Materials and Methods .. ..........
2.2.1 Lattice Model of Intraguild Predation
2.2.2 Mean Field Approximation ......
2.2.3 Pair Approximation .. ........
2.2.4 Invasibility Analysis .. ........
2.2.5 Individual Based Model ........
2.2.6 Heterogeneous Environment .....
2.3 R results . . . . . .
2.3.1 Mean Field Approximation ......
2.3.2 Pair Approximation .. ........
2.3.3 Unequal Neighborhood Sizes .....
2.3.4 Quantitative Comparison Between Spa
M odels . . . . .
2.3.5 Heterogeneous Environments .....
2.4 D discussion . . . . .


STRUCTURED SPECIES
.. . 1 1


. . .






tialand on-.atia










2.4.1 Effects of Spatial Structure on the Basic Results
M odels . . . . . .
2.4.2 Quantitative Effect of Spatial Structure .....
2.4.3 Effect of Spatial Heterogeneity .. ........


of X 'i-1, ii i


3 NITROGEN LIMITATION IN CANNIBALISTIC JUMPING SPIDERS


3.1 Introduction .. ............
3.2 Materials and Methods ........
3.2.1 Experimental Treatments .
3.2.2 Effect on Growth .......
3.3 R results . . . . .
3.4 Discussion .. ... .. .. .. .


4 EVOLUTIONARILY STABLE STRATEGY OF PREY ACTIVITY IN
A SIMPLE PREDATOR-PREY MDOEL ....... . 38

4.1 Introduction .................. ............ .. 38
4.2 The Model .................. ............. .. 40
4.3 Results ........... .. ....... ........ 43
4.3.1 Evolutionarily Stable Strategy (ESS) of Foraging Effort .. 43
4.3.2 Incorporating ESS into the Community Dynamics . 46
4.3.3 Comparison with the Quantitative Genetics Model . 49
4.3.3.1 Behavior of the system with fast evolution . 50
4.3.3.2 Behavior of the system with slow evolution . 54
4.4 Discussion ............... . . .... 55
4.5 Appendix A: Derivation of the ESS ................. .. 60

5 ON THE QUANTITATIVE MEASURES OF INDIRECT INTERACTIONS 63

5.1 Introduction .................. ............ .. 63
5.2 Quantifying Indirect Effects .................. ..... 64
5.2.1 Standard Experimental Design ................ .. 64
5.2.2 Indices of Indirect Effects ................... .. .65
5.2.3 Decomposing Total Effects .................. .. 66
5.2.4 Incommensurate Additive Metrics . . ..... 68
5.3 Complications .............. . . .... 68
5.3.1 Biological Complexities: Short-term . . ..... 68
5.3.2 Biological Complexities: Long-term . . ...... 70
5.4 Summary .................. ............. .. 72

6 ADAPTIVE BEHAVIOR IN SPATIAL ENVIRONMENTS . ... 74


6.1 Introduction .. ........
6.2 The Model .. .........
6.2.1 Lattice Simulations .


6.2.1.1 Direct effects: performance of foragers .....


.............
.............
.............









6.2.1.2 Indirect effects ................ 80
6.3 Results ..................... ....... .... 81
6.3.1 Direct Effects: Performance of GIFs and LIFs . ... 81
6.3.2 Indirect Effects .................. ...... 81
6.4 D discussion . . . . . . . .. .. 81

7 PROLONGED EFFECTS OF PREDATOR ENCOUNTERS ON THE
JUMPING SPIDER, PHIDIPPUS AUDAX (ARANAE: SALTICIDAE) 87

7.1 Introduction .................. ............ 87
7.2 Materials and Methods .................. ...... 91
7.2.1 Study System .................. ....... 92
7.2.1.1 Predator treatment ................. .92
7.2.1.2 Ambiguous visual stimuli . . ..... 93
7.2.2 Behavioral Measures .................. ..... 93
7.2.3 Statistical Analysis .................. .. 94
7.3 Results . . . . . . ..... 94
7.4 Discussion . . . . . . . .. .. 97

8 SUSTAINED EFFECTS OF VISUAL STIMULI ON RESTING METABOLIC
RATES OF JUMPING SPIDERS .................. .... 101

8.1 Introduction .................. ............ 101
8.2 Materials and Methods .................. .... 102
8.2.1 Experimental Treatments ................ 103
8.2.2 Oxygen Measurement .................. ... 104
8.2.3 Statistical Analyses .................. .. 105
8.3 Results ................... ..... ........ 106
8.4 Discussion . . . . . . . .. .. 107

9 ACTIVITY MODES OF JUMPING SPIDERS . . 110

9.1 Introduction ................... .. ........ 110
9.2 Part I: Quantification of Spider States in the Field . .... 111
9.2.1 Materials and Methods .................. .. 112
9.2.2 Results . . . . . . .... 113
9.2.3 D discussion . . . . . . .. .. 114
9.2.4 A Simple Model .................. .... 114
9.3 Part II: Examining the Simple Model .. . . 115
9.3.1 Materials and Methods .................. .. 115
9.3.1.1 The treatment ................ . 116
9.3.1.2 Behavior measure .................. 116
9.3.1.3 Statistical analysis ................. 117
9.3.2 Results .............. . ..... ....... 118
9.4 Part III: Individual Based Model .................. 120
9.4.1 The M odel ... .. .. .. .. ... .. .. .. ... .... 120
9.4.2 Parameter Estimation .................. ... 121









9.4.3 R results . . . . . . . .
9.5 C conclusion . . . . . . . .

10 INTRAGUILD PREDATION IN A JUMPING SPIDER COMMUNITY:
A SY N TH ESIS . . . . . . . .


10.1 Introduction .. ..........
10.2 The Model .............
10.2.1 Activity of Spiders ...
10.2.2 Foraging Activity .....
10.2.3 Exploitation Competition .
10.2.4 Intraguild Predation .
10.2.5 Reproduction .. .....
10.3 Results . . . .
10.4 Discussion .. ...........
10.4.1 Spatial Structure .....
10.4.2 Biphasic Activity .....
10.4.3 Adaptive Behavior .
10.4.4 Modelling Behavior in Comn


munity Ecology


REFEREN CES . . . . . . . . .


BIOGRAPHICAL SKETCH .............















LIST OF TABLES
Table page

3-1 Nutrient and energy contents of prey (standard errors in parentheses). .. 33

4-1 Equilibrium analysis. The c specified is the choice of ESS in Region III
required for a nonzero equilibrium to exist. ............... 49

5-1 Existing studies that have explicitly compared TMII and DMII. . 67

6-1 Parameter values used for the simulations. For the description of parameters,
see the text. .................. ............... ..79

7-1 Estimated parameters from the random effect model. SE (standard error)
and RE (random effect). .................. ...... 95

7-2 Statistical results for with and without ambiguous stimuli. RE indicates
the random effect describing the standard deviations. Parameters with
missing RE values are fixed effects. .................. .... 96

8-1 AIC for each model. p = number of parameters. . . ..... 107

9-1 Estimated parameters of the reduced model. The model is an hierarchical
logistic regression model with the intercept al + a2j and slope b1 + b2j,
where j is the di, of the experiment. .................. 119















LIST OF FIGURES
Figure page

1-1 Building up an intraguild predation system. Starting from a consumer-resource
(N-R) interaction with an independent predator population (P), addition
of predation (left column to right column) and competition (top row to
bottom row) are necessary to make IGP. Direction of the allows indicate
the direction of energy flow. .................. .... 2

1-2 Spatial distribution of predators P and prey N. In the right figure, each
species is well mixed. In the left figure, the two species are spatially segregated. 4

1-3 Hypothetical activity dynamics. The top figure shows the dynamics of
predator density. The bottom figure shows the corresponding foraging
activity of adaptive (solid) and non-adaptive (dashed) foragers. . 5

1-4 Trait-mediated indirect interaction (TMII) and density-mediated indirect
interaction (DM II) ............... ........ 7

1-5 Spatial distribution of predators P and prey N. Two prey individuals are
labeled as A and B. ................... ........ 8

2-1 Examples of random binary landscapes based on different patch scales.
Patch scale refers to the number of times the procedure diffuse was
applied (see text). ............... ......... .. .. 19

2-2 Parameter regions indicating the outcome of IGP in a non-spatial model. 20

2-3 Results of invasion analysis in the pair approximation model. ....... ..21

2-4 Results of invasion analysis in the pair approximation model. Spatial
scale of IGpredators was fixed at zp = 4 while that of IGprey varied. .22

2-5 Parameter intervals resulting in expansion and reduction of the coexistence
interval. The line indicates the contour at Ispatial/Inon-spatial = 1. When
this ratio is greater than 1, spatial structure increased the size coexistence
intervals ....... ............ ................ .. 23

2-6 Persistence probability (squares) and density of IGprey (circles) and IGpredators
(triangles).. .................. ............... 24

2-7 Persistence probability (squares) and density of IGprey (circles) and IGpredators
(triangles). Densities of consumers are presented as fraction of total cells
occupied by the species. .................. .... 25









2-8 Persistence probability (squares) and density of IGprey (circles) and IGpredators
(triangles). Densities of consumers are presented as fraction of total cells
occupied by the species. .................. ...... .. 26

3-1 Growth in carapace width of jumping spiders in each instar. ...... ..34

3-2 Duration of each instar of P. audax. .................. .... 35

4-1 Solutions for the ESS for c in each of three regions of the nondimensionalized
NP-plane (s ahN and r P). In both Region I and Region II there
is precisely one ESS function. In Region III there are three possibilities
for an ESS. For the expressions for yi and Y2 in terms of r and s, see
Appendix A. ............... ........... .. .. 44

4-2 A complicated ESS function, where Region III is split into many subregions,
with each subregion associated with one of the three possible basic ESSs. 45

4-3 The three basic ESS functions determined by which of the three strategies
is chosen uniformly in Region III. Top (c = 1), bottom left (c = y ),
bottom right (c = y2). Plots of the ESS functions are shown on the
nondimensionalized NP-plane (s = ahN and r = P). . .... 46

4-4 Apparent functional responses of predators when P = 10, a = h
1,m = 0.1, mp = 0.1. In this parameter region, there are three ESSs
(Figure 4-1). These functional responses were plotted assuming that the
three ESS values are distinct strategies. Left: c = yi. Middle: c = y2.
Right: c l. ............... ........... ..47

4-5 Evolutionary dynamics of foraging effort (dc/dt versus c) in Region I (left),
Region III (middle), and Region II (right), under the assumption of fast
rate of evolution g. ............... ........... .. 51

4-6 Simulation of the dynamics of predators and prey plotted on the effort
diagram (Figure 4-1). s = ahN and r = (a/b)P. The gray line indicates
r = s and the dotted line indicates r = (1 + s)/2. The area between
these two curves with s > 1 indicates the region where there are multiple
ESSs. Left figure: c = yi, middle figure: c = y2, and right figure: c = 1. 54

4-7 Foraging effort as a function of N and P for an Evolutionary stable strategy
(left) and for a strategy that maximizes population fitness (right). . 58

6-1 Schematic representation of foraging efforts. .............. 77

6-2 Proportion of prey surviving, average number of offspring, and fitness of
GIFs (G) and LIFs (L). .................. .... ... .. 82

6-3 Effect size for TMII (T) and DMII (D) with variable number of predators
(P ) . . . . . . . ... ... . 8 3









8-1 Average change ( SE) in oxygen consumption of spiders (pl/hr/individual).
Average weight of individual spiders was 2.26 mg. ........... .106

9-1 Box plots for the food deprivation degrees of the spiders in the field. Spiders
were classified based on sex and the location where they were found: outside
retreat (active) or inside retreat (inactive). ............... 114

9-2 Proportion of individuals that attacked a fly. ............... .118

9-3 Treatment effect parameter estimates. Solid line and two dashed lines
indicate the mean and 95'. credible regions of the reduced model. Squares
indicate the means for the full model. ................. 120

9-4 Relationship between mass and food deprivation degree. The estimated
function is wt 0.145(t + 1)-0.063 ........ ... . ....... 122

9-5 Simulation of an individual. Circles and squares are realizations of the
simulation corresponding with active and inactive phase, respectively.
Solid line traces the deterministic prediction of the stochastic IBM. Horizontal
lines are LA-I (top) and LI-A (bottom). ................ 123

9-6 Results of the IBM based on 1000 individuals. 125 individuals are active.
. . . . . . . . . . 1 2 4

10-1 Effect of productivity level on the persistence of IGP community under
various degrees of spatial structure. The smaller the neighborhood size
U, the stronger the spatial structure. Local reproduction and non-adaptive
behavior (i.e., a = P = 0) are assumed. .................. 132

10-2 Average persistence with and without adaptive behavior. Adaptive behavior
indicates that both a and 3 are positive (see text). U = 7. Local reproduction
is assumed. .. .. .. .............. . . ..... 133

10-3 Average persistence with and without ballooning. U = 7. Adaptive
behaviors are included. ............... ......... 133















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

MAINTENANCE OF INTRAGUILD PREDATION IN JUMPING SPIDERS

By

Toshinori Okuyama

i ,v 2006

C'! n: Be i ,::i:: M. Bolker
Major Department: Zoology

Intraguild predation (IGP), defined as predation among individuals of

the same guild, is predicted on theoretical grounds to destabilize ecological

communities. However, IGP is found in most communities. In an attempt to

resolve this discrepancy between theory and observation, I explored mechanisms

that may maintain IGP in ecological communities, focusing on two ecological

factors: animal behavior and spatial structure. Jumping spiders were used as

model organisms because IGP is common among them and because behavioral

observations are tractable with these animals.

Theoretical investigations predict that both spatial structure and behavior

can have independent positive or negative effects on the persistence (ability of all

species to coexist) of an IGP community. Empirical evidence -Ii--:. -I that real

systems lie in a parameter region of the model where spatial structure enhances

IGP persistence relative to its non-spatial counterpart.

Simultaneous examination of behavior and spatial structure indicate that

commonly used analyses in community ecology overestimate the magnitude of

behavioral effects on community dynamics. Empirical examinations of antipredator









behavior in jumping spiders confirms this prediction; although spiders change

their behavior based on experiences such as predator encounters, such changes in

behavior do not persist long enough to induce the magnitude of behavioral effect

typically predicted by the models.

Field observations of spider activity reveals the potential for complex activity

dynamics. In particular, jumping spiders exhibit distinct active and inactive phases

(biphasic activity) where inactive individuals stay in a retreat even during the day

when they are normally active. Thus, activity dynamics of jumping spiders is more

complex than its representation in previous models, where individuals alter activity

only with respect to the external variables such as density of predators or density of

prey.

An individual-based model built on the theoretical and empirical results of this

study shows that typical adaptive behavior, such as lowering activity level after

encountering a predator, had only weak influences on overall community dynamics.

In contrast, biphasic activity allowed the IGP community to persist.

Taken together, this work i.-. -I I that common models linking behavioral and

community dynamics neglect important behavioral details. Both within-individual

behavioral variation (e.g., biphasic activity) and between-individual behavioral

variation (e.g., caused by spatial structure) need to be carefully examined when

attempting to scale behavioral dynamics up to the level of communities.















CHAPTER 1
GENERAL INTRODUCTION

This dissertation examines the dynamics of ecological communities that

include intraguild predation. This introduction will first define (and describe)

intraguild predation and outline the general questions addressed in this study. This

problem description is followed by an outline that describes each chapter in relation

to the overall goal of the project.

1.1 Intraguild Predation

Intraguild predation (IGP) is defined as predation among individuals of

the same guild, i.e., individuals from species that share similar resources. Thus,

broadly speaking, cannibalism is considered a form of IGP unless there is a distinct

ontogenetic niche shift that differentiates the resource profile of cannibals and their

victims (but see Wise, 2006). For example, most spiders are generalist predators

that feed on a variety of prey items such as mosquitoes and flies, making them

members of the same guild. However, spiders also eat other spiders; we count this

cannibalism as intraguild predation. IGP commonly involves larger individuals

feeding on smaller individuals (Polis, 1988). We call the victim intraguild prey

(IGprey) and the predator intraguild predator (IGpredator). IGP is common

in nature and is found in a vi ,i, I r of taxa (Polis, 1981; Polls et al., 1989; Polls

and Holt, 1992; Williams and Martinez, 2000; Arim and Marquet, 2004). One

characteristic of IGP is the simultaneous existence of competitive and trophic

interactions between the same species (Figure 1-1).

Theoretical models predict that coexistence of IGpredators and IGprey is

difficult (Holt and Polis, 1997), because IGprey experience the combined negative

effects of competition and predation. In systems with competition only, IGprey







Independent Predator-prey
predator population interaction
N P N-P
t t
R R

Competitive Intraguild predation
interaction
N P N-P
t/ t/
R R
Figure 1-1. Building up an intraguild predation system. Starting from a
consumer-resource (N-R) interaction with an independent predator
population (P), addition of predation (left column to right column)
and competition (top row to bottom row) are necessary to make IGP.
Direction of the allows indicate the direction of energy flow.


suffer no predation; in standard predator-prey interactions without competition,
IGprey suffer no exploitative competition from the IGpredator (Figure 1-1).
Thus, IGP is more stressful for the intermediate consumer (IGprey) than either
exploitative competition or trophic interaction alone (Figure 1-1).
The theoretical difficulty in explaining IGP persistence and its observed
ubiquity have identified IGP as an ecological puzzle (Holt and Polis, 1997) and led
to a series of studies that have attempted to resolve this puzzle. These studies have
considered factors such as top predators (food web topology) (Yurewicz, 2004), size
structure (\!ylius et al., 2001; Borer, 2002; MacNeil and Plavoet, 2005), habitat
segregation (\ I Neil and Platvoet, 2005), metacommunity dynamics (\I. I I 1
and Bascompte, 2002), intraspecific predation (Dick et al., 1993), and adaptive
behavior (Krivan, 2000; Krivan and Diehl, 2005). However, it is not clear whether
the most important interactions have yet been identified, and consensus has yet to









emerge from these studies about the relative frequency and strength of different

possible mechanisms in maintaining IGP.

Understanding the dynamics of IGP has practical importance because

IGP occurs in agricultural systems and also affects species of conservation

concern (M\!ill. r and Brodeur, 2002; Harmon and Andow, 2004; Koss and Snyder,

2005; Harmon and Andow, 2005). Despite the need for solid understanding about

IGP in order to manage these systems successfully, the unresolved puzzle -i.-i-. -1-

that we still do not understand how IGP communities persist in nature. Thus,

in this dissertation, I examine how an IGP system can persist by examining

both theoretical and empirical issues, using jumping spiders as model organisms,

in an attempt to resolve the discrepancy. I focused on two classes of ecological

phenomena, animal behavior and spatial structure, as possible explanations; each is

further discussed below.

1.2 Jumping Spiders

Jumping spiders are the largest spider family (Salticidae) (Coddington and

Levi, 1991). They are generalist predators that prey primarily on arthropod

species, including other spiders (Jackson and Pollard, 1996). The frequency

of IGP is known to be high in some species. For example, approximately 21' .

of Phidippus audax's diet consists of other spider species that also consume

similar resources (Okuyama, 1999). Intraguild predation among jumping spiders

is alv--x size-dependent, with large-bodied individuals consuming smaller

individuals (Okuyama, 1999). Nevertheless, smaller species of jumping spiders

appear to coexist in local communities with large species of jumping spiders for

many years, posing the puzzle of persistence discussed above.

The focus of this study on behavior makes jumping spiders particularly good

study subjects. These spiders are visual foragers and their complex foraging tactics

are often compared to those of vertebrate species (Land, 1972; Hill, 1979; Jackson









and Pollard, 1996). In fact, the 1i ii i ly of studies on jumping spiders concern

their behavior, rather than their ecological dynamics. Complex behavior does not

occur in microcosms of microorganisms (or is difficult to examine at the individual

level when it exists), whose rapid dynamics otherwise make them well suited for

community level studies.

1.3 Spatial Structure

Although spatial processes are well recognized as an important factor in

ecology (Kareiva and Tilman, 2000), the ini i, i ily of theoretical models of IGP,

including those that pose the discrepancy with observation (e.g., Holt and Polls,

1997), are non-spatial. Non-spatial models assume that individuals are mixed

homogeneously in space, ignoring various forms of spatial structure that are

ubiquitous in nature (Figure 1-2). For example, except in landscapes that have

been artificially homogenized by humans (lawns, crop fields), we observe distinct

spatial vegetation patterns in all terrestrial communities.

Mixed Segregated
NPNPNPNPNP NNNNNPPPPP
PNPNPNPNPN NNNNNPPPPP
NPNPNPNPNP NNNNNPPPPP
PNPNPNPNPN NNNNNPPPPP
NPNPNPNPNP NNNNNPPPPP
PNPNPNPNPN NNNNNPPPPP
NPNPNPNPNP NNNNNPPPPP
Figure 1-2. Spatial distribution of predators P and prey N. In the right figure, each
species is well mixed. In the left figure, the two species are spatially
segregated.



Regardless of whether spatial structure is generated exogenously or endogenously (e.g.,

Bolker, 2003), it can have a large impact on species interactions. If species are well

mixed (Figure 1-2, left), the community's dynamics can be well approximated

by traditional non-spatial models. However, if there is a spatial pattern in

animal distribution and if individuals interact only with individuals in a local






5



neighborhood, spatial models describing the true dynamics are more accurate (Bolker

et al., 2000; Iwasa, 2000; Sato and Iwasa, 2000). In C'! lpter 2, I examine the role

of spatial structure in a simple IGP community. The analysis presented in Chapter

2 shows that the nutrient content of IGprey is an important model parameter;

C'! lpter 3 explores this parameter experimentally.

1.4 Animal Behavior

Animal behavior is an ecological factor that is considered to p1 l a pivotal

role in ecological communities (Fryxell and Lundberg, 1998). While traditional

models such as the Lotka-Volterra model and its numerous variants assume that

behavior (e.g., foraging activity) of individuals is constant and independent of

environmental factors, there is considerable evidence for dynamic variation in

behavior. In particular, there is a large body of evidence that animals alter their

foraging activity with respect to predation risk (Caro, 2005) (Figure 1-3).



U)



o
I I I I I
0 2 4 6 8
Time

SNon-adaptive

0
r. o
0)



0 2 4 6 8
Time


Figure 1-3. Hypothetical activity dynamics. The top figure shows the dynamics of
predator density. The bottom figure shows the corresponding foraging
activity of adaptive (solid) and non-adaptive (dashed) foragers.









Community models that include behavior typically assume that animals forage

optimally, balancing the risks of starvation and predation (Stephens and Krebs,

1986). As a consequence, these models predict that foragers decrease their foraging

effort (e.g., search rate) when predation risk is high (Figure 1-3). This type of

behavioral analysis has been done for a variety of food web modules (Bolker et al.,

2003).

However, the inclusion of animal behavior in community dynamics has

overlooked many important aspects of behavior. For example, theoretical models

that include adaptive behavior have largely ignored intraspecific interactions.

In other words, the model shown in Figure 1-3 overlooks the possible effects of

changes in forager densities over time, despite the fact that animals are known to

alter their behavior based on the behavior and density of their peers (Giraldeau

and Caraco, 2000; Caro, 2005). In chapter 4, I examine the evolution of adaptive

foraging behavior in a simple predator-prey model based on Evolutionarily Stable

Strategy (ESS) analysis. This model incorporates a Holling type II functional

response. When even this basic ecological detail is included in an otherwise

standard model of behavioral responses, it results in infinitely many ESSs due

to the evolution of intraspecific interaction caused by the predator's handling

constraint. This result cautions us in the interpretation of results from existing

models and slr.;. -I that consideration of behavior in existing models may be too

simplistic.

Empirical ecologists, however, do not tend to focus on individual behavior.

Even studies that directly examine behavior have collected behavioral data at

the level of populations rather than measuring individual responses (e.g., Anholt

et al., 2000). Instead, much of the effort of examining animal behavior has focused

on indirect effects generated by such behavior. For example, trait-mediated

indirect interactions (T\ ITT1) are induced by changes in a trait (behavior) of an









intermediate species. In a three-species food web, the intermediate consumers
may reduce their foraging effort in response to predation risk, reducing their

consumption rate of resources (Figure 1-3). Thus, predators have a positive
indirect effect on resource (Figure 1-4). Another class of well-recognized indirect
interactions, density-mediated indirect interactions, are transmitted via changes in

density of intervening species rather than via trait changes.
(a) None (b) DM11 (c) TMII

S NNp N P
4 t t""
R R R
Figure 1-4. Indirect interactions. Black arrows indicate the consumption of one
species by another (lethal/direct density effect). Thickness of the arrow
represents the rate of consumption. Gray arrow indicates a non-lethal
effect (direct trait effect). The community in (a) includes only a
single consumer species and the resource, and thus has no indirect
interactions. In (b), the predator species P consumes the consumer
species thus decreasing the density of consumers (depicted by the
small font). Because the consumer density is smaller, the consumer
population removes fewer resources. In (c), although they do not
consume the consumer, predators induce antipredator behavior by the
consumer, which decreases the consumption of resource by consumer.


Although TMII and DMII are widely described (Werner and Peacor, 2003),

quantifying them is not straightforward. In C'! lpter 5, I examine indices of indirect
effects that are commonly used in ecological experiments, focusing on how they
can be used to facilitate the connection between indirect effects and community
dynamics.
1.5 Adaptive Behavior Under Spatially Structured Environments

Although I demonstrate that spatial and behavioral factors are both important

when considered independently in the previous chapters, this does not guarantee
that both components are still important when they operate together. For example,









in the presence of spatially structured interactions, behavior may have little

effect on the dynamics of community and vice versa (e.g., if they interact in a

non-additive manner).

Individual A
^P P
PNP P
P P P
P P
P N
P P Individual B

N N
N

Figure 1-5. Spatial distribution of predators P and prey N. Two prey individuals
are labeled as A and B.



A simple scenario illustrates the possible effect of spatial structure on adaptive

behavior. In Figure 1-5, the distribution of predators is concentrated in the upper

left corner. Prey are randomly distributed. In this scenario, prey individual A

may forage much less than prey individual B because its perceived predation

risk is higher (Figure 1-5). Thus spatial structure leads to considerable spatial

variation in individual behavior, variation that the common non-spatial models

neglect (Abrams, 2001). C'! lpter 6 examines how adaptive behavior and spatially

structured species interactions can produce qualitatively different outcomes in

community dynamics.

1.6 Dynamics of Jumping Spider Activity

The chapters described above establish that the dynamics of activity is an

important component and is sensitive to details of the models such as spatial

structure. Thus, in Chapters 7, 8, and 9, I examine experimentally the assumptions

of the models and the dynamics of jumping spiders to identify how their behavior

should be incorporated into a community model, and whether the details of their









natural history tell us anything further about how to model their community

dynamics.

First, consider Figure 1-3 again. Although antipredator behavior is widely

described and thus the qualitative pattern we observe in the figure makes sense,

behavioral tracking of the environment in this way implies that individuals can

maintain information on the changing state of the environment. If environmental

cues indicating predator density are not continuously present and prey can respond

only to direct encounters with predators, foraging effort is unlikely to track

predator density as cleanly as shown in Figure 1 3. How individuals exhibit

antipredator behavior in the absence of immediate threats and how their behavior

affected by the environmental variables such as the density of predators? I address

this question in a series of three studies, each focusing on a specific time scale. In

C'!i lpter 7, I examine the behavior of jumping spiders after an encounter with a

predator, behavior that has the potential to produce tracking behavior such as that

shown in Figure 1-3 (short time scale). In ('!i lpter 8, I examine how the resting

metabolic rates of jumping spiders are affected by their previous experience with

predators or prey during the previous d-,i (intermediate time scale). In chapter 9, I

examine the general activity level of jumping spiders in the field (long time scale).

These results, combined with the results from previous chapters, -Ii--:: -1 that the

commonly used modelling framework is inappropriate for studying the community

dynamics of jumping spiders.

1.7 Synthesis

In the final chapter (C'!i lpter 10), I synthesize the findings obtained from

each piece of this project and discuss how they apply to the problem of long-term

persistence of jumping spider communities. To assist in this synthesis, I create

an individual based model that combines many of the factors discussed and

measured in the previous chapters. I also consider an additional detail of spider









behavior, which is not specifically examined in this project (i.e., ballooning, [Bell

et al., 2005]), because it strongly affects the spatial structure of the model and

exclusion of the behavior may result in an unrealistic degree of spatial structure.

Taken together, this model demonstrates that the activity patterns of jumping

spiders that are described in this project p1 i, key roles in allowing the two species

of jumping spiders that exhibit IGP to coexist. This result incorporates natural

history characteristics of spiders such as ballooning, further strengthening the

validity of this conclusion. Without the simultaneous consideration of spatial and

behavioral factors together, it would not be possible to derive this conclusion.

Although the focus of the study is IGP, my results about the relationship between

behavior and community ecology are more general. Based on the findings of this

project, a general discussion about behavioral modelling in community ecology is

also provided to facilitate reexaminations of relationships between behavior and

community ecology.















CHAPTER 2
INTRAGUILD PREDATION WITH SPATIALLY STRUCTURED SPECIES
INTERACTIONS

2.1 Introduction

Intraguild predation (IGP) is a common and important species interaction

in many ecological systems (Polis and Holt, 1992; Arim and Marquet, 2004).

Effective implementation of biological control, for example, must take IGP into

consideration (Harmon and Andow, 2004; Koss and Snyder, 2005). IGP has also

affected the success of conservation and wildlife management programs (e.g.,

Palomares and Caro, 1999; Longcore, 2003). It is now well established that IGP

dynamics have strong implications for both basic and applied ecology.

Theoretical studies of IGP sl.---- -1 that the coexistence of species in IGP food

webs is difficult. Due to the double pressure of competition and predation from

intraguild predators (IGpredators), models predict that intraguild prey (IGprey)

will be eliminated in a wide range of parameter space. Because IGP is ubiquitous

in nature (Arim and Marquet, 2004), there is a discrepancy between theory and

observations. This discrepancy continues to puzzle ecologists (Holt and Polls, 1997;

Krivan and Diehl, 2005).

Simple models show that

1. IGprey must be better at exploiting the basal resource than IGpredators in
order to coexist.
2. At low productivity levels, IGprey can outcompete IGpredators. When
productivity is high, IGpredators will drive IGprey to extinction. At
intermediate productivity levels, the two species may coexist.
3. As productivity increases within the range that allows coexistence, the
equilibrium IGprey density decreases while the equilibrium of IGpredator
density increases.









All these predictions have been empirically verified in parasitoid systems

(predictions 1 and 2, Amarasekare (2000); prediction 3, Borer, Briggs, Murdoch,

and Swarbric (2003)), and in microcosms (prediction 1, Morin (1999); all

predictions, Diehl and Feissel (2000)).

These empirical studies confirm that simple models capture some qualitative

properties of IGP interactions. However, the real challenge to theory lies not in the

qualitative (im)possibility of IGP coexistence but in its predicted improbability.

The narrow parameter space that models -i-i.;. -1 could allow coexistence does

not seem to support the ubiquitous occurrence of IGP in ecological communities.

Several ecological factors (e.g., size-structure and adaptive behavior) have been

examined to see whether they allow an increased probability of coexistence ( \ylius

et al., 2001; Kfivan and Diehl, 2005); the coexistence parameter region may or may

not expand depending on the details of the models. Although these factors are

important, it is likely that we still lack some important ecological components in

IGP models.

One factor that has not been examined in IGP models is spatial structure

(but Amarasekare (2000a,b) concluded that the competition-dispersal tradeoff is

not important in the coexistence of a parasitoid community that includes IGP

in a patchy environment). Spatially explicit modelling (e.g., distinguishing local

and global interactions) has generated a number of new hypotheses (Amarasekare,

2003a). Furthermore, spatial structure is known to stabilize simple predator-prey

models (Keeling et al., 2000). As all organisms in an IGP food web inherently

interact with other species in a spatially structured manner, this is an important as

well as a realistic axis to examine. In this study, I use pair approximations (Sato

and Iwasa, 2000) and an individual based model (IBM) to examine a simple IGP

food web in a spatially structured environment. The so-called pair approximation

keeps track of local dynamics as well as global dynamics, while the mean field









approximation makes no distinction (non-spatial); pair approximation reduces

to the mean field approximation in its non-spatial limit (discussed below). Thus,

using pair approximation allows one to examine the effect of local interactions

by comparing the results with the analogous mean field model. With IBMs, I

examine the effects of spatial heterogeneity in productivity, which Holt and Polis

(1997) -ir.-.- -I, .1 should be important in IGP systems. The three main questions

are (1) how the qualitative predictions of IGP models are affected by taking space

into account, (2) whether spatial structure expands the possibility of coexistence,

and if so, under what conditions, and (3) how spatial heterogeneity in resource

distribution affects IGP dynamics.

2.2 Materials and Methods

To examine the effect of spatial structure on IGP dynamics, a simple

IGP community was constructed in a lattice environment (see below). Both

homogeneous and heterogeneous resource distributions were examined. For

homogeneous resources, pair approximations worked well and allowed for simple

comparisons to a standard non-spatial model. The heterogenous model makes the

pair approximation model complex and thus was analyzed with individual based

computer simulations. This section first describes the spatially structured IGP

community, then describes how it can be analyzed using either pair approximation

or an individual based model.

2.2.1 Lattice Model of Intraguild Predation

We consider two predators of the same guild interacting on a lattice space

(i.e., IGpredator P and IGprey N). IGpredators and IGprey consume the same

basal resource and IGpredators also eat IGprey. Basal resources are not explicitly

represented in this model: instead, each cell is characterized by a fixed resource

level, R. Because resources are fixed in space and are not depleted, competition is

for space in lattice cells (IGprey and IGpredators cannot coexist in a cell).









Each lattice site has z neighbors (e.g., hexagonal neighborhoods would have

z = 6) and is either occupied by P or N, or is vacant (E). The two species can

have distinct neighborhood sizes (ZN for IGprey and zp for IGpredators); however,

unless otherwise stated, we assume that IGprey and IGpredators have the same

neighborhood sizes (z = ZN = zp). The fraction of sites in state P, N, and E

are called global densities and are designated pp, PN and PE. We define qij as the

local density of sites in state i with a neighbor in state j. For example, qp/N is the

fraction of P sites that are next to an N site (i.e., the probability that a randomly

chosen N site is located next to a P site).

IGprey (N) and IGpredators (P) can reproduce (at a rate depending on

basal resource consumption) only if they are .,,li i,:ent to a vacant site, and their

reproductive rate per vacant site is RbN/z and Rbp/z, respectively. Therefore, RbN

and Rbp are the maximum rates of reproduction in an empty neighborhood. Hence,

the reproduction rate of a randomly chosen IGprey is z(RbN/Z)qE/N = RbNqE/N

the product of the maximum birth rate and the expected fraction of vacant sites in

the neighborhood. For simplicity, I assume that spatial movement occurs only by

means of reproduction.

IGprey die due to predation by IGpredators at a rate A, which together with

conversion efficiency (e) also determines the reproduction of IGpredators. We

define A as the maximum predation rate which is attained when the IGpredator is

completely surrounded by IGprey.

Based on these rules, the equations for the global population densities are

dN =pN(RbNqE/N AqP/N- MN) (2-1)
dpp
dp =pp(RbpqE/p + eAqN/p mp). (2-2)


where mr and mp represent the background mortality rate for IGprey and

IGpredators, respectively.









2.2.2 Mean Field Approximation

By making the approximations qi/j a pi (i.e., the probability that an .,I.i ient

cell occupied by i is the same as the global distribution of the species i) and

applying the identity pE = 1 PN PP, we obtain the following non-spatial mean

field approximation model,


PN =RbNpN(1 PN p) mrNPN APNpp (2-3)
dt
dpp
=Rbppp(l pN pP) mppp + e~pNpp. (2-4)

This mean field model is essentially the same as previously studied non-spatial

IGP models: in particular eqs. 2-3 and 2-4 represent a special case of the model

described in Figure 4 of Polis et al. (1989) when the intensities of inter- and

intra-specific competition are the same. Although competition is for space rather

than for a resource with explicit within-cell dynamics, the non-spatial version of the

model matches a model derived with resource competition in mind.

2.2.3 Pair Approximation

As discussed above, pair approximation keeps track of local densities in

addition to the global densities described above. To derive the dynamics of local

densities, we first need to resolve the dependencies among possible variables. For

example, there are three global densities (PN, Pp, PE) and nine local densities

(qN/N, qIN/p, ..., qE/E), but because of the following constraints, they are not
independent:

PN + PP + PE 1

qN/i + qp/i + qE/i =1 (i = N, P, or E)

qi/jpj =qj/iPi (i = N, P, or E).









Due to these dependencies, we have only five independent variables, which

can be chosen arbitrarily. We will choose PN, PP, qN/N, qp/p and qplN as the

independent variables and express all the others in terms of these five variables

based on the above constraints.

In order to calculate the dynamics of local density, for example qN/N, we first

derive the dynamics of the doublet density PNN (i.e., two cells that are .il1i ient to

each other are both occupied by IGprey).

dpNN 1 + (z 1)qN/EN (z 1)qP/NNN
-d = -2mNpNN + 2RbN PEN 2A PNN.
dt z z

The first term on the right hand side indicates the loss of an IGprey doublet

(i.e., NN -+ NE or EN) due to density independent mortality. The second

term indicates the gain of an IGprey doublet from an EN or NE doublet due to

reproduction either by the IGprey within the doublet or by any other potential

IGprey individual located in one of the (z 1) neighbors of the vacant cell. qi/jk

indicates that i is located .,l1i ient to the doublet jk. The last term describes the

loss of the IGprey doublet, NN, due to intraguild predation by an IGpredator

located in the neighborhood of the doublet.

The equations for the other two doublets are:

dppp 1 i CD! + (z 1)qP/EP\ fl 1)qP/NP\
-- 2mpppp + 2Rbp ( + PEP + 2eA ( + (z 1-- PNP
dt z z
dPNP (z 1)qP/EN (- 1)qN/EP
d- (mN + mnP)PNP + Rbp E PEN + RbN P EP
dt z z

t + (z 1)qYPI PNP + eA ( 1)qPN PNN


Pair approximation substitutes triplets for pair densities (i.e., qi/jk

qi/j), assuming conditional independence between pairs of points. Interested
readers should refer to Sato and Iwasa (2000) and Iwasa et al. (1998) for

discussions/details. These doublet densities are transformed to conditional









probabilities. For example, the dynamics of qp/p are found to be,

dqp/p d(ppp/pp) ppp 1 dppp
dt dt p4 pp dt

2.2.4 Invasibility Analysis

The pair approximation model is too complex to analyze algebraically, so

we used invasion analysis to examine the outcomes of community dynamics.

Invasibility was examined by asking whether one of the species could increase its

population from low initial densities when the other species was present at its

equilibrium density. For example, to examine the possibility of IGprey invasion,

we evaluated the values for pp and qp/p when PN 0. Based on these equilibria,

we then obtained qN/N and qp/N that in turn were used to examine whether PN

increased based on equation 2-1. The outcome was classified into one of four cases:

IGprey can invade IGpredators, but IGpredators cannot invade IGprey (IGprey

win), IGpredators can invade IGprey, but IGprey cannot invade IGpredators

(IGpredators win), each species is able to invade the other (coexistence), and

neither IGprey nor IGpredators can invade the other (bistability) (ili rell and

Law, 2003).

2.2.5 Individual Based Model

The pair approximation described above assumes a homogeneous environment,

characterized by constant productivity R in every cell. In order to examine the

role of a heterogeneous environment, an individual based model that corresponds

to the pair approximation model was created. For simplicity, simulations were run

in discrete time; thus, the model did not exactly match the differential equation

model (e.g., it ignores action of offspring within the time step), but it maintained

the qualitative behavior of the model.

At the beginning of each iteration, the number of IGprey and IGpredators that

died due to density-independent mortality were determined as Poisson deviates.









Reproduction and death events were randomly ordered for each individual in each

iteration; an animal that died in a time step may or may not have reproduced

before death. The potential fecundity for each individual was simulated as a

Poisson deviate with mean RbN (IGprey) or Rbp (IGpredators) where R is

the productivity of the cell where individuals reside (i.e., either RL or RH:

see below for the description). For each potential offspring, the probability of

actual reproduction was fraction of vacant .,i1] i:ent cells (e.g., the probability

of converting one potential reproduction to an actual offspring is 1/4 if only

one .,ili i ent cell is empty). Reproduction of offspring was realized sequentially,

allowing for depletion of free space in the neighborhood. IGpredator updating

includes predation, which resulted in a Poisson reproduction process (number of

offspring) with mean e.

A 51-by-51 square lattice with periodic boundaries was used as the environment.

Simulations alv--,v- began with 200 IGprey and 100 IGpredators, both randomly

distributed in the environment. Persistence was defined as fraction of simulations

out of 50 resulting in P > 0 and N > 0 at t = 5000. The model was implemented

in Netlogo (Wilensky, 1999).

2.2.6 Heterogeneous Environment

Each cell in the environment was assigned a random productivity value R

from [0,1]. Spatial correlation was generated by letting each cell share 50'-. of its

productivity value with its neighboring cells; this procedure is called diffuse and

is a standard function in Netlogo. Increasing the number of sequential uses of the

diffuse function increases the spatial correlation of productivity values. We call

the number of diffuse iteration the patch scale. Two different productivity values

(high and low) were assigned based on whether a patch was higher or lower than

the median value. To alter the mean productivity, the high productivity cells were

ah--,v-b set to RH = 1 while the low productivity patch value, RL, varied. Because









half the cells were assigned to high or low, the average environmental productivity

was alh-i- (RH + RL)/2. Examples of patches of different scales are shown in

Figure 2-1.

-20-10 0 10 20
L I I I I I I I


-D

S00 10 20 -20-10 0 10 20








Figure 2-1. Examples of random binary landscapes based on different patch scales.
Patch scale refers to the number of times the procedure diffuse was
applied (see text).



2.3 Results

2.3.1 Mean Field Approximation

In addition to the trivial equilibrium where no species can survive (which

occurs when R < mN/bN and R < mp/bp), four outcomes are possible (Figure

2) in the mean field approximation model (eqns 2-3 and 2-4): IGprey win (when

R < cl and R < c2), IGpredators win (when R > cl and R > c2), coexistence (when

R < ci and R > c2), and bistability (when R > ci and R < c2) where

mpA mNeA
C1 bp(mN + A) mpbN' C2 bN(eA mp) + bprN

Although the basal resource is not explicitly modelled here, the model's properties

are equivalent to those of a standard non-spatial IGP model with explicit

resources (Holt and Polis, 1997).









O
Coexistence

S\ IG predator win
e \

S Bistability


S-IGpreywin


1.0 1.5 2.0 2.5

Productivity, R

Figure 2-2. Parameter regions indicating the outcome of IGP in a non-spatial
model. Parameters: mp = 0.3, m = 0.2, bp 0.5, bN 0.8, e 0.4.
When R < 0.25, neither species can survive.


For example, IGprey will outcompete IGpredators when the productivity

level is low and IGpredators will win when the productivity level is high provided

A > 0.5 (Figure 2-2). At intermediate productivity, both species can coexist. When

both species coexist, the amount of resource in vacant cells, RpE, (analogous to

the standing stock of unused basal resource in an explicit resources model) is at an

intermediate proportion between that with IGprey alone and that with IGpredator

alone. When the species coexist, increasing productivity will decrease the density

of IGprey while increasing that of IGpredators. The coexistence condition based

on the productivity level above (i.e., R < cl and R > c2) can be rewritten

as bN/mN > bp/mp. Thus for species with equal background mortality rates,

coexistence is only possible when IGprey is a better competitor for resource than

IGpredators (i.e., bN > bp).

2.3.2 Pair Approximation

The qualitative results of pair approximation based on one dimensional

(z = 2), square lattice (z = 4), and hexagonal lattices (z = 6) are similar to those of









the mean field approximation model (Figure 2-3). As the neighborhood size grows

(e.g., Figure 2-3, z = 9999), the dynamics of the pair approximation approaches
the limiting case, the mean field approximation model. In a bistability region, one

species (N or P) can win depending on the initial densities (i.e., founder control).

1.0 1.5 2.0 1.0 1.5 2.0
II I | II I I I
z=2 z=4 z=6 z=9999

S0.9
0.8
0.7
S0.6
0.5
0.4 bil
1.0 1.5 2.0 1.0 1.5 2.0
Productivity, R

Figure 2-3. Results of invasion analysis in the pair approximation model. The
parameters used are the same as in Figure 2-2. When neighborhood
size (z) is large, the results are indistinguishable from the non-spatial
model (Figure 2-2).



2.3.3 Unequal Neighborhood Sizes

The above results assume that neighborhood sizes of IGprey and IGpredators

are the same (i.e., z = ZN = zp). To examine the effect of unequal neighborhood

sizes (i.e, ZN > zp), the neighborhood size of IGpredators was fixed at zp = 4 and

ZN was varied, and the same invasion analysis was applied (Figure 2-4).

Both reproduction and mortality parameters for the IGprey and IGpredators

were fixed at the same value (bN = bp and mN = mp; see the figure caption for the

actual values). This condition prevents IGprey from persisting in the non-spatial

model (see introduction) or when both IGprey and IGpredators had the same

neighborhood size (i.e., ZN = zp = 4 resulted IGpredator dominance in all the

parameter regions in Figure 2-4). However, as IGprey's neighborhood size became









1.4 1.5 1.6
I I I I I I I I I
I z=6 zN =8 I ZN =9999










1.4 1.5 1.6 1.4 1.5 1.6
Productivity, R

Figure 2-4. Results of invasion analysis in the pair approximation model. Spatial
scale of IGpredators was fixed at zp = 4 while that of IGprey varied.
When ZN = 4 (i.e., ZN = zp), IGpredators dominate in the entire
parameter space shown. Parameters: mN = mp bN= bp 1= e=
1.


greater than that of IGpredators, coexistence between IGprey and IGpredators

became possible.

2.3.4 Quantitative Comparison Between Spatial and Non-spatial
Models

However, the pair approximation and mean field approximation make

different quantitative predictions about coexistence as a function of environmental

productivity (Figure 2-3). For example, in Figure 2-4, when A = 0.6 for the range

of productivity examined, strong spatial structure (z = 2) predicts coexistence is

impossible while coexistence may be possible in the case of weaker spatial structure

such as (z = 6). In the other words, the interval of productivity levels that allows

for coexistence changes with z.

Thus, to make a quantitative comparison between spatial and non-spatial

models, we compared the range of productivity levels that allows coexistence in

the two models. Let Ispatial and Inon-spatial be the coexistence interval in productivity

for spatial and non-spatial model, respectively (The parameter values used to










obtain Ispatial and Inon-spatial are described in the caption of Figure 2-5). The ratio

of intervals, Ispatial/Inon-spatial, were examined: values greater than 1 indicate that

spatial structure enhanced the probability of coexistence with respect to the

non-spatial model.
SI I I

Spatial structure increases
the coexistence interval
< 0.8-



a 0.6-
< 7-

0.4- Spatial structure decreases
the coexistence interval
I I I I
0.2 0.4 0.6 0.8
Conversion efficiency, e

Figure 2-5. Parameter intervals resulting in expansion and reduction of
the coexistence interval. The line indicates the contour at
spatial/ non-spatial = When this ratio is greater than 1, spatial
structure increased the size coexistence intervals. Parameters:
mp 0.3,N = 0.2,bp 0.3, bN 0.6,R E (0.1, 10),z 4.



Depending on the parameter values, spatial structure can either decrease or

increase the coexistence interval (Figure 2-5). High conversion efficiency e and

a high attack rate of IGpredators A meant that spatial structure increased the

probability of coexistence (Figure 2-5). Although the comparison between the

spatial model with z = 4 and the non-spatial model is shown, the results for other

neighborhood sizes (e.g., z = 6, z = 8) are similar.

2.3.5 Heterogeneous Environments

The IBM model captures the same qualitative characteristics as the differential

equation models in terms of the dominance of IGprey and IGpredators along the









productivity gradient: IGpredators are eliminated at low productivity levels and

IGprey are eliminated at high productivity levels (Figure 2-6).
> I I I I I I
'* 1.0 -*- -i -n -i-
S,' Persistence "
S0.8-

S0.6- IGprey density ,
c & A IGpredator
u *4 density
0.4 -' A

S02 ,," I

0.0- a A...
-. I I I I I I
0.65 0.70 0.75 0.80 0.85 0.90
Productivity

Figure 2-6. Persistence probability (squares) and density of IGprey (circles) and
IGpredators (triangles). Parameters: mp = 0.2, m = 0.2, bp
0.5, bN = 0.8, A = 0.9, e = 0.9, z = 4.



In heterogeneous environments (i.e., each cell is assigned either RH or RL),

when the productivity of a low resource patch is small (e.g., if the environments

were homogeneous at this productivity, even IGprey alone could not persist), a

small patch scale was favorable to IGprey and IGpredators went extinct quickly.

When the patch scale was large, however, IGprey were eliminated. At intermediate

patch scales, both species coexist. Because average productivity at different patch

scales is the same, this si r-. -. -- that the spatial configuration of patches may

strongly affect the outcome of IGP. This relationship, however, flipped as the

productivity of low-resource patches (RL) increased. When RL was relatively high,

persistence of the IGP system was higher when the patch scale was either low or

high. Persistence probability was lowest at an an intermediate level of patch scale.

Spatial heterogeneity also modified the effect of productivity level on numerical

dominance. For example, the non-spatial model predicts that when IGprey and












1.0-
E 0.8-
| 0.6-
0
U 0.4-
o
o 0.2-
* 0.0-


0 5 10 15 20
I I I I I I I I I I I I I I I
Mean productivity = 0.65 Mean productivity = 0.8 Mean productivity = 0.835
"E i .o s
Persistence r
%,

It IGpredator density '
S- .OG A-AAA-A-A A A AAA hAdens
*l., *, O SO-O- 4- OIGprey density OO


0 5 10 15 20 0 5 10 15 20
._
Patch scale

Figure 2-7. Persistence probability (squares) and density of IGprey (circles)
and IGpredators (triangles). Densities of consumers are presented
as fraction of total cells occupied by the species. Parameters:
mp = 0.2, mN = 0.2, bp 0.5, bN 0.8, A 0.9, e= 0.9, z 4.



IGpredators coexist, as productivity increases, IGprey will decline in density.

However, when spatial heterogeneity is introduced, IGprey density may remain

constant as productivity increases (Figure 2-8).

2.4 Discussion

2.4.1 Effects of Spatial Structure on the Basic Results of Nonspatial
Models

The homogeneous environment model (i.e., pair approximation) maintained

the qualitative predictions of non-spatial models. As si-L'-- .1' by the non-spatial

model, resource utilization ability of IGprey had to be be greater than that of

IGpredators in order for the two species to coexist when they have the same

neighborhood sizes. Nonetheless, the pair approximation model predicts that

IGprey and IGpredators can coexist even when the resource utilization condition

(i.e., bN > bp) is not met as long as the spatial scale for IGprey is larger than

that of IGpredators (Figure 2-4). Amarasekare (2000) considers this phenomenon

a dispersal-colonization tradeoff. Recognizing this potential tradeoff is important

because laboratory measurement of parameters such as bN and bp overlooks the









SI I I I I I I
-. -- -----

0 0.8 /-
0. Persistence ,

E 0.6-
S',
0
V 0.4 -
oV IGpredator density A--
U Ar
0.2 ,
S--A-- -
.^ e--4--*---*---*.,
S0.0- IGprey density "*---' -
0- I I I I I I I
0.2 0.3 0.4 0.5 0.6 0.7 0.8
Mean productivity

Figure 2-8. Persistence probability (squares) and density of IGprey (circles)
and IGpredators (triangles). Densities of consumers are presented
as fraction of total cells occupied by the species. Parameters:
mp = 0.2,mN = 0.2,bp = 0.5,bN = 0.8, A 0.9,e e 0.9, = 4,
and Spatial scale = 7.



differences in spatial scale of foraging, which may be essential in order to tease

apart mechanisms of coexistence in IGP communities (Amarasekare, 2003b).

Explicitly considering these factors, Amarasekare (2000a,b) concluded that

local resource utilization differences (e.g., bN > bp) were more important than

dispersal-colonization tradeoff (e.g., ZN > Zp) in coexistence of a parasitoid

community.

Spatial heterogeneity can overturn the second prediction (species dominance

shifts from IGprey to coexistence, and then to IGpredators as productivity

increases). For example, even when average productivity is low (i.e., the homogeneous

model model predicts that IGprey will exclude IGpredators), IGpredators can still

outcompete IGprey when resources are distributed at particular patch scales

(Figure 2-7). Thus, although the homogeneous model predicts a dominance shift

(i.e, IGprey dominance coexistence IGpredator dominance), this prediction

can be violated in the presence of resource heterogeneity if patch scale is large in









low productivity environments and small in intermediate and high productivity

environments. In other words, without knowing how the spatial scale of resource

changes, we cannot reliably predict changes in species dominance with increasing

average productivity.

Furthermore, spatial heterogeneity affects the qualitative prediction that the

density of IGprey will decrease while that of IGpredator increases as productivity

level increases. On the contrary, we see that density of IGprey may remain roughly

constant as productivity increases in the coexistence region (Figure 2-8) in a

heterogenous environment. In the field, Borer et al. (2003) observed the same

phenomenon that density of IGprey was unaffected by the resource level. Thus the

spatial model can potentially explain unresolved results observed in nature. The

results shown in Figure 2-8 assume a constant patch scale. If the patch scale varied

(e.g., different scales for each productivity value), we could potentially see many
different trends.

2.4.2 Quantitative Effect of Spatial Structure

The qualitative results of the mean field approximation and pair approximation

models were similar (e.g., Figure 2-3), but the models gave different quantitative

predictions of coexistence probability. As spatial structure becomes stronger (i.e.,

z decreases), the coexistence region expands while shifting in the parameter

space. For example in Figure 2-3, as z decreased, lower values of A (attack

rate of IGpredators on IGprey) that previously allowed for coexistence (e.g.,

R = 1, A = 0.8) instead allowed IGprey dominance. In the parameter region of

coexistence near IGprey dominance, IGpredators become less effective in utilizing

IGprey as z decreases. At the same time, the parameter region that allowed

IGpredator dominance but was near the boundary of coexistence parameter

region (e.g., R = 1.5, A = 0.6) shifted to coexistence because IGprey became less

vulnerable to intraguild predation. The parameter e has a similar influence in the









model because reproduction due to IGP is realized only if IGpredators can capture

IGprey.

In general, given a high attack rate of IGpredators (A) and high profitability

of IGprey (e) (i.e., intraguild predation is more beneficial to the IGpredators than

resource consumption), spatial structure favors the persistence of the IGP system.

One hypothesis for the evolution of IGP is based on stoichiometry (Denno and

Fagan, 2003); IGpredators consume IGprey because IGprey have the right balance

of nutrients (i.e., e is large). For example, Matsumura et al. (2004) documented

that wolf spiders grow better if other spiders (i.e., IGprey) were included in their

diet than when they were raised on a diet that did not include other spiders

as diet. Thus, the parameters of natural systems are likely to lie in the region

where spatial structure favors IGP persistence. This relationship between the

benefit of IGP and spatial structure -ir-.-- -I-; that understanding the proximal

consequences and determinants of IGP (e.g., Matsumura et al., 2004; Rickers and

Scheu, 2005) and the roles of spatial structure should facilitate our understanding

of the ecological and evolutionary significance of IGP.

2.4.3 Effect of Spatial Heterogeneity

Individual based model simulations in spatially heterogeneous environments

revealed that even at the same average productivity of the environment, changes

in patch scale can result in different IGP dynamics. When patch scale is small,

each cell's productivity is independent of the neighboring cells, and individuals will

experience the same average productivity level no matter where they are (as long as

their neighborhood includes at least few different cells). Therefore, the prediction

of IGP coincides with the case from a homogeneous environment. However, as

patch scale increases, the expected productivity level experienced by individuals

will diverge depending on their location. If a parent is in a low resource patch,

for example, its offspring will experience a lower than average productivity level.









Accordingly, predictions of IGP outcomes in a heterogeneous environment depart

from predictions when the environment is homogeneous.

When average productivity level was low (e.g., 0.65 in Figure 2-7), IGP

dynamics changed from IGprey dominance to coexistence and then to IGpredator

dominance as patch scale increased. In this example, the low patch productivity

was very low (RL = 0.3), and neither IGprey or IGpredators could persist

in a large scale low patch. The high resource patches, in contrast, were very

productive (RH = 1) so that IGpredators dominated (Figure 2-7). Therefore,

as patch scale increased and patch productivity diverged, neither species could

persist in the low-resource patches while IGpredators won in the high-resource

patches. At intermediate patch scales, patch heterogeneity created an environment

that allowed both species to coexist, creating a hump shaped relationship in

persistence. This hump-shape was flipped at high mean productivity levels (e.g.,

0.835 in Figure 2-7 corresponding to RL = 0.67). In high-resource patches (RH),

IGpredators dominated, and in low-resource patches (RL), IGprey dominated;

habitat segregation resulted, causing an increase in persistence. If patch scale

was further increased, persistence would eventually approach 1. For habitat

segregation to be effective, each patch type must be sufficiently large. For example,

continuous "spill ((.1, i of IGpredators from the high productivity patch can wipe

out a small low productivity patch with IGprey. This result sl--.-, -1-I that details

of landscape configuration may significantly alter characteristics of community

stability. Further investigations on how landscape structure affects movement of

species and community dynamics is needed (van Dyck and Baguette, 2005).

Although spatially structured species interaction and spatially heterogeneous

environment are well recognized factors in ecology, systematic exploration of this

axis has only begun recently (Bolker, 2003; Hiebeler, 2004a,b), and we do not yet

have clear general hypotheses about the effects of space even in simple models.






30


Further explorations of the roles of both spatial structure and spatial heterogeneity

are needed. Furthermore, although spatial interactions may be difficult to analyze,

some spatial data are relatively easy to collect once we know exactly what to

collect. In fact, field studies have often collected these data as auxiliary information

even when their analyses ignored space. The development of spatial theories will

create more testable hypotheses and increase our ability to utilize data more

efficiently, which may resolve some of the discrepancies between theory and data.















CHAPTER 3
NITROGEN LIMITATION IN CANNIBALISTIC JUMPING SPIDERS

3.1 Introduction

The nutrient content of food resources has a direct influence on consumer

behavior, fitness and population dynamics. For example, dietary nitrogen

level influences the demographic parameters (i.e., growth and fecundity) of

grasshoppers (Joern and Behmer, 1997); feeding activity and growth rate of

zooplankton are strongly influenced by phosphorus levels in their algal diet (Plath

and Boersma, 2001). Theory sl.-.- -I that animals should forage selectively

to maximize their nutrient requirements (e.g., Simpson et al., 2004). These

considerations are important not only for understanding the behavioral and

physiological mechanisms of foraging but also for understanding the dynamics

of ecological communities. N li il i. i directly affect the functional and numerical

responses of species interactions (Andersen et al., 2004). In recent years, analysis

of community models with nutrient specific interactions has became common

(e.g., usually called ecological stoichiometry (Loladze et al., 2004) or nutrient
homeostasis (Logan et al., 2004)) and these studies have helped to understand

previously unexplained patterns in nature reviewd in Moe et al., 2005).

Intraguild predation (IGP), predation within a guild (i.e. between members

of different species at the same level in a food chain), has been -i r--.- --1. to be

a response to the mismatch in the ratio of carbon to nitrogen (C:N) between

predators and herbivorous prey (Denno and Fagan, 2003). C:N decreases as trophic

level increases (Fagan et al., 2002). This type of predation (IGP, or omnivory more

broadly) benefits the consumers because consuming other nitrogen-rich predators

(and thus decreasing the C:N imbalance) helps satisfy their nutrient requirements









and facilitates growth (Fagan and Denno, 2004). If IGP facilitates the growth of

intraguild predators, it will directly affect the size-structure of individuals within

the community because occurrence of IGP is size-dependent (Polis, 1988). However,

direct examination of this hypothesis is rare (but see Matsumura et al., 2004).

In this study, I examined whether the nitrogen content of prey affects the

growth of jumping spiders. Specifically, I showed that spider growth rate is

facilitated by nitrogen content of prey.

3.2 Materials and Methods

Eggs of the jumping spiders Phidippus audax were collected in the field

near the campus of the University of Florida, Gainesville, FL, USA. First instar

spiderlings do not eat before becoming second instars. Once they moulted to

second instar, spiders were assigned to one of two treatment groups: control

and N-rich. Individual spiders were reared separately in plastic cup containers

(62 mm in diameter and 30 mm in height; approximately cylindrical). Forty spiders

were used in each treatment. The mean (sd) carapace widths of the spiders in

the beginning of the experiment were 0.830.05mm (N-rich) and 0.840.04mm

(control) and were not statistically different (t-test: p = 0.1633).

3.2.1 Experimental Treatments

Spiders in the two treatment groups received prey (Drosophila i,, .1,.;1. I-. r)

that were reared on different media to alter their nutrient profiles (\! ivntz and

Toft, 2001). In the control group, fruit flies were raised on Drosophila medium

(Carolina Biological Supply). In the N-rich group, blood meal (Pennington

Enterprizes, Inc) was added to the medium (3:1 = medium:blood meal).

In order to examine potential confounding factors of the treatment (i.e.,

treatment may create difference in aspects of prey in addition to N level), the

energetic content of prey was also quantified based on a whole-animal ..-- ,i, with a

dichromate oxidation method described in McEdward and Carson (1987).









3.2.2 Effect on Growth

The spiders were kept in a controlled environmental chamber (28 Celsius,

light:dark = 14:8 hrs). One fruit fly was given every three d-,i,- during the second

instar. Two fruit flies were given every three d-,i,- during the third and the fourth

instars. Spiders were able to consume all the prey within a di although prey

were given only once every three d-,,- Water was supplied daily in the form of a

water-soaked sponge. Carapace widths of spiders were recorded within 24 hours

of moulting to examine the treatment effect on size, while durations in each instar

were recorded to examine the effect on growth rate.

One spider in the N-rich group died during the experiment and was excluded

from the analyses. Thus, for all the following analyses, the sample size for the

control group was 40 while the sample size for the N-rich group was 39. Treatment

effects on growth (size and instar duration) were analyzed with t-tests.

3.3 Results

The treatment created prey individuals that were statistically different in

their nitrogen content but were equivalent in energetic content and were similar in

composition to spiderlings (Table 3-1).

Table 3-1. Nutrient and energy contents of prey (standard errors in parentheses).
Mean energy contents of prey were not significantly different (ANOVA,
F2,12 = 0.8808,p = 0.4396). Nitrogen content ('. N) of flies was
higher in the N-rich (blood) treatment (Welch two sample t-test,
t1.183 9.2783,p 0.048).

Prey N'_ C0. Energy (J)
flies (control) 7.64 (0.08) 50.38 (0.26) 2.20 (0.48)
flies (blood) 10.36 (0.28) 49.59 (0.94) 1.71 (0.36)
spiderlings 10.06 (0.57) 35.88 (1.27) 1.50 (0.26)


Figure 3-1 shows the carapace widths of spiders for each treatment. The

N-rich group had wider carapaces on average for all the instars examined, but the

differences were not statistically significant (Figure 3-1, t-test: p > 0.05 in all






34



cases). The size differences were insignificant even when the cumulative difference

in growth was considered (i.e., the changes in size from the second instar to the

fifth instar).


2nd instar
0


0

Co
0

(N
0


0
CD

17
CD


--


o


I I
N c


3rd instar


o


I I
N c


4th instar


0
N-
N c


Treatments


Figure 3-1.


Growth in carapace width of spiders. nth instar data indicate the
difference in size between (n + l)th and nth instar. No significant
differences were found between treatments, for any instar. Treatments:
N-rich (N) and control (c). Top and bottom lines of box indicate the
7".', quartile and 25'. quartile of sample, respectively. The horizontal
bar in the box indicates the median. Top and bottom bars around the
box indicate 911' quartile and 10' quartile, respectively. The upper
and lower notches corresponds to the upper and lower 95'. CI about
the median.


The mean durations of instars were alv--i shorter for the N-rich group

(Figure 3-2); thus, spiders grew faster while moulting between instars at the

same sizes. The differences in duration were statistically significant for the 2nd

instar (t-test: p = 0.0006) and 4th instar (p = 0.0003), but not for the 3rd instar

(p= 0.839).














O 2rd instar 3th instar 4th instar
0 *
0 ,




o 0
0" o



C- R

0 0
o-0
I I I I I I
N c N c N c

Treatments


Figure 3-2. Duration of each instar. Durations of 2nd instar and 4th instar
were significantly smaller for the N-rich treatment (indicated by *).
Treatments: N-rich (N) and control (c).


3.4 Discussion

In support of Denno and Fagan's (2003) hypothesis, the consumption of

N-rich prey facilitated the growth of jumping spiders, which would provide a

selective advantage to IGP behavior. Furthermore, because the outcome of IGP

and cannibalism in these spiders is size-dependent (i.e., large individuals eat

small individuals), the degree to which this predation occurs will have direct

consequences for their size-structured dynamics (de Roos et al., 2003). Because

occurrence of IGP depends on the size-structure, the growth consequences of

stoichiometry may have strong implications for IGP community dynamics.

Future work should consider the prey's nutrient profile more carefully.

Matsumura et al. (2004) have done experiments similar to this study examining

the effects of prey type on the growth level of wolf spiders (genus Pardosa), finding









that spiders that fed on intraguild prey (i.e., other spiders) alone did not enhance

their growth rate. Yet, they found that a mixed diet (i.e., addition of intraguild

prey to herbivorous prey) facilitated the growth of wolf spiders. Researchers

have found similar results (i.e., advantages of mixed diets) in other ecological

systems (Agrawal et al., 1999; Cruz-Rivera and Hay, 2000). We still do not clearly

understand the optimal nutrient requirements for these carnivores, nor how

those nutrients are distributed among prey in the field. Nor do we know whether

simplifying the description of stoichiometry to a single C:N ratio, or C:N:P (Logan

et al., 2004), is adequate for understanding community dynamics. For example,

Greenstone (1979) found that wolf spiders forage selectively to optimize amino acid

makeup, which -,i.i.- -I that more complex stoichiometric descriptions may be

necessary if we hope to study stoichiometric community ecology.

In this experiment, the food (i.e., fruit fly medium) of prey items was varied

to manipulate the nitrogen content of prey (\! ,yntz and Toft, 2001). We do not

know whether the level of nitrogen difference between treatment and control groups

was created as a result of nitrogen assimilation into fly tissues or blood meal in

their gut content. This difference is not crucial to the interpretation of this study

as spiders nonetheless consumed nitrogen rich prey and increased their growth

rate (Figure 3-2). However, the results have other implications. For example,

nitrogen content varies greatly among plants (\! ill-i. oi 1980). Anthropogenic

environmental changes (e.g., increased CO2 and soil pollution) alter nutrient levels

of plants (Newman et al., 2003). The blood meal used in this study is a common

agricultural fertilizer. If herbivores that consume different plants of different

qualities influence predators as shown in this study, the effect of stoichiometric

interaction on systems with IGP could occur at very large temporal and spatial

scales. By carefully examining the nutrient requirement of organisms as well as the

flow of nutrients, we may obtain deeper insights not only into a specific ecological






37


community with IGP but also into general properties of the persistence of complex

food webs.















CHAPTER 4
EVOLUTIONARILY STABLE STRATEGY OF PREY ACTIVITY IN A SIMPLE
PREDATOR-PREY MDOEL

4.1 Introduction

Incorporating adaptive traits into community models has shed light on

a number of ecological problems such as coini 1. :;:il --il diiliy and phases of

population cycles (e.g., Kondoh, 2003; Yoshida et al., 2003). Antipredator behavior

(e.g., activity level) is one of the most well studied classes of adaptive traits

both empirically and theoretically; it is widely observed in nature (Werner and

Peacor, 2003; Benerd, 2004; Preisser et al., 2005; Luttbeg and Kerby, 2005) and

its community level consequences can be significant (Fryxell and Lundberg, 1998;

Bolker et al., 2003).

One of the earliest approaches to the study of evolutionary adaptation (\! 'ynard

Smith and Price, 1973) goes under the general name of evolutionary game theory.

This approach seeks to identify the set of all strategies (trait values) that are

evolutionarily stable by applying an ESS (Evolutionarily Stable Strategy) criterion.

A strategy is called evolutionarily stable if a population of individuals adopting

this strategy cannot be invaded by a mutant strategy. The usual indicator for

measuring invadability is the fitness (contribution to the next generation's gene

pool) of the individual (Roughgarden, 1996). Thus, a strategy is called an ESS

if when it is adopted by almost all members of a population, then any mutant

individual will have a lesser fitness than that of an individual of the general

population. One shortcoming of this approach to studying ecological dynamics

is that while the ESS criterion makes good intuitive sense, it is based on a static

analysis of the population and does not indicate how the population may have









come to evolve to such an ESS. In fact, in has been shown that the non-invadability

of a trait value (a particular ESS) does not imply that a population with a nearby

different trait value will evolve to the ESS over time (Taylor, 1989; C'!i -1 in -, i,

1991; Takada and Kigami, 1991). In other words, if we regard evolution as a

dynamic process, there can exist strategies that are evolutionarily stable according

to the ESS criterion, but that are not attainable in the dynamics of evolution.

The dynamical approach to evolution of a trait C is commonly modelled by

including the following equation:

dC aw(C, C)
d g a (4-1)
dt QC
c=C

Here, W(C, C) is the fitness of an individual with trait value C in a population

that is numerically dominated by individuals with resident trait values C. The

coefficient g scales the rate of evolutionary change. Adaptive dynamics of the of the

kind given by Eq. 4-1 have been motivated by the general principle that regards

evolution as a gradient-climbing process on an adaptive landscape (Gavrilets,

2004), and by similar principles (Brown and Vincent, 1987, 1992; Rosenzweig

et al., 1987; Takada and Kigami, 1991; Vincent, 1990; Abrams, 1992; Abrams

et al., 1993). It has also been shown that one can obtain an equation similar to the

dynamics (Eq. 4-1) as a limiting case of results from quantitative genetics (Lande,

1976; Abrams, 2001). In the derivation of Eq. 4-1 by quantitative genetics it is

assumed that the trait in question is determined by a large number of genetic loci,

each contributing a small additive effect. In this setting, the rate of evolution g

may be interpreted as the ratio of additive genetic variance to population mean

fitness (Iwasa et al., 1991; Abrams, 2001).

When the focus of study is ecological dynamics, we assume that foragers

behave optimally (with an evolutionarily stable strategy) and we study the

consequences of this behavior to community dynamics (e.g, Abrams, 1992; Krivan,









1996; Krivan and Sirot, 2004). Thus, instead of incorporating the evolutionary

trait equation, Eq. 4-1, an optimal solution (ESS) for the trait C is calculated by

the ESS criterion and then substituted into the ecological dynamics equations.

In other words, we assume that evolution has already taken place to shape the

adaptive behavior, and that an evolutionarily stable value for C is in place. An

understanding of the relationship between genes and behavior is not necessary

when using this approach. However, as we shall see, even in a simple model, such

an optimal behavior may be very complex.

In this paper, using a simple Lotka-Volterra type predator-prey model with a

type II functional response (Holling, 1959; Royama, 1971; Jeschke et al., 2002) in

which the prey have a density-dependent foraging effort, we analytically derive the

ESS of prey activity, as defined by the ESS criterion. Specifically, we show that at

particular densities of predators and prey, there are multiple ESSs. To examine the

relationship between the ESSs and trait evolution, we also examine the common

dynamical model of evolution (i.e., Eq. 4-1). To examine ecological implications

of adaptive behavior, we explore the differences that may arise in community

dynamics between the evolutionary dynamical approach and the situation where

any one of the multiple ESSs of prey behavior is fixed in the base ecological model.

4.2 The Model

The base model we consider is the Lotka-Volterra model with a type II

predator functional response.



d N be a c2P (4 2)

dP P 3 ac2 \
f _QP +a 2hN -TP) (4-3)
dt 1 + ac2 hN

where P is predator density, N is prey density, and c e [0,1] is a dimensionless

quantity interpreted as the foraging effort of prey. When c = 1 (or a constant),









the model reduces to the standard Lotka-Volterra model with a type II functional

response. The parameters b, 3, h, a, mN, mp we regard to be constants, but the

foraging effort c we regard to be a function c(N, P), so that c is modelled here

as an adaptive behavior of the prey that is dependent on the densities N and P.

We interpret b as the maximum rate of benefit (reproduction) of prey when they

forage maximally. Handling time, h, is the time required for predators to consume

a prey. The density-independent death rates of prey and predators are denoted

by mN and mp, respectively. The search efficiency, a, is a characteristic of the

predators that measures their success rate of finding prey. The search efficiency

a of the predators is modified by the vulnerability 0(c) of the prey. The usual

assumption is that vulnerability of the prey increases with their foraging effort

as a convex function of c. The reason for choosing a linearly increasing function

of c to modify the prey benefit rate b and a convex increasing function to modify

predation efficiency a is so that the risk of predation does not outweigh the benefit

of enhanced reproduction when increasing foraging effort from c = 0 in the presence

of a large predator population. No matter what the densities N and P are, there

will J.i. ,-, be some positive value of c which is better for the prey than c = 0. For

simplicity we use the vulnerability function 0(c) = c2. For example, if the prey

decrease their effort from c = 1 to c = 0.5, then the effective search efficiency of

the predators decreases from a to 0.25a while effective benefit decreases from b to

0.5b. Food assimilation efficiency in converting ingested prey into new predators

is denoted by 3. Note again that we assume the foraging effort has bounded

values. Since we assume c(N, P) E [0, 1], c may be interpreted as a fraction of the

maximum foraging effort.

A method for calculating all possible ESSs for an adaptive behavior is based

on the following









ESS criterion: A strategy is an ESS if when used by almost all
members of the population, results in the fitness of an individual from
the general population being greater than or equal to the fitness of any
mutant individual in the population.

We remark that this criterion in and of itself does not guarantee the existence

of an ESS and also does not rule out the possibility that more than one (or

infinitely many) ESS can exist. Furthermore, it assumes that the population is

homogeneous in the sense that almost all individuals in the population use the

same strategy.

In our model, suppose a mutant prey individual (e.g., c = B) emerges in a

population where every other prey individual employs the foraging effort c = A.

The fitness WN of an individual in the general population is derived from Eq. 4-2:

aA2p
WN bA maN.
1 + aA2hN

The fitness W, of a mutant individual (with c = B) is very closely

approximated by
aB2p
W, bB re2.
1 + aA2hN

The mutant fitness is of this form, because in the expression for a type II

functional response, the denominator 1 + ac2hN (a dimensionless quantity) is the

factor by which the risk of predation is reduced due to the time that a predator

spends handling prey, and by assumption essentially all prey that are being handled

are non-mutant (c = A). On the other hand, the numerator is the risk to the focal

(i.e., mutant) prey type, which has c = B.

The ESS criterion for adaptive foraging in our model is:

An effort E E [0, 1] is an ESS if and only if

aPx2 aPE2
bx ahNE < bE- a E2 for all x c [0, 1].
1 + ahNE2 -1 + ahNE2









We remark again that we regard such strategies E and x to be density-dependent

functions E(N, P) and x(N, P).

4.3 Results

4.3.1 Evolutionarily Stable Strategy (ESS) of Foraging Effort

The derivation of evolutionarily stable strategies (ESSs) for the trait c is shown

in Appendix A. There are three possible functional types of ESS: c = y, c = y2 and

c= 1, where

I 1 P P2 hN 2 1 p2 hN
bhN a a 2 bhN a

Each of the three ESSs is valid only in a certain region of the NP-plane, as

shown in Figure 4-1.

When predator density is relatively low (Region I, Figure 4-1), the only ESS

is for prey to forage with the maximal effort of c = 1. When predator density

is high relative to the prey density (Region II, Figure 4-1), the only ESS is for

prey to forage with effort of c(N, P) = y (N, P). The value of y, throughout

most of Region II is generally low (< 1), although c = y, agrees with c = 1 at

the boundary between Region I and Region II. At all points in the NP-plane of

intermediate predator density (Region III, Figure 4-1), each of c = y, c = y2 and

c = 1 is an ESS.

When we refer to a particular ESS, c(N, P), it is with the understanding that

at each point (N, P) of the NP-plane c has a well defined value that is among the

possible values given above. The existence of multiple ESS values in Region III

implies that there do exist complicated ESS, because the criterion does not require

that just one of c = y, c = y2 and c = 1 must apply uniformly to all points in

Region III. One such complicated, and perhaps unlikely, strategy is depicted in

Figure 4-2, where Region III is divided into many subregions, with each subregion

associated to one of the three possible ESSs. For simplicity in the subsequent






















(ESS = 1)
1



1 S

Figure 4-1. Solutions for the ESS for c in each of three regions of the
nondimensionalized NP-plane (s = ahN and r -= P). In both
Region I and Region II there is precisely one ESS function. In Region
III there are three possibilities for an ESS. For the expressions for yi
and y2 in terms of r and s, see Appendix A.


analysis we will consider three basic ESSs, one for which c = yl is chosen uniformly

for all points in Region III, and similarly those for which c = y2 and c = 1 are

chosen uniformly in Region III. No matter which of the three possible ESSs is

chosen uniformly for Region III, there will be a discontinuity of the ESS function.

If c = 1 is chosen for Region III, then there is a discontinuity at all points on the

boundary between Regions III and II (Figure 4-3); if c = yl is chosen for Region

III, then there is a discontinuity at all points on the boundary between Regions

III and I; if c = y2 is chosen then there is a discontinuity at all points on both

boundaries of Region III.

We note that the strategy c = y2 (in Region III) is a strategy that is counter

to intuition in the sense that for fixed prey density N, as P increases then y2

increases in value, so that a prey individual that has adopted the strategy of c = y2

in Region III would increase its foraging effort as the predator density increases.










A prey individual that has adopted the strategy of c = yi in Region III would

decrease its foraging effort as the predator density increases.

For a fixed predator density, similar characteristics of the three ESS functions

are observed as prey density increases from N = 0 (Figure 4-3).





C = y'
= Y1 c= 1
1 C = Y2

C=1


1C=




1 S


Figure 4-2. A complicated ESS function, where Region III is split into many
subregions, with each subregion associated with one of the three
possible basic ESSs.


The functional response of predators will be very different, depending on

the strategy that prey employ in Region III. Considering each of yl, y2, and 1 as

a strategy employ, ,1 uniformly in Region III by the prey, the type II functional

responses appear as shown in Figure 4-4. The choice of c = 1 while in Region III,

naturally yields a response that is equivalent to a standard type II response. The

choice of c = y2 however yields a functional response that is opposite in trend to

the standard response increasing prey density results in decreased kill rate for the

predators while in Region III.
















C


4 4
66 6 5
00 0

Figure 4-3. The three basic ESS functions determined by which of the three
strategies is chosen uniformly in Region III. Top (c = 1), bottom
left (c = yi), bottom right (c = y2). Plots of the ESS functions are
shown on the nondimensionalized NP-plane (s ahN and r -= P).


4.3.2 Incorporating ESS into the Community Dynamics

The calculation of ESS in the previous section was done without any

consideration of the population dynamics, other than to use Eq. 4-2 in deriving the

ESS criterion. In this section we take the basic ecological model (Eqs. 4-2 and 4-3)

and replace the foraging effort c in these equations with one of the three basic

ESS strategies. Thus the right-hand sides of equations (4-2) and (4-3) are now

formulated as three-part functions, since there are three functional forms for a basic

ESS for c, depending on which of three regions of the NP-plane the point (N, P)

lies in. In Region III we choose just one of the three possible ESSs to incorporate

into the system, and we look at each of these three choices in turn to compare the






47





ro
0


o 00
ci- 6 0



30 50 70 30 50 70 30 50 70




Prey density
Figure 44. Apparent functional responses of predators when P a ,





t, mN = 0.1, mp = 0.1. In this parameter region, there are three ESSs


c = y2. Right: c -= .


effects on the ecological dynamics with these choices. Conceptually, we are now
30 50 70 30 50 70 30 50 70








Provoking at community dynamics with the assumption that evolution has already

Figurtaken place and4. Apparent functional responses of pree basic ESSs. The analysis in this







section includes equilibrium and stability results of the community dynamics.

For any fixed positive values of the parameters a, h, mN and mp, if both b

and 0 are sufficiently large, then there is exactly one nonzero equilibrium (N*, P*)
possible and this equilibrium is guaranteed this parameter for region,e of there are three choices of
(Figure 4 1). These functional responses were plotted assuming that









theESS in Region This nonzero equilibrium is locally stable only if both b and dle:
c Y 2. Right: c 1.










effects on there sufficiently large. In particular, with these choices. Conceptually, we arge, a locally
looking at community dynamics with the assumption that evolution has already

taken place and has arrived at one of the three basic ESSs. The analysis in this







section includesable equilibrium will occur if and stability results of the ESS community dynamics.

For any fixed positive values of the pasramtegrs a, the ecological equilibrium

and*, P*) is re sufficiently large, th the effort c* exactly one nonzeroat this equilibrium, i.e.*

possible and this equilibrium is guaranteed to occur P*).r one of the three choices of
ESS in Region III. This nonzero equilibrium is locally stable only if both b and 3

are further sufficiently large. In particular, with b and f3 sufficiently large, a locally

stable equilibrium will occur if and only if c 1y is the ESS in Region III.

In the results listed below, for given strategy c(N, P), the ecological equilibrium

(N*, P*) is recorded along with the effort c* evaluated at this equilibrium, i.e.

c* c(N*,P*).






48


1. If b < mN or 3/h < mp, there is no nonzero equilibrium.

If b < mN, then both species will go extinct.

If b > mN and 3/h < mp, then P will die, but N will grow without
bound.

2. If mN < b < 2mN and 3/h > mp, then an equilibrium is guaranteed to exist
if c = 1 is chosen for Region III. This equilibrium is given by

N mp p, 3(b mN) 1
a(3 hmp)' a(3 hmp)'

This equilibrium is unstable. The location of this equilibrium (N*, P*)
may lie in either Region I or Region III, depending further on the values of
b, 3/h, mN, mp.


3. If b > 2mN and mp < K < 2mp, then an equilibrium is guaranteed to exist
if c y2 is chosen for Region III. The location of this equilibrium and the
corresponding value of the ESS at equilibrium is given by

Smpb2 Ob32 2mN
N* P c* y
4amN(3 hmp)' 4amN(3 hmp)' b
This equilibrium is unstable. The location of this equilibrium (N*, P*) lies in
Region III.


4. If b > 2mn and > 2mp, then an equilibrium is guaranteed to exist if c = y1
is chosen for Region III.
The location of this equilibrium and the corresponding value of the ESS at
this equilibrium is given by

Smpb2 P* O3b2 c* 2mN
N* PN c by
4am(3 hmp) 4aN( hmp)' 1 b

Note that this expression for the equilibrium is the same as in item (3)
above, but the stability is different. This equilibrium may be locally stable or
unstable:









Table 4-1. Equilibrium analysis. The c specified is the choice of ESS in Region III
that guarantees the existence a nonzero equilibrium.



b < mN mN < b < 2mN b > 2nN


< mp none none none

c= 1 c=y 2
mp < < 2mp none
p < unstable unstable

c = yi
C 1 C c=
h > 2mp none unable locally stable iff
h > max{2mp, mp + mfN}



If, in addition to the above conditions, K < mp + -'nN, then the
equilibrium is unstable.

If, in addition to the above conditions, 2 > mp + mwN, then the
equilibrium is locally stable.
The location of this equilibrium (N*, P*) may lie in either Region II or
Region III, depending on the values of b, 3/h, mnN, mp.


Table 1 summarizes the equilibrium analysis results.

4.3.3 Comparison with the Quantitative Genetics Model

The quantitative genetics (QG) model analyzes the community dynamics

along with the evolutionary adaption of c by taking the ecological system and

incorporating the dynamics of c. To the ecological system of Eqs. 4-2 and 4-3

we add the following differential equation, which is Eq. 4-1 applied to the fitness

function of the prey.
de acP
dt 1 + ac2hN)

where g indicates the rate of evolution. The usual QG model (e.g., Matsuda and

Abrams, 1994) assumes that c can take on an arbitrarily large value and so the

system of three differential equations alv--i- has well defined solutions, although









solutions will generally have unbounded c values. In our model, we assume that the

value of c is restricted to c c [0, 1], and so we restrict the third differential equation

to limit the growth of c.
acN acP
g (b acP ) if c < 1 or g (b 2 acP < 0
A I + a C h N 1 + a C h N ( 4 -4 )

I l+ashN
0, ifc 1 and g(b-2 1cr )>0

In any solution to this system, if the value of c ever evolves to the value c = 1,

then it will remain at c = 1 until the population densities reach levels at which

- < 0, where 7 is given by Eq. 4-4.

In a stability analysis of the system (Eqs. 4-2, 4-3, 4-4), solving the the

equation 0 easily confirms that the equilibria values c* for the QG system are

precisely the same as the ESSs derived by the ESS criterion in section 3.1. Thus

we find, as expected, that the equilibria (N*, P*, c*) for the QG dynamical system

are the same as those derived in section 3.2 from analyzing the ecological dynamics

with ESS inserted.

4.3.3.1 Behavior of the system with fast evolution

C'!I..-. ig to have a large value in the QG system assures that evolution of

the trait c occurs rapidly, and so in the QG model the value of c is aliv-- at, or

very near, an ESS for the current densities of N and P. For example, if the QG

system is in a state such that the current value of (N, P) lies in Region I, then,

with the assumption that the rate g is fast relative to the ecological dynamics,

we can conclude that foraging must be at c = 1 (or will evolve very quickly

to c = 1), because there is only one equilibrium c-value in Region I. Similarly,

if the current value of (N, P) lies in Region II, then we may conclude that the

value of c is at c = yi(N, P) since evolution occurs quickly and there is only one

equilibrium in Region II. But if the current value of (N, P) lies in Region III,

then it is not immediately clear at which level c will be (or will quickly evolve to).






51













40 .T1 i
U

II I
-I ******* *.***. ...
S .................... I
I I -I+ 4-1* I
0 1 Yl Y2 1 Y 1

Foraging effort

Figure 4-5. Evolutionary dynamics of foraging effort (dc/dt versus c) in Region I
(left), Region III (middle), and Region II (right), under the assumption
of fast rate of evolution g.


Which of the three ESS in Region III is favored by evolution can be determined

by examining the phase plane for c with fixed N and P. We may assume N and

P to be essentially constant as c evolves, because we have assumed a fast rate of

evolution g. A phase plane diagram (plot of dc/dt versus c) with value of (N, P)

in Region III is shown in Figure 4-5. The diagram shows that in the dynamics of

the QG model, c = yi is stable, c = y2 is unstable and c = 1 is stable when the

system is in Region III. (Note that c = 1 is bounded above, so that it has nowhere

to evolve but down, but the dynamics of c will cause any small perturbation to a

lesser c-value to quickly return to c = 1.) In particular, if the system is in a state

such that (N, P) is in Region III, then a value of c that is greater than y2(N, P)

will quickly evolve to c = 1 while a value of c that is less than y2(N, P) will quickly

evolve to c = yi(N, P).









Which of the three ESSs is evolved to while in Region III is dependent on the

trajectory of the ecological system. If the trajectory of the system in the NP-plane

enters Region III with trait value c greater than y2, then the trait will converge

to the ESS c = 1, while if the trait value is below y2 when entering Region III,

then c will evolve to c = yi. It is well known that the predator-prey dynamics of

a Lotka-Volterra system involves counterclockwise trajectories in the NP-plane.

The same is true of this system. Unless the parameters are such that one or both

species are dying out, trajectories proceed in a counterclockwise fashion in the

NP-plane. This means that if the initial state of the system is such that (N, P)

lies in Region I, the only way that the resultant trajectory of the system in the

NP-plane may enter Region III is by crossing the boundary between Region I

and Region III. In general, spiral trajectories that pass through each of the three

regions proceed in counterclockwise cyclic order of (I, III, II). As noted before, any

trajectory that passes through Region II quickly evolves to c = y1 while in Region

II, and any trajectory that passes through Region I quickly evolves to c = 1 while

in Region I. Thus for any initial state (N, P, c), unless the the ecological dynamics

are such that the resulting trajectory converges to an ecological equilibrium

without ever entering Region I, it is necessarily the case that the trajectory will

enter Region III with its trait value fixed at c = 1. As a general principle, we can

-Ji that in the QG model with fast g, any trajectory that involves a spiral or a

cycle passing through Region I takes on the value c = 1 while in Region III.

It was shown in Section 3.2 that stability of the ecological system is possible

only for certain values of the parameters, and only if the ESS being used has c = y1

fixed in Region III. The above analysis shows however that even if the parameters

are favorable to ecological stability, if the rate of evolution is fast (large g), then

the potential ecological stability may not be realized, because the system evolves to

c = 1 whenever Region III is entered on a trajectory that passes through Region I.









On the other hand, it is also possible that the trajectory of a stable ecological

system may be contained entirely within Regions II and III, in which case a fast

rate of evolution would not affect the stability.

Figure 4-6 shows typical trajectories in a simulation of the base ecological

system when the three different ESSs in turn were fixed in Region III. (The

simulation shown is for the base system without the dynamics of c incorporated.)

For the simulation in Figure 6, where c = y2 is fixed in Region III, there are

periodic outbreaks of prey. Where c = 1 is fixed in Region III, the system system

is unstable with oscillations with very high magnitude (prey growth is much more

rapid). Where c = yi is used in Region III, the result is a limit cycle that passes

through each of the three regions. This stable limit cycle is possible if we regard

evolution as having already occurred and further consider that evolution has

terminated with the basic ESS that fixes < 'u in Region III. But this same cycle is

not possible in the dynamics of the QG model with fast g, because the cycle passes

through the three regions in cyclic order (II, I, III), thus resulting in the fixing of

c 1 in Region III.

Another interesting consequence of the above a n i1 -i of the evolutionary

dynamics is that we have identified the basic strategy c = y2 as unstable in the

dynamics of evolution. Thus we have, at first glance, the seemingly paradoxical

existence of an unstable strategy that is evolutionarily stable. The possible

confusion lies in the two v--,v' that the word "-I Ii.1, is being used in this sentence.

The strategy c = y2 is stable relative to invasion by mutants. If the general

population adopts the basic ESS strategy that fixes c = y2 in Region III, then the

population is not invadable by a small number of mutants. However this strategy

is not stable relative to small shifts in the behavior of the general population. If by

some happenstance the entire population experienced a small shift in behavior, due

for example to environmental change or to a large scale mutation, then the entire















0 1 2 3 4 -2 2 6 10 -5 5 15 25
0
6i 0 0



0 1 2 3 4 -2 2 6 10 -5 5 15 25

log(s)

Figure 4-6. Simulation of the dynamics of predators and prey plotted on the effort
diagram (Figure 4-1). s = ahN and r = (a/b)P. The gray line
indicates r = / and the dotted line indicates r = (1 + s)/2. The area
between these two curves with s > 1 indicates the region where there
are multiple ESSs. Left figure: c = yi, middle figure: c = y2, and right
figure: c 1=. h 1.6, b = 1.25, rN = 0.8, mp = 0.5, 3 = 1.5.


population would evolve away from the strategy y2 towards one of the dynamically

stable strategies with c = yi or c = 1 in Region III. Similarly, we can i that

the basic ESS that fixes c = 1 in Region III is stable relative to small shifts in the

behavior of the population with fast g, but this behavior does not allow ecological

stability.

4.3.3.2 Behavior of the system with slow evolution

C'! -....i g to have a small value in the QG system relative to the ecological

time scale assures that evolution of the trait c occurs slowly. In such a situation,

if the community is not initially at ESS, then evolution to an ESS is not possible

unless the community dynamics allow an ecological equilibrium of some sort,

simply because the community must persist in order for evolution to take place.

Assuming that a community is initially at ecological equilibrium and assuming

that an initial non-ESS strategy is employ, -1 then as evolution occurs (slowly), the

ecological equilibrium will change, because mutants with superior genetics (superior

foraging behavior c) will successfully invade the population, thus changing the









ecological dynamics. To attempt to calculate all possible effects of slow evolution

on the community dynamics would be more difficult for this model, because we

would need to characterize those foraging functions c(N, P) that allow for an

ecological equilibrium, and then determine for which of these initial foraging

functions the QG dynamics will maintain stability of the community as evolution of

c occurs. We will leave such calculations for a future paper, but note that it is not

implausible that slow evolution could lead to the destruction of ecological stability.

It is also plausible that slow evolution could converge to an ESS that supports

ecological stability and that is more complicated than one of the three basic ESSs

(i.e. an ESS which is piecewise defined on several subregions of Region III, as in

Figure 4-2)

4.4 Discussion

When studying dynamics of the evolution of an adaptive behavior it is

essential to consider evolutionary dynamics along with ecological dynamics. Certain

behaviors can be determined to be evolutionarily stable strategies without those

behaviors being compatible with a stable ecological system. In a simple but

commonly used predator-prey model, we found that the quantitative genetics

model of evolution with fast rate of evolution g often may converge to an ESS that

promotes instability of the ecological system. In a wide parameter region for this

same model, we found the existence of multiple ESSs (Figure 4-1). Among these

we identified one simple such strategy that allows a stable ecological equilibrium.

However, a system that is initially in ecological equilibrium and employing this

ESS, may well evolve under the quantitative genetics model to an ESS that

destabilizes the community dynamics. Even without considering the issue of

ecological equilibrium, it is apparent that depending on which of the multiple ESS

is adopted, community dynamics can be very different (Figure 4-6), and so we need

to be cautious about the implications derived from results of these models.









Most community models with a component of adaptive behavior do not

include intraspecific interactions such as the one imposed by a type II functional

response. This study shows that the inclusion of one of the most commonly used

functions in community ecology (i.e., type II functional response) induces the

existence of multiple evolutionarily stable strategies of adaptive behavior. Any

one of these ESS is by definition, a strategy that is stable relative to invasion by

mutants. However, not all ESSs are attainable in the evolutionary dynamics that

are derived from specific assumptions about how genes influence behavior.

The Type II functional response is generally considered to be a destabilizing

factor in community ecology ( 1,irdoch et al., 2003). Indeed, the same Lotka-Volterra

system as used in this study, but without the adaptive foraging behavior c

incorporated, is known to be a globally unstable system without limit cycles.

The result of the stability analysis of our system gives some evidence for the notion

that inclusion of adaptive behavior along with a Type II functional response may

have a stabilizing effect on community dynamics.

The analysis of the ESSs for this model was confined to what we referred to

as the three i- ESSs, that is strategies that uniformly fix one of the three

functional forms for an ESS throughout Region III. However, the existence of

these three basic ESSs implies that there are theoretically infinitely many ESSs

for the model (e.g., Figure 4-2). The analysis of apparent functional response

(Figure 4-4) was done just for the three basic ESSs. However, a complicated ESS

such as is shown in Figure 4-2 is a possible behavior, and if prey were to adopt

such a strategy, the functional response would appear very erratic. The community

dynamics associated with such an irregular, non-basic ESS would also be very hard

to predict or analyze. The biological feasibility of a strategy set is difficult to assess

with confidence. However, it is important to note that simple considerations (i.e.,

type II functional response) in a simple predator-prey model led to potentially









infinitely many ESSs, -ii-.-, -ii.-; that ,in ii-- of behavioral data on activity level

can be very difficult.

The present study also has implications for functional response studies. If

multiple ESSs are possible, the apparent functional response can look very different

depending on which ESS is emploi-, .1 by the prey (Figure 4-4), even though the

underlying mechanism of predator activity is in each case the type II functional

response (Holling, 1959; Royama, 1971). When faced with experimental functional

response data that differs from the classical Type II curve, a type II functional

response can still be fitted, provided that information about the about activity

behavior of the prey can be incorporated into the model. Furthermore, although

certain functional responses may look unfamiliar (e.g., Figure 4-4, middle), without

carefully examining the intraspecific behavior of prey in the field, we should not

dismiss such responses as possibilities. For example, jumping spiders are known

to stay in their retreat even when they are starved (Okuyama, unpublished

manuscript). Although, the mechanism behind this behavior is still unknown, if

we were to estimate a functional response of predators of these jumping spiders by

including the inactive individuals in the analysis, we may see a relationship that is

very different from the case when only active individuals are used in the analysis.

In laboratory experiments, such inactive behaviors of prey are often not recovered

due to the use of a small arena, and so such analyses may artificially lead to the

usual type II functional response. It is important to examine how activity level is

really expressed in a natural environment.

In many theoretical investigations of adaptive behavior, no upper bound is

imposed on the trait value (e.g. Matsuda and Abrams, 1994; Abrams, 1992). In

the model studied here, if the foraging effort c were allowed to take on arbitrarily

large values, then there would be no ESS for c possible in Region I. At all other

points (in Region II and III), both of the strategies yi and y2 would be ESSs,










and y2 would take on very large value in Region II. The quantitative genetics

model with no bound on c would also yield drastic instability in Region I, since

any trajectory that entered Region I would result in c evolving to higher and

higher, unbounded values. Although unbounded trait value is a commonly used

assumption for its simplicity, it can have a strong influence on the conclusions that

are made about community dynamics. In our model, the trait c is a dimensionless

quantity interpreted as foraging effort and so the only realistic interpretation is

with c taking values between 0 and 1.

Kfivan and Sirot (2004) investigate community dynamics with type II predator

functional response under the assumption that prey effort is determined by

maximizing population fitness rather than by an ESS criterion for individual

fitness. The dependence of prey effort on the densities of N and P is a very

different relation when the criterion is maximizing population fitness compared to

when the criterion is ESS. (Figure 4-7). When analyzing adaptive behavior in a


ESS non-ESS
40 40
1 1
: 35 :235
V) -08 08
30 30
06 -06
-25 025
o o
-' -04
20 04 20






Figure 4-7. Foraging effort as a function of N and P for an Evolutionary stable
strategy (left) and for a strategy that maximizes population fitness
(right). a2 h =, b 1.



community ecological context, for example when considering trait mediated indirect

interactions, the criterion for .i i i b. !i i, i i needs to be considered carefully

and stated explicitly. For a fixed prey density, an increasing predator density would









increase the effect size of the trait-mediated indirect interaction in the case of an

ESS, but may decrease it in a non-ESS solution (Figure 4-7).

As far as the an the interface of evolution and ecology is concerned, the

current study highlights the importance of understanding the genetic basis

of behavior. While more progress in this field has been made in recent years

(e.g., Greenspan, 2004), we still have little information about the mechanisms

of behavior for most traits, and thus about how behavior comes to fixation. It

has not been well established that the assumptions of the quantitative genetics

approach (Abrams, 2001) are appropriate for the study of adaptive behavior.

Until the genetic basis of behavior is more well grounded, evolutionary ecological

modelling remains highly phenomenological, even if the model is based on the

mechanistic genetics argument. While research in behavioral genetics is already

recognized as an exciting research front, the field of community ecology also awaits

its exciting progress.









4.5 Appendix A: Derivation of the ESS

ESS Problem: Find E E [0, 1] such that

aPx2 aPE2
bx < bE for all x e [0, 1]
1 + ahNE2 1 + ahNE2

(Assume all parameters and variables a, b, h, N, P have positive values.)

r(x2 2)
Let r = (a/b)P, s = ahN, and let F(x, y) (y- x) + r 2
1 + sy2
Then the ESS problem is equivalent to:

Find E E [0,1] such that F(x, E) > 0 for all x e [0,1].

It is clear for any E, that F(E, E) = 0, so the problem is equivalent to:

Find E E [0,1] such that min F(x, E) 0.
xe[O,1]
For any fixed .i-v-,iu, E, the function F(x, E) is quadratic in x so it is easy

to locate the value xo that yields the min value for F(x, E). That value is xo =
(l+sE2)
2r
There are two cases to consider, because the value of xo may or may not lie in

the interval [0,1]. In particular, xo E [0, 1] = E2 < (2r 1)/s.

1. If E2 < (2r 1)/s, then the min value of F(x, E) occurs at x = xo, and so the
problem requires in this case that

1
F(xo, E) ((1 + sE2) 2rE)2 = 0.
4r(1 + sE2)
If E is to be an ESS in this case, then E must satisfy

sE2 2rE + 1 0.


2. If E2 > (2r 1)/s, then the min value of F(x, E) occurs at x = 1, and so the
problem requires in this case that

F(1,) (E- 1)((1 + s2) r(1 + ))
F(l'^ = )= 0.
(1+ sE2)
It is not hard to show, with the assumption E2 > (2r 1)/s, that the only

solution of this equation for E in the interval [0, 1] is E = 1.









In summary, suppose E is an ESS. Then the two cases to consider are:

1. If E2 < (2r 1)/s, then E must satisfy sE2 2rE + 1 0.


2. If E2 > (2r l)/s, then E must satisfy E = 1.


Next consider the polynomial equation given in Condition (1). The solutions of

the equation sy2 2ry + 1 are:


r 2-Ar s
Y

Thus an ESS for condition (1) is possible only if r2 > s. Furthermore, if we let


r pT-2 s r+ P2 -s
Y1 = r and Y2
s s

then both Yl, Y2 are positive and we have the following requirements for y, and Y2

to lie in the interval [0, 1]:

y, e [0, 1] r r > and (r < s or r > (1 + s)/2)

y2 [0, 1] r > 2 and (r < s and r < (1+ s)/2.)

Formulas for ESS can now be given in terms of conditions involving just the

parameters r and s. It turns out that results are dependent on whether s < 1 or

s > 1. There are a few subcases for each of these two cases. Here is a summary of

the formulas:


Assuming s > 1 : (so s > -- > /s.)

If r > then y1 is the only ESS. (y2 > 1 in this case.)

If < r < (+, then y1 < Y2 < 1 are both ESS, and y = 1 is also ESS.

If r < /s, then y = 1 is the only ESS. (y/, /2 are not real in this case.)






62


Assuming s < 1 : (so s < s < I.)

* If r > (", then yi is the only ESS. (yi < 1 and Y2 > 1 in this case.)

* If r < (1 then y 1 is the only ESS. (Both yi, Y2 are either > 1 or not
real in this case.)















CHAPTER 5
ON THE QUANTITATIVE MEASURES OF INDIRECT INTERACTIONS

5.1 Introduction

Indirect interactions, whether density-mediated (DMII) or trait-mediated

(TMII), can profoundly alter community dynamics (Werner and Peacor, 2003;

Bolker et al., 2003). Two well-known phenomena in community ecology, trophic

cascades and :. I. :iw predation, illustrate the importance of both trait and

density effects (Schmitz, 1997; Wissinger et al., 1999; Schmitz et al., 2004);

TMII can also promote coexistence in ecological communities (e.g., Damiani,

2005). Ecologists have quantified the strengths of TMII and DMII in a variety of

systems (Werner and Peacor, 2003; Preisser et al., 2005), typically concentrating on

the relative strengths of the two types of indirect interactions, and their effects on

long-term community dynamics (Krivan and Schmitz, 2004; van Veen et al., 2005).

In order for us to make progress in this area, however, we must quantify indirect

interactions in v--i- that are accurate, consistent among studies, and consistent

with the underlying community dynamics.

Here, we point out that the methods used in previous studies have been

inconsistent and may inaccurately estimate the relative strength of trait and

density effects, one of the main goals of these studies. We explore the strengths

and weaknesses of different metrics using the example of a three-species linear

food chain (predators-foragers-resources). Predators both kill foragers (density

effects) and induce antipredator behavior in foragers (trait effects), in both cases

reducing the absolute rate at which the forager population consumes resources

and thus increasing the density of resources. We find that ratio-based metrics

typically quantify TMII and DMII most consistently, although other metrics may









be required in specific cases where absolute differences in resource density are of

interest or where the community is observed over a long time scale.

5.2 Quantifying Indirect Effects

5.2.1 Standard Experimental Design

Studies that aim to quantify the strength of TMII and DMII are typically

short-term, usually much shorter than a generation time, with negligible reproduction

or regrowth of any of the species in the community. Thus researchers typically

quantify indirect effects based on the change in resource density between the

beginning and end of the experiment (which is equivalent to the total resource

consumed by foragers if the regrowth of resource is negligible).

Previous attempts to quantify the strength of indirect effects have used some

or all of the following treatments.

1. The true predator treatment includes unmanipulated predators, foragers, and
resource, mimicking the natural system;
2. The threat predator treatment includes predators (or predator cues), inducing
antipredator traits in foragers, but prevents predators from consuming
foragers (e.g., predators are disabled or caged);
3. The no predator treatment contains only foragers and resources, and thus
eliminates indirect effects.
4. The culling treatment removes foragers in a way that matches the predation
rate in the true predator treatment in the absence of predators.

While the first three treatments are standard, culling is rarer (Peacor and

Werner, 2001; Griffin and Thaler, 2006). We will discuss the importance of culling

below; we simply note here that the accuracy of the culling treatment (i.e.,

the degree to which it mimics the natural removal of foragers by predators) is

important (Griffin and Thaler, 2006). Experimenters must record the number of

surviving foragers in the true predator treatment at frequent intervals and remove

foragers in a no-predator treatment to match the population trajectory in the

predator treatment.









5.2.2 Indices of Indirect Effects

Suppose we run a short-term experiment with abundant resources, minimizing

(1) the effects of both forager and resource depletion (and resulting variation

in forager death and resource uptake due to functional responses of predators

and foragers (Juliano and Williams, 1987)); (2) variation in forager strategy over

time (Luttbeg et al., 2003); (3) differential mortality due to costs of antipredator

behavior; and (4) intraspecific interference. Then we can define F and f as the

per-forager uptake in the absence and presence of predators; since antipredator

behavior generally reduces foraging effort or efficiency, we suppose f < F.

Similarly, if N and n are the average numbers of foragers alive during the

experiment in the absence and presence of predators, we expect n < N. The total

uptake in different treatments, which should approximately equal the difference in

resources between the beginning and end of the experiment, is:

no pred. threat culling true
(5-1)
FN fN Fn fn

We can quantify TMII, DMII, and the total indirect effect (TII) by contrasting

these treatments:
T TIT,_ no pred threat = N(F f)

TMII,+ cull true = n(F f)

DMII- no pred cull = F(N n) (5-2)

DMII+ = threat true = f(N n)

TII no pred true = FN fn

In the subscripts, a denotes .'lii, 1,v and the + and indices refer to the

presence or absence of the other effect; for example, DMIIa+ is the additive effect

of predator-induced changes in density (present in the true predator treatment but

not the threat treatment) in the presence of antipredator behavior (present in both

treatments).






66



Alternatively, we can quantify the indirect effects based on proportional

changes (ratios):

TMIIr_ no pred F
S threat f
TMIIr+ cull



DMIIrt threat N
true r
TIIr no pred FN
true fn
or similarly (as used by all existing studies):


TMII2_ 1 no pred 1 F
threat f

TMIIr2+ 1 cull 1 F
true f
DMIIr2- 1 no pred 1 N (5-4)
cull t

DMIIr2+ 1 threat 1 N
true n
TII no pred 1 FN
true fn

All existing studies that used ratio based indices have used eq. 5-4 rather than

eq. 5-3. Using the ratio-based indices, the difference between + and disappears

(e.g., TMII, TMIIr_ = TMIIr+ and DMII, DMIIr_ = DMIIr+). However, as

we discuss below, the difference between indices with different subscripts (e.g., r+

vs. r-) can become important in some circumstances.

Existing studies vary widely (Table 5-1), using both additive (eq. 5-2) and

ratio (eq. 5-4) indices. In addition, some studies have calculated the indices of

TMII and DMII directly from the contrasts shown above (direct method), while

others have quantified TMII using the contrasts but derived DMII by subtracting

TMII from the overall size of indirect effects: we discuss this indirect method

further below.

5.2.3 Decomposing Total Effects

Our first criterion for metrics of indirect effects is that they should neatly

decompose total indirect effects into trait- and density-mediated components. The









Table 5-1. Existing studies that

Study
Huang and Sih (1991)
Wissinger and McGrady (1993)
Peacor and Werner (2001)
Grabowski and Kimbro (2005)
Wojdak and Luttbeg (2005)
Griffin and Thaler (2006)


metrics defined above lead to


TII,

TII
TII,2


have explicitly compared TMII and DMII.

Additive/Ratio Direct/Indirect Culling
Additive Indirect No
Additive Indirect No
Additive Direct Yes
Ratio Indirect No
Ratio Direct No
Ratio Direct Yes


FN fn = DMII,_ + TA\ 11,
FN DMII, TMII,
fn
1 1 -- (1 DMII,2)(1
frn


DMII,+ + TA\ ii,


(5-5)


TMIIr2).


While one can decompose total effects in any of the three frameworks shown

above, the ratio framework is simplest, and for some purposes can be simplified

further by taking logarithms: log TII, log DMII, + log TMII,. Furthermore, the

decomposition of total additive effects into components with different subscripts is

problematic: we discuss this further below. Although some studies have used ratio

measures (Griffin and Thaler, 2006), the general importance of assessing contrasts

on an appropriate scale does not seem to have been appreciated as it has in the

closely analogous problem of detecting multi-predator interactions (Billick and

Case, 1994; Wootton, 1994).

However, additive indices may be preferable when the goal is to quantify

the absolute change in resource depletion instead of the relative size of TMII and

DMII. For example, in a study of eutrophication one might want to know the

absolute change in phytoplankton in a lake due to TMII or DMII; in this case,

TS IIT,+ and T\ II ,_ will quantify the change in resource depletion due to the

antipredator behavior if we fixed the the density of foragers to that of the true

predator and no predator treatments, respectively.









5.2.4 Incommensurate Additive Metrics

Comparing additive metrics with different subscripts (e.g., DMIIa_ vs.

T TT, ) is problematic. This invalid comparison arises when one tries to quantify

DMII indirectly by subtracting (additive) TMII from the total (additive) indirect

effects (eq. 5-5). For example, Huang and Sih (1991) quantified metrics similar

to T1TT\I,_ and TII, and estimated DMII, which corresponds to DMII,+, by

subtracting trait effects from the total. To see the problem, suppose that predators

reduced both the average density and the average uptake rate of foragers by

a proportion p, in which case we would probably like to conclude that the

magnitudes of DMII and TMII are equal. Carrying through the equations above

with f = (1 p)F, n = (1 p)N shows that trait effects (TT II ,_) are in ';,

estimated to be 1/(1 -p) times larger than density estimates in this case (DMII,+).

(The problem still applies if F and N are reduced by the same absolute amounts

although it would be hard to interpret this scenario in any case since F and N have

different units.) Similarly, if one tries to use additive metrics without having run

a culling treatment, one can only estimate TA\ II,_ and DMII,+. Indirect methods

can work -for example dividing TII, by TMII, should give a consistent estimate

of DMII but only in the case where all the simplifying assumptions stated

above (no depletion, no intraspecific competition, etc.) hold.

5.3 Complications

5.3.1 Biological Complexities: Short-term

What if biological complexities such as depletion of resources or intraspecific

interference do occur? Restating eq. 5-1 more generally as

no pred. threat culling true
(5-6)
F1NI fiN2 F2n1 f22









highlights our implicit assumptions above. For example, by assuming that fl = f2,

we are assuming that antipredator behavior is independent of population density;

by assuming that F = F2, we are assuming that per capital foraging success in

the absence of predator cues is independent of forager density (Luttbeg et al.,

2003). Assuming N1 = N2 is safe unless significant numbers of foragers die due

to the costs of antipredator behavior (easily detected in an experiment); assuming

n1 = n2 may be reasonable since it is an explicit goal of the culling treatment.

In the standard experimental design without culling, we have three treatments

with which to test two contrasts, and no remaining information with which to test

our assumptions. The culling treatment provides a second pair of contrasts that

were initially supposed (eq. 5-3) to be equivalent. Continuing in the tradition

of the multiple-predator-effects literature (Billick and Case, 1994; Wootton,

1994), we may be able to use the log-ratio indices and interpret non-additivity or

interaction terms as evidence for additional ecological mechanisms. For example,

we can think of prey relaxing antipredator behavior under high conspecific density

as an interaction between density and trait effects, in both the ecological and

statistical sense: this phenomenon could be quantified (if F = F2) as log f2fl

log TMIIr_ log TMIIr+. Unfortunately, as Peacor (2003) -i -i-. -1. 1 conspecific

density may also change forager behavior even in the absence of predators, meaning

F1 / F2. While the available contrasts do not provide enough information to

disentangle all of the possible effects, at least the presence of an interaction tells

us that something interesting may be happening. Auxiliary measurements of

behavioral proxies for uptake, or measurements of resource uptake at a range of

different forager densities, are more detailed potential solutions to the problem of

additional interactions.

We have also assumed so far that the absolute rate of forager consumption

is independent of the amount of resource available -given enough time, foragers









will reduce the resource density linearly to zero, which may be reasonable in small

experimental arenas. If alternatively foragers deplete resource exponentially (so we

can redefine F and f as predation probability of one unit of resource per forager in

the absence and presence of predators respectively), then the change in the amount

of resource (e.g. in the no predator treatment is proportional to (1 (1 F)N). We

can define yet another set of indices in this case as (e.g.)

log(threat)
DMII,= (5-7)
log(true)

where (threat) and (true) are the proportional reduction of resources with respect

to the previous time step. We call these "log-log-ratio metrics", because the

decomposition log TIIr3 = log TMIIr3 + log DMII,3 involves taking the logarithm of

the response variables twice. The equivalence of the + and indices, and the clean

decomposition of TII into trait and density effects, still holds in this case.

Ecological systems are diverse, and we have certainly not covered all of the

possible scenarios. For example, strongly nonlinear dynamics (e.g. self-competition

among the resource) could, like most strongly nonlinear interactions, lead to

peculiar results for example, resource densities dropping as forager densities

or foraging efforts decreased (Abrams, 1992). If -1 i i- such dynamics should be

obvious from unusual signs or magnitudes of the indices (e.g. F/f < 1); if weak,

they could throw off interpretations of data. The only preventive measures we can

sir---- -I are common sense (avoid using resources with potential for such strong

self-suppression) and auxiliary observations (behavior proxies) or experiments

(ranges of forager densities).

5.3.2 Biological Complexities: Long-term

So far we have assumed that indirect-interaction experiments were run over

short time scales -to estimate "instantaneous" effects, and to avoid potential

complications of resource regrowth or variation in resource or forager densities.









However, indirect interactions clearly act over longer time scales as well. Luttbeg et

al. (2003) have pointed out that forager strategies may vary even over the course of

a fairly short-term experiment where densities are held constant, and of course the

densities of predators, foragers, and resource may all vary over longer time scales.

If we are to try to understand the longer-term dynamics of ecological communities,

whether empirically or theoretically, we will eventually need to think about how to

quantify indirect interactions that run over long enough time scales that population

density and behavior vary significantly.

If we run an experiment over T time steps and simply add together the

log-ratio indices from (eq. 5-5), we do preserve the decomposition of indirect

effects:
T T T
Slog(TII,)t log(TMII,)t + Ylog(DMII,)t (5-8)
t=1 t 1 t=1
However, using the ratio indices we cannot expect that computing TMII and

DMII from the total amount of resources consumed between the beginning and

end of the experiment will give us the same answer as computing TMII and DMII

period-by-period and adding them, because, e.g.

(no pred) Et(no pred)
t (threat) E,(threat)
period-by-period overall

(see e.g. Earn and Johnstone (1997) for other biological implications of the fact

that sums of ratios are not equal to the ratios of sums).

This difference can cause a large difference in the relative sizes of TMII and

DMII even over a fairly short experiment. In this case the product of TMII, and

DMII, computed from the endpoint data (the difference between beginning and

ending resource levels) will no longer satisfy the decomposition given in eq. 5-5,

and the + and ratio indices will no longer be equivalent (TMII,+ / TMII_,

DMIIr+ / DMII,_). Another consequence is that a culling treatment will be









necessary in order to compare trait- and density-mediated effects accurately. Griffin

and Thaler (2006) found large differences between TMII+, and TMII_ as well

as between DMII+r and DMII_r in a 3-d ,- experiment; while differences between

the TMII indices could be caused by intraspecific interactions as sl--:.- -1. 1 above,

differences in DMII are more constrained and may reflect the effects of variation in

density and behavior over time.

A few possible solutions to these difficulties are to:

Use additive metrics, including a culling treatment to avoid comparing unlike
subscripts. Since the total amount of resource consumed over the course
of the experiment equals the sum of the period-by-period consumption,
period-by-period and endpoint calculations of indices are consistent.

Use log-log-ratio metrics, i.e. assuming geometric depletion of resources.
If we define the response variables in each treatment as the proportion
of the resource being consumed in each time step, then the values of
TII TMII+DMII calculated at each time step sum to the value calculated at
the endpoint (i.e. based on the ratio of initial to final resource densities).

Collect p <' .: .-1/,;;-i' riod data, with frequency depending on the study system.
For example, if behavior of foragers changes in a systematic way on a fast
time scale (e.g., morning vs. afternoon), subsampling the data (e.g., collecting
data only once a d, i) would result in bias. Similarly, if the forager density is
depleted more than a few percent by predators, collecting data that does not
reflect this forager depletion will also give biased answers.

Model the system: while we have focused on very simple dynamics here
(few interactions, simple functional responses, etc.) it is clear that we have
neglected many possibilities. In the absence of detailed period-by-period data,
the only way to estimate the effects of time-varying densities and behaviors
is to build a simple model of resource dynamics, predation rate, and forager
responses and parameterize it from the system (van Veen et al., 2005). Here
even a little bit of period-by-period data, even if the sampling frequency is
too slow to capture the details of the dynamics, can be enormously useful for
validating the functional forms incorporated in the model.

5.4 Summary

While some of the metrics we have presented here appear to be generally


in most cases it appears that ratio-based indices will


better than others









more clearly and consistently decompose total indirect effects into trait- and

density-mediated components -it is also clear that significant complexities lurk

once we go beyond short-term, highly controlled experiments in small arenas.

However, these complexities are actually the signature of interesting ecological

dynamics, representing the next stage beyond the now-familiar questions of i,.

trait-mediated effects detectable?" and v.-!h I is the relative magnitude of trait-

vs density-mediated effects?" (Werner and Peacor, 2003; Preisser et al., 2005).

We s-l--:- -1 that, as in studies of multiple predator effects, ratio-based indices

should probably be the default, but that empiricists interested in quantifying

indirect effects should (1) consider metrics that are most appropriate for their

particular system and question (e.g. additive vs. log-ratio vs. log-log-ratio, linear

vs. geometric resource consumption); (2) report iv.-' measures (e.g. resource

densities or consumption rates) to allow readers to calculate different indices

from the data; (3) incorporate culling treatments in their experiments and use

the additional contrasts to test for and interpret interactions between trait and

density effects; and (4) consider running longer experiments, despite the potential

added complexities, to gain information on a larger and richer set of ecological

phenomena.















CHAPTER 6
ADAPTIVE BEHAVIOR IN SPATIAL ENVIRONMENTS

6.1 Introduction

Optimal foraging theory has proven useful in analyzing feeding behaviors in

a variety of contexts (Stephens and Krebs, 1986) as well as in understanding how

those behaviors affect community dynamics (reviewed in Bolker et al., 2003). In the

scenario where foragers adjust their foraging activity level based on their perception

of the environment, the simplest case assumes a single homogeneous foraging arena

with a known density of predator and prey (e.g., Abrams, 1992). In these models,

foragers are assumed to react to the average predation risk of the environment.

This behavior introduces trait interactions into the community (Abrams, 1995),

which influence the dynamics of the community in important manner (Werner

and Peacor, 2003). However, predation risk can vary spatially based on exogenous

factors (e.g., microhabitats) (Schmitz, 1998; Bakker et al., 2005) and endogenous

factors (Keeling et al., 2000; Liebhold et al., 2004). Thus, models assuming that

a population of foragers responding to an average (i.e., spatial average) risk of

predation may give inaccurate results if animals respond to spatially variable local

cues (e.g., encounter with a predator) (Jennions et al., 2003; Hemmi, 2005b; Dacier

et al., 2006).

Spatial properties of foragers and predators can influence the resulting

species interactions. For example, while visual foragers can detect predators

that are located within their perceptual range at any moment (Cronin, 2005),

chemosensory foragers (Cooper, 2003; Greenstone and Dickens, 2005) may detect

the presence of predators based on cues that may or may not be closely associated

with predator's actual location depending on how the predator's chemical cues









travel the environment and how long the chemical cues persist. Thus, ecological

communities with different spatial properties may exhibit different outcomes in

species interactions. For example, a iin I i- i1 1Ji -; by Preisser et al. (2005) showed

that trait-mediated effects are stronger in aquatic than in terrestrial systems.

Whether or not this difference can be attributed to the spatial characteristics (e.g.,

physical properties of the predator cues) discussed here is not clear, but most

community ecological studies that examined trait-effect of chemical foragers are

based on aquatic systems (Werner and Peacor, 2003), -i.-. i i-; the possibility

that the observed trend is influenced by the spatial properties.

In this paper, I examined how spatial consideration may affect the strength

of species interactions by constructing two types of foragers in a simple three

species linear food chain (resource forager -predator). The first type of foragers,

Global Information Foragers (GIFs), represent the commonly used modelling

framework (e.g., Krivan, 2000) where foragers detect the average predation risk of

the environment regardless of their current activity (e.g., even when foragers are

hiding) or the actual locations of the predators. This scenario may be appropriate

if predator cues (e.g., chemical) diffuse rapidly in the environment. For example,

aquatic chemical foragers can detect predator density based on the concentration of

diffusing chemical cues (i.e., actual presence of predators is not required to induce

antipredator behavior) (Holker and Stief, 2005). The second type of foragers, Local

Information Foragers, only detect local predator cues that are associated with

the actual predators. LIFs develop their perception of predator density based on

their experience of encounters with predators (C'i lpters 7 and 8). The difference

between GIFs and LIFs is not only the spatial range over which they estimate the

predator density but also how they obtain the information. While GIFs can detect

the predator density passively even when they stay in a refuge, LIFs must leave

their refuge and sample the environment to gain information about predators.









I examined two important determinants of the fate of ecological communities;

direct interactions (i.e., the performance of foragers) and indirect interactions

(e.g., interactions between predators and resources). To quantify direct and

indirect species interactions, I solved for the foraging effort for GIFs and LIFs that

maximizes their fitness under their respective biological and physical constraints

by using dynamic state variable models (Clark and Mangel, 2000). The solutions

were then simulated in a spatially explicit lattice environment. This procedure

allowed me to examine the performance (i.e. survival and reproduction) of

foragers with different sensory properties. I also examined how these two different

foraging strategies affect indirect species interactions. Specifically density- and

trait-mediated indirect interactions (DMII and TMII, respectively) of predators on

the resource population were examined. DMII is the indirect effect of predators

on the foragers' resource through reductions in forager density, while TMII is the

effect of predators on the resource through reductions in forager activity (i.e., due

to antipredator behavior) (Werner and Peacor, 2003).

6.2 The Model

A model similar to Luttbeg and Schmitz's (2000) dynamic optimization

model was developed for a K x K square lattice space with periodic boundary

conditions (i.e., edges of the environment are connected to the opposite edges). The

model is a three species linear food chain where predators consume foragers, while

foragers consume resources. Each cell is occupied by a predator or a forager or is

empty. Thus, predators and foragers have explicit spatial locations. Resources are

randomly distributed across space, and are instantaneously renewed -hence they

are only represented implicitly in the model. Predators and foragers reproduce at

the end of one 40-d, foraging season. The following fecundity rule from Luttbeg









and Schmitz (2000) was used for the foragers;


Number of offspring =


where x is the energy state of foragers. The exponent reflects the allometric

constraint of reproduction (Luttbeg and Schmitz, 2000).

The foraging effort C of foragers is described by the number of lattice

cells searched each di-. If a forager searches more cells, it is more likely to find

resources, but it also becomes more vulnerable to predators. There are six possible

levels of foraging effort C ranging from 0 to 80 (Figure 6-1).


UEE HEEHE
HEN MENEM EMEMEME
UEE HEEHE


Figure 6-1. Schematic representation of foraging efforts. The black center square is
the forager's location. The gray squares indicate cells in which the
forager will seek food i.e., C = 0, 4, 12, and 28 from left to right
respectively. Foraging effort of C = 48 and 80 can be similarly
characterized (not shown).


Given a probability w of finding a resource in a single cell, and that resources

are assumed to be independent between cells, the probability of finding a resource

for a given level of effort is A = 1 (1 r)C.

Foragers expend energy on metabolism at a rate of a per d4'; if a forager

finds a resource, it increases its energy state by Y. The maximum energy state

obtainable was set to 40. Foragers starve to death if their energy state falls below

1.









GIFs' perception of the probability of encountering a predator by searching

one cell is
P
Prob0 (predator) =K
K2

where P is the actual number of predators in the entire lattice space (e.g., average

risk). LIFs base their estimate of predator probability on past experience: based

on the number of predators encountered (p) while foraging in k cells over the

past m time steps, foragers predict the encounter probability based on a binomial

distribution

p ~ Binomial(k, ProbL (predator)).

where ProbL(predator) is the perception of forager about the encounter probability.

LIFs have a prior knowledge about this encounter probability, which is set as

Beta(a, /) where a and 3 constitute innate knowledge of the foragers (i.e., priors)

about the environment. I chose a weak prior that corresponds to an intermediate

predator density (a = 0.01, = 0.99). This prior is weak and is equivalent

to a single prior observation in a binomial process with 1 of probability of

encountering a predator for a given cell (e.g., ,25 predators in the environment).

These specifications lead to the posterior distribution for ProbL(predator), Beta(a+

p, f + k p), which is used by LIFs to determine their optimal strategies.

The perception of the probability of surviving a given foraging effort, C, for

GIFs is approximated by


ProbG(survive)= (1 d Probe(predator))c


where d is the probability of being killed given an encounter with a predator. LIFs'

perception of this probability is approximated by


ProbL(survive) (1 dU)c, U ~ Beta(a + p, / + k p).









The fitness functions F(x,t) (for GIFs) and F(x,p, k,t) (for LIFs) are defined

as the maximum expected reproductive success between div t and the end of the

forager's life given that its current energy state is x and that it has encountered p

predators while searching k cells in past m time steps.

The dynamic optimization rules can be described by


F(x, t) =Probc(survive) {AF(x + Y a, t + 1) + (1 A)F(x a, t + 1)}

F(x,p, k, t)= ProbL(survive)Beta(u; a, /3,, k)

x {AF(x + Y -a, t + 1) + (1 )F(x a, t + )}du


Then we can solve for the optimal foraging effort C by using the backward iteration

procedure (Clark and Mangel, 2000). Table 1 shows the parameters used for the

backward solutions.

Table 6-1. Parameter values used for the simulations. For the description of
parameters, see the text.

Parameter Notation Value
Lattice K 51
P(food|cell) Tr 0.05
Memory n 3
Predation d 0.5
Resource value Y 3
Dispersal D 1,2,3,4,5
Metabolism a 1



6.2.1 Lattice Simulations

6.2.1.1 Direct effects: performance of foragers

After the behavioral solutions for GIFs and LIFs were found, spatially explicit

simulations were conducted with 100 foragers with an initial energy state of 5 units.

Predators and foragers were randomly distributed over the lattice space at the

beginning of the simulation.









The number of predators was varied from 5 to 50 in increments of 5. Predators

are considered encountered if they are found in the cells that were searched

(Figure 6-1) in accordance with the foraging solution. Foragers were set to their

initial location throughout the season (i.e., they foraged around a random fixed

location) whereas predators relocated daily. Predators dispersed randomly to an

empty cell within a radius of D. For each possible parameter set (Table 6-1), 30

simulations were conducted. At the end of each simulation, the number of surviving

foragers, fecundity of the survivors, and the depletion of resource by the forager

population were recorded. The survival and fecundity represent the direct effect

of predators on foragers, while resource uptake is used to quantify indirect effects

(discussed below).

6.2.1.2 Indirect effects

In order to compare the strength of various indirect effects, I used three

treatments in the simulation (C'!i lpter 5). All the treatments contained the basal

resource and foragers, but differed in the type of predators they included. The

true-predator treatment incorporated unmanipulated predators that could both kill

foragers and induced changes in traits of foragers. The threat-predator treatment

contained predators that induced forager antipredator behavior but did not kill

foragers. In the culling treatment, no predator was introduced, but foragers were

artificially removed from the environment at the rate at which they were removed

by predation in the true predator treatment.

Based on these treatments, TMII and DMII were quantified as follows

(C'!i Ipter 5),


DMII =Resource eaten(threat) Resource eaten(true)

TMII Resource eaten(culling) Resource eaten(true)


where "Resource eaten" indicates the cumulative amount of resource consumed









by the forager population at the end of the season. Thus, these measurements are

approximations to the actual TMII and DMII that occur in the system throughout

the season (C'!i plter 5).

6.3 Results

6.3.1 Direct Effects: Performance of GIFs and LIFs

When predator density was high (e.g., 50 predators), GIFs survived better

than LIFs (Figure 6-2). On the other hand, the average fecundity of surviving

LIFs was .li-- i,- higher than that of GIFs. Fitness (i.e., the product of survival

and fecundity) of LIFs was uniformly higher when predators' movement range was

small, but as the dispersal range D of predators increased, their advantage over

GIFs diminished (Figure 6-2).

6.3.2 Indirect Effects

At any parameter and variable combinations, TMII for GIFs were .i.- ,'

larger than TMII of LIFs (Figure 6-3). DMII was uniformly larger than TMII

in LIFs. In GIFs, the relative strength of TMII and DMII changed depending on

predator dispersal and resource level (Figure 6-3).

The strength of DMII decreased with increasing resource level while it

increased with increasing predator dispersal D and density P. The strength of

TMII for GIFs was greater when the predator density was high than when the

predator density was low, but was relatively unaffected by predators' dispersal

range.

6.4 Discussion

The current study indicates that differences in the way predator cues

propagate leads to considerable differences in forager performance and in indirect

effects on resource uptake. Because the model included only random predator

movement (excluding behaviors such as .-I- regation and area-concentrated

foraging (Kareiva and Odell, 1987; Schellhorn and Andow, 2005)), the spatial









0.02 0.06 0.10
I I I I| I I I I I


dispersal=l dispersal=2 dispersal=31 dispersal=4 dispersal=5
-LLLLLLLLL LLLLLLL LLLLLL L LLLLL LLLL
L L L L
L L L L
GGGGGGGGL GGGGGGGGG L GGGGGGGG GGGGGGG GGGGG
G GGGG GG LGGGG GGG
G 0
I I I I I 1 I I I I I I I 1 1I I I
0.02 0.06 0.10 0.02 0.06 0.10 0.02 0.06 0.1
I I I I I I II I I I I I I I I I


0.4-
0.3-
0.2-
0.1-
0.0-


1.5-
1.0-
0.5-
0.0-


0


I I I I I I I I I I I I I I I I I I I I I I I
0.02 0.06 0.10 0.02 0.06 0.10 0.02 0.06 0.10


I I I I I I I I I 1 I I I I I I
0.02 0.06 0.10 0.02 0.06 0.10 0.02 0.06 0.10
Resource level

Figure 6-2. Proportion of prey surviving, average number of offspring, and fitness
of GIFs (G) and LIFs (L). Number of predators = 50.


effects seen were due to the sampling error (i.e., random variability) of predation

risk in the spatial environment. For example, if the environment contains a

single predator, then a location near the predator and another location far

from the predator have very different actual predation risk. This difference

diminishes as predator density increases because every location becomes closer

to a predator. Thus, sampling error is largest when the density of predators is low.

When predator density is low, even when predator dispersal is high, LIFs have

higher fitness than GIFs (results not shown). Dispersal of predators also acts to

homogenize the predation risk in the environment. If dispersal is unlimited, the

model loses its spatial characteristics. Limited dispersal of predators enhanced the


dispersal=l dispersal=2 dispersal=3 dispersal=4 dispersal=5


dispersal=1 dispersal=2 dispersal=3 dispersal=4 dispersal=5
LL
L
LL L
L GG LL
LGGG L GGG G

^ ____ 60; cc Ge'go L


0.02 0.06 0.10
I I I I I I I I I


I I I I I


f










Predator dispersal=1
0.02 0.06 0.10
LIF LIF
P=5 P=50

DDDD DD
DDDDD


Predator dispersal=5
0.02 0.06 0.10
LIF LIF
P=5 P=50
DDDDDDDD

DDDDD D
D DDDn


DDDDDDDDYD 0 U DDDD
.N -TTTTTTTTTTT'TTTTTTT .N TTTTTTTTT TTTTTTTTTTT
GIF GIF GIF GIF
SP=5 P=50 0 P=5 P=50
$1200 T
LU 1000- T LU
800 TT TT D TTT DDDDDDDD
600- T T DD DDDDD
400 T 500 DDDDDDTMD TTTTTTTT
200 6DDDDDDDDDT T TT
0- 0
0.02 0.06 0.10 0.02 0.06 0.10
Resource level Resource level

Figure 6-3. Effect size for TMII (T) and DMII (D) with variable number of
predators (P).


sampling errors of predation risk in the environment and gave an advantage to LIFs

(Figure 6-2).

In a spatially structured environment (e.g., with limited predator dispersal),

GIFs survived better but sacrificed fecundity compared to LIFs (Figure 6-2).

Because survival and fecundity represent direct density and trait effects of

predators on foragers respectively, we can interpret that the different mechanisms

(i.e., GIF vs. LIF) result in the tradeoff between direct trait and density effects.

This result may be consistent with Preisser et al's (2005) meta-analysis, which

found that trait-mediated effects are stronger in aquatic system than in terrestrial

systems. In aquatic systems, predator cues may diffuse in the environment more

readily and/or persist longer and thus foragers cannot respond to the actual

location of predators; they must act like GIFs. Consequently, aquatic chemical

foragers may exhibit high levels of antipredator behavior even when actual

predation risk is low.


F









The effect of the spatial structure on indirect interactions was large (Figure 6-3).

Like direct trait effects, trait-mediated indirect interactions were generally stronger

in GIFs than in LIFs. In particular, TMII of LIFs is almost negligible throughout

the parameter space, indicating antipredator behavior alone does not produce

much effect. This is because predation probability used in the simulation was

relatively high (d = 0.5). Under the highly efficient predators, LIFs become more

opportunistic and the value of antipredator behavior becomes small. Antipredator

behavior induced through experience, as in LIFs, has value only when foragers

have sufficiently good chance of surviving the encounter (Sih, 1992). When the

probability of surviving an encounter is small, there is little chance of learning

from the experience. If the predation risk is lowered (d = 0.25), the effect size of

TMII increases, but the general characteristic discussed here is not affected by this

change.

In LIFs, DMII was alh-w stronger than TMII. On the other hand, in LIF,

the relative magnitude of TMII and DMII were sensitive to the predator dispersal,

the resource availability, and number of predators. GIFs change their behavior

based on the number of predators in the environment, not where predators are

located, thus predator dispersal does not affect trait expression. On the other hand,

predators with a high dispersal ability can more effectively deplete foragers in the

environment. Therefore, when predator dispersal is high, parameter region where

DMII is greater than TMII becomes wide. Previously, the relative size of indirect

effects were discussed potentially as an important index that helps determine

community stability (Werner and Peacor, 2003). If true, this result indicates that

we should consider spatial structure as well as the perception of foragers in these

models as they can qualitatively alter such a relationship.

The results from direct and indirect effects also have implications for

experimental designs that are commonly used to quantify trait effects (Werner









and Peacor, 2003). In experiments where foragers detect predators based on the

cues that may spread far from predators (e.g., GIFs), artificial arenas may cause

prey to : '-r'---ite their trait expression For example, in aquatic system with

chemosensory foragers, antipredator behaviors are often studied by introducing

water that held predator species because it contains chemical cues used for

identifying the existence of predators by foragers (Holker and Stief, 2005) or

introducing caged predators (Anholt and Werner, 1998). However, no study has

examined how the chemical cue diffuses in water or how rapidly it decays. Thus

although there is evidence that water that contained more predators is more

effective in inducing antipredator behavior (Holker and Stief, 2005), emerging

spatial interactions will be strongly affected by such unknown physical details. For

example, chemical foragers in aquatic and terrestrial environment would mediate

very different trait effects because of the differences between the physical properties

of water and air. If the cue is quickly homogenized in the environment, the system

becomes similar to GIFs examined in this paper. Relatively small arenas used in

experiments may potentially create a bias because it prohibits foragers from moving

to areas where the chemical cue is absent (e.g., eventually predator cues may fill up

the arena).

To date, most community models with adaptive foraging behaviors have not

incorporated spatial structure (Abrams, 1993; Fryxell and Lundberg, 1998; Krivan,

2000; Abrams, 2001). Thus, we do not understand how these adaptive behaviors

result in community dynamics in a spatially explicit environment or the possible

role of the physical environment. Furthermore, because conventional non-spatial

models give results similar to GIFs, it is possible current general understanding

about the effect of trait change on community dynamics (Bolker et al., 2003)

may apply only to specific scenarios. Behaviorists have long known that physical

environment affects behavior through sensory mechanisms (Endler, 1992), and






86


these specificities indeed seem to act distinctively in real c -i--, I (Preisser

et al., 2005). Although more studies are needed, investigation of adaptive behavior

through sensory constraints may be a fruitful way to further advance the interface

of adaptive behavior and community dynamics.















CHAPTER 7
PROLONGED EFFECTS OF PREDATOR ENCOUNTERS ON THE JUMPING
SPIDER, PHIDIPPUS AUDAX (ARANAE: SALTICIDAE)

7.1 Introduction

Prey species respond to the effect of predation risk on fitness by expressing

a variety of antipredator traits (e.g., Lima and Dill, 1990; Eisner et al., 2000;

Mappes et al., 2005; Caro, 2005). One common antipredator trait is vigilance

behavior, where animals increase their ability to detect predators at the cost of

reduced resource intake (e.g., Bertram, 1980; Bekoff, 1995; Bednekoff and Lima,

2002; Randler, 2005). Community ecologists have been increasingly interested

in this type of behavior because it is known to affect the dynamics of ecological

communities (Werner and Peacor, 2003; Bolker et al., 2003).

Most community models with adaptive behavior include a variable describing

the level of foraging effort. The nature of this variable varies among studies. Some

studies are vague about foraging effort (! I iiuda and Abrams, 1994; Abrams, 1992;

Luttbeg and Schmitz, 2000); some identify it with a measure of foraging intensity

such as search speed (Leonardsson and Johansson, 1997); and others define it as

frequency, the fraction of total time available that foragers spend foraging (Abrams,

1990). Empirical studies indicate that foragers can respond to environmental cues

by changing both intensity and frequency of foraging (Johansson and Leonardsson,

1998; Anholt et al., 2000). The community implications of predator-induced

changes in foraging effort depend on whether foragers change their intensity or

frequency of foraging. Indeed, one of the central foci of community models are

the indirect effects that arise on the prey's resources in response to changes in the

forager's behavior induced by the predator.




Full Text

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Anumberofpeoplehelpedmetocompletethisproject,whichmakesthisdissertationsomuchmoremeaningfultome.Idon'tlistthemhere,butIhopemyappreciationiswellunderstood.StillImustacknowledgemycommittee(BenBolker,JaneBrockmann,CraigOsenberg,JimHobert,BobHolt,andStevePhelps)fortheirvaluablecriticismandencouragementthroughout,whichIvalueverymuch.IamextremelygratefulabouttheTeachingAssistantshipandResearchAssistantshipopportunitiesaswellastheCLASfellowshipfortheirsupportandexperiences.ComplexSystemsSummerSchooloftheSantaFeInstitutealsoprovidedanextremelyenjoyableenvironmentinwhichIwasabletoinitiateapartofthisdissertation.Lastly,interactionsIhavehadwithBenBolkerhavebeenmymostvaluableexperiencehereatUF.Benimprovednotonlymyprojectbutalsomywayofapproachingecologicalproblems.IfIaccomplishedanythingworthwhileinthefuture,itisbecauseIwasfortunateenoughtoworkwithhim. iii

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page ACKNOWLEDGMENTS ............................. iii LISTOFTABLES ................................. viii LISTOFFIGURES ................................ ix ABSTRACT .................................... xii CHAPTER 1GENERALINTRODUCTION ....................... 1 1.1IntraguildPredation .......................... 1 1.2JumpingSpiders ............................ 3 1.3SpatialStructure ............................ 4 1.4AnimalBehavior ............................ 5 1.5AdaptiveBehaviorUnderSpatiallyStructuredEnvironments .... 7 1.6DynamicsofJumpingSpiderActivity ................. 8 1.7Synthesis ................................. 9 2INTRAGUILDPREDATIONWITHSPATIALLYSTRUCTUREDSPECIESINTERACTIONS ............................... 11 2.1Introduction ............................... 11 2.2MaterialsandMethods ......................... 13 2.2.1LatticeModelofIntraguildPredation ............. 13 2.2.2MeanFieldApproximation ................... 15 2.2.3PairApproximation ....................... 15 2.2.4InvasibilityAnalysis ....................... 17 2.2.5IndividualBasedModel ..................... 17 2.2.6HeterogeneousEnvironment .................. 18 2.3Results .................................. 19 2.3.1MeanFieldApproximation ................... 19 2.3.2PairApproximation ....................... 20 2.3.3UnequalNeighborhoodSizes .................. 21 2.3.4QuantitativeComparisonBetweenSpatialandNon-spatialModels .............................. 22 2.3.5HeterogeneousEnvironments .................. 23 2.4Discussion ................................ 25 iv

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.............................. 25 2.4.2QuantitativeEectofSpatialStructure ............ 27 2.4.3EectofSpatialHeterogeneity ................. 28 3NITROGENLIMITATIONINCANNIBALISTICJUMPINGSPIDERS 31 3.1Introduction ............................... 31 3.2MaterialsandMethods ......................... 32 3.2.1ExperimentalTreatments .................... 32 3.2.2EectonGrowth ........................ 33 3.3Results .................................. 33 3.4Discussion ................................ 35 4EVOLUTIONARILYSTABLESTRATEGYOFPREYACTIVITYINASIMPLEPREDATOR-PREYMDOEL ................. 38 4.1Introduction ............................... 38 4.2TheModel ................................ 40 4.3Results .................................. 43 4.3.1EvolutionarilyStableStrategy(ESS)ofForagingEort ... 43 4.3.2IncorporatingESSintotheCommunityDynamics ...... 46 4.3.3ComparisonwiththeQuantitativeGeneticsModel ...... 49 4.3.3.1Behaviorofthesystemwithfastevolution ..... 50 4.3.3.2Behaviorofthesystemwithslowevolution ..... 54 4.4Discussion ................................ 55 4.5AppendixA:DerivationoftheESS .................. 60 5ONTHEQUANTITATIVEMEASURESOFINDIRECTINTERACTIONS 63 5.1Introduction ............................... 63 5.2QuantifyingIndirectEects ...................... 64 5.2.1StandardExperimentalDesign ................. 64 5.2.2IndicesofIndirectEects .................... 65 5.2.3DecomposingTotalEects ................... 66 5.2.4IncommensurateAdditiveMetrics ............... 68 5.3Complications .............................. 68 5.3.1BiologicalComplexities:Short-term .............. 68 5.3.2BiologicalComplexities:Long-term .............. 70 5.4Summary ................................ 72 6ADAPTIVEBEHAVIORINSPATIALENVIRONMENTS ........ 74 6.1Introduction ............................... 74 6.2TheModel ................................ 76 6.2.1LatticeSimulations ....................... 79 6.2.1.1Directeects:performanceofforagers ....... 79 v

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.................... 80 6.3Results .................................. 81 6.3.1DirectEects:PerformanceofGIFsandLIFs ........ 81 6.3.2IndirectEects ......................... 81 6.4Discussion ................................ 81 7PROLONGEDEFFECTSOFPREDATORENCOUNTERSONTHEJUMPINGSPIDER,PHIDIPPUSAUDAX(ARANAE:SALTICIDAE) 87 7.1Introduction ............................... 87 7.2MaterialsandMethods ......................... 91 7.2.1StudySystem .......................... 92 7.2.1.1Predatortreatment .................. 92 7.2.1.2Ambiguousvisualstimuli .............. 93 7.2.2BehavioralMeasures ...................... 93 7.2.3StatisticalAnalysis ....................... 94 7.3Results .................................. 94 7.4Discussion ................................ 97 8SUSTAINEDEFFECTSOFVISUALSTIMULIONRESTINGMETABOLICRATESOFJUMPINGSPIDERS ...................... 101 8.1Introduction ............................... 101 8.2MaterialsandMethods ......................... 102 8.2.1ExperimentalTreatments .................... 103 8.2.2OxygenMeasurement ...................... 104 8.2.3StatisticalAnalyses ....................... 105 8.3Results .................................. 106 8.4Discussion ................................ 107 9ACTIVITYMODESOFJUMPINGSPIDERS .............. 110 9.1Introduction ............................... 110 9.2PartI:QuanticationofSpiderStatesintheField .......... 111 9.2.1MaterialsandMethods ..................... 112 9.2.2Results .............................. 113 9.2.3Discussion ............................ 114 9.2.4ASimpleModel ......................... 114 9.3PartII:ExaminingtheSimpleModel ................. 115 9.3.1MaterialsandMethods ..................... 115 9.3.1.1Thetreatment .................... 116 9.3.1.2Behaviormeasure ................... 116 9.3.1.3Statisticalanalysis .................. 117 9.3.2Results .............................. 118 9.4PartIII:IndividualBasedModel ................... 120 9.4.1TheModel ............................ 120 9.4.2ParameterEstimation ...................... 121 vi

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.............................. 123 9.5Conclusion ................................ 124 10INTRAGUILDPREDATIONINAJUMPINGSPIDERCOMMUNITY:ASYNTHESIS ................................ 127 10.1Introduction ............................... 127 10.2TheModel ................................ 128 10.2.1ActivityofSpiders ....................... 129 10.2.2ForagingActivity ........................ 130 10.2.3ExploitationCompetition .................... 130 10.2.4IntraguildPredation ...................... 131 10.2.5Reproduction .......................... 131 10.3Results .................................. 132 10.4Discussion ................................ 134 10.4.1SpatialStructure ........................ 134 10.4.2BiphasicActivity ........................ 135 10.4.3AdaptiveBehavior ....................... 135 10.4.4ModellingBehaviorinCommunityEcology .......... 136 REFERENCES ................................... 140 BIOGRAPHICALSKETCH ............................ 153 vii

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Table page 3{1Nutrientandenergycontentsofprey(standarderrorsinparentheses). .. 33 4{1Equilibriumanalysis.ThecspeciedisthechoiceofESSinRegionIIIrequiredforanonzeroequilibriumtoexist. ................. 49 5{1ExistingstudiesthathaveexplicitlycomparedTMIIandDMII. ..... 67 6{1Parametervaluesusedforthesimulations.Forthedescriptionofparameters,seethetext. .................................. 79 7{1Estimatedparametersfromtherandomeectmodel.SE(standarderror)andRE(randomeect). ........................... 95 7{2Statisticalresultsforwithandwithoutambiguousstimuli.REindicatestherandomeectdescribingthestandarddeviations.ParameterswithmissingREvaluesarexedeects. ..................... 96 8{1AICforeachmodel.p=numberofparameters. .............. 107 9{1Estimatedparametersofthereducedmodel.Themodelisanhierarchicallogisticregressionmodelwiththeintercepta1+a2jandslopeb1+b2j,wherejisthedayoftheexperiment. .................... 119 viii

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Figure page 1{1Buildingupanintraguildpredationsystem.Startingfromaconsumer-resource(N-R)interactionwithanindependentpredatorpopulation(P),additionofpredation(leftcolumntorightcolumn)andcompetition(toprowtobottomrow)arenecessarytomakeIGP.Directionoftheallowsindicatethedirectionofenergyow. ......................... 2 1{2SpatialdistributionofpredatorsPandpreyN.Intherightgure,eachspeciesiswellmixed.Intheleftgure,thetwospeciesarespatiallysegregated. 4 1{3Hypotheticalactivitydynamics.Thetopgureshowsthedynamicsofpredatordensity.Thebottomgureshowsthecorrespondingforagingactivityofadaptive(solid)andnon-adaptive(dashed)foragers. ..... 5 1{4Trait-mediatedindirectinteraction(TMII)anddensity-mediatedindirectinteraction(DMII). .............................. 7 1{5SpatialdistributionofpredatorsPandpreyN.TwopreyindividualsarelabeledasAandB. .............................. 8 2{1Examplesofrandombinarylandscapesbasedondierentpatchscales.Patchscalereferstothenumberoftimestheprocedurediffusewasapplied(seetext). ............................... 19 2{2ParameterregionsindicatingtheoutcomeofIGPinanon-spatialmodel. 20 2{3Resultsofinvasionanalysisinthepairapproximationmodel. ....... 21 2{4Resultsofinvasionanalysisinthepairapproximationmodel.SpatialscaleofIGpredatorswasxedatzP=4whilethatofIGpreyvaried. .. 22 2{5Parameterintervalsresultinginexpansionandreductionofthecoexistenceinterval.ThelineindicatesthecontouratIspatial=Inon-spatial=1.Whenthisratioisgreaterthan1,spatialstructureincreasedthesizecoexistenceintervals. .................................... 23 2{6Persistenceprobability(squares)anddensityofIGprey(circles)andIGpredators(triangles). ................................... 24 2{7Persistenceprobability(squares)anddensityofIGprey(circles)andIGpredators(triangles).Densitiesofconsumersarepresentedasfractionoftotalcellsoccupiedbythespecies. ........................... 25 ix

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........................... 26 3{1Growthincarapacewidthofjumpingspidersineachinstar. ....... 34 3{2DurationofeachinstarofP.audax. ..................... 35 4{1SolutionsfortheESSforcineachofthreeregionsofthenondimensionalizedNP-plane(s=ahNandr=a bP).InbothRegionIandRegionIIthereispreciselyoneESSfunction.InRegionIIItherearethreepossibilitiesforanESS.Fortheexpressionsfory1andy2intermsofrands,seeAppendixA. ................................. 44 4{2AcomplicatedESSfunction,whereRegionIIIissplitintomanysubregions,witheachsubregionassociatedwithoneofthethreepossiblebasicESSs. 45 4{3ThethreebasicESSfunctionsdeterminedbywhichofthethreestrategiesischosenuniformlyinRegionIII.Top(c=1),bottomleft(c=y1),bottomright(c=y2).PlotsoftheESSfunctionsareshownonthenondimensionalizedNP-plane(s=ahNandr=a bP). .......... 46 4{4ApparentfunctionalresponsesofpredatorswhenP=10,a=1;h=1;mN=0:1;mP=0:1.Inthisparameterregion,therearethreeESSs(Figure 4{1 ).ThesefunctionalresponseswereplottedassumingthatthethreeESSvaluesaredistinctstrategies.Left:c=y1.Middle:c=y2.Right:c=1. ................................. 47 4{5Evolutionarydynamicsofforagingeort(dc=dtversusc)inRegionI(left),RegionIII(middle),andRegionII(right),undertheassumptionoffastrateofevolutiong. .............................. 51 4{6Simulationofthedynamicsofpredatorsandpreyplottedontheeortdiagram(Figure 4{1 ).s=ahNandr=(a=b)P.Thegraylineindicatesr=p 54 4{7ForagingeortasafunctionofNandPforanEvolutionarystablestrategy(left)andforastrategythatmaximizespopulationtness(right). .... 58 6{1Schematicrepresentationofforagingeorts. ................ 77 6{2Proportionofpreysurviving,averagenumberofospring,andtnessofGIFs(G)andLIFs(L). ............................ 82 6{3EectsizeforTMII(T)andDMII(D)withvariablenumberofpredators(P). ...................................... 83 x

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............. 106 9{1Boxplotsforthefooddeprivationdegreesofthespidersintheeld.Spiderswereclassiedbasedonsexandthelocationwheretheywerefound:outsideretreat(active)orinsideretreat(inactive). ................. 114 9{2Proportionofindividualsthatattackeday. ................ 118 9{3Treatmenteectparameterestimates.Solidlineandtwodashedlinesindicatethemeanand95%credibleregionsofthereducedmodel.Squaresindicatethemeansforthefullmodel. ................... 120 9{4Relationshipbetweenmassandfooddeprivationdegree.Theestimatedfunctioniswt=0:145(t+1)0:063. ..................... 122 9{5Simulationofanindividual.Circlesandsquaresarerealizationsofthesimulationcorrespondingwithactiveandinactivephase,respectively.SolidlinetracesthedeterministicpredictionofthestochasticIBM.HorizontallinesareLA!I(top)andLI!A(bottom). .................. 123 9{6ResultsoftheIBMbasedon1000individuals.125individualsareactive. ......................................... 124 10{1EectofproductivitylevelonthepersistenceofIGPcommunityundervariousdegreesofspatialstructure.ThesmallertheneighborhoodsizeU,thestrongerthespatialstructure.Localreproductionandnon-adaptivebehavior(i.e.,==0)areassumed. ................... 132 10{2Averagepersistencewithandwithoutadaptivebehavior.Adaptivebehaviorindicatesthatbothandarepositive(seetext).U=7.Localreproductionisassumed. .................................. 133 10{3Averagepersistencewithandwithoutballooning.U=7.Adaptivebehaviorsareincluded. ............................ 133 xi

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xii

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xiii

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Wise 2006 ).Forexample,mostspidersaregeneralistpredatorsthatfeedonavarietyofpreyitemssuchasmosquitoesandies,makingthemmembersofthesameguild.However,spidersalsoeatotherspiders;wecountthiscannibalismasintraguildpredation.IGPcommonlyinvolveslargerindividualsfeedingonsmallerindividuals( Polis 1988 ).Wecallthevictimintraguildprey(IGprey)andthepredatorintraguildpredator(IGpredator).IGPiscommoninnatureandisfoundinavarietyoftaxa( Polis 1981 ; Polisetal. 1989 ; PolisandHolt 1992 ; WilliamsandMartinez 2000 ; ArimandMarquet 2004 ).OnecharacteristicofIGPisthesimultaneousexistenceofcompetitiveandtrophicinteractionsbetweenthesamespecies(Figure 1{1 ).TheoreticalmodelspredictthatcoexistenceofIGpredatorsandIGpreyisdicult( HoltandPolis 1997 ),becauseIGpreyexperiencethecombinednegativeeectsofcompetitionandpredation.Insystemswithcompetitiononly,IGprey 1

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Figure1{1. Buildingupanintraguildpredationsystem.Startingfromaconsumer-resource(N-R)interactionwithanindependentpredatorpopulation(P),additionofpredation(leftcolumntorightcolumn)andcompetition(toprowtobottomrow)arenecessarytomakeIGP.Directionoftheallowsindicatethedirectionofenergyow. suernopredation;instandardpredator-preyinteractionswithoutcompetition,IGpreysuernoexploitativecompetitionfromtheIGpredator(Figure 1{1 ).Thus,IGPismorestressfulfortheintermediateconsumer(IGprey)thaneitherexploitativecompetitionortrophicinteractionalone(Figure 1{1 ).ThetheoreticaldicultyinexplainingIGPpersistenceanditsobservedubiquityhaveidentiedIGPasanecologicalpuzzle( HoltandPolis 1997 )andledtoaseriesofstudiesthathaveattemptedtoresolvethispuzzle.Thesestudieshaveconsideredfactorssuchastoppredators(foodwebtopology)( Yurewicz 2004 ),sizestructure( Myliusetal. 2001 ; Borer 2002 ; MacNeilandPlavoet 2005 ),habitatsegregation( MacNeilandPlatvoet 2005 ),metacommunitydynamics( MelianandBascompte 2002 ),intraspecicpredation( Dicketal. 1993 ),andadaptivebehavior( Krivan 2000 ; KrivanandDiehl 2005 ).However,itisnotclearwhetherthemostimportantinteractionshaveyetbeenidentied,andconsensushasyetto

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emergefromthesestudiesabouttherelativefrequencyandstrengthofdierentpossiblemechanismsinmaintainingIGP.UnderstandingthedynamicsofIGPhaspracticalimportancebecauseIGPoccursinagriculturalsystemsandalsoaectsspeciesofconservationconcern( MullerandBrodeur 2002 ; HarmonandAndow 2004 ; KossandSnyder 2005 ; HarmonandAndow 2005 ).DespitetheneedforsolidunderstandingaboutIGPinordertomanagethesesystemssuccessfully,theunresolvedpuzzlesuggeststhatwestilldonotunderstandhowIGPcommunitiespersistinnature.Thus,inthisdissertation,IexaminehowanIGPsystemcanpersistbyexaminingboththeoreticalandempiricalissues,usingjumpingspidersasmodelorganisms,inanattempttoresolvethediscrepancy.Ifocusedontwoclassesofecologicalphenomena,animalbehaviorandspatialstructure,aspossibleexplanations;eachisfurtherdiscussedbelow. CoddingtonandLevi 1991 ).Theyaregeneralistpredatorsthatpreyprimarilyonarthropodspecies,includingotherspiders( JacksonandPollard 1996 ).ThefrequencyofIGPisknowntobehighinsomespecies.Forexample,approximately20%ofPhidippusaudax'sdietconsistsofotherspiderspeciesthatalsoconsumesimilarresources( Okuyama 1999 ).Intraguildpredationamongjumpingspidersisalwayssize-dependent,withlarge-bodiedindividualsconsumingsmallerindividuals( Okuyama 1999 ).Nevertheless,smallerspeciesofjumpingspidersappeartocoexistinlocalcommunitieswithlargespeciesofjumpingspidersformanyyears,posingthepuzzleofpersistencediscussedabove.Thefocusofthisstudyonbehaviormakesjumpingspidersparticularlygoodstudysubjects.Thesespidersarevisualforagersandtheircomplexforagingtacticsareoftencomparedtothoseofvertebratespecies( Land 1972 ; Hill 1979 ; Jackson

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andPollard 1996 ).Infact,themajorityofstudiesonjumpingspidersconcerntheirbehavior,ratherthantheirecologicaldynamics.Complexbehaviordoesnotoccurinmicrocosmsofmicroorganisms(orisdiculttoexamineattheindividuallevelwhenitexists),whoserapiddynamicsotherwisemakethemwellsuitedforcommunitylevelstudies. KareivaandTilman 2000 ),themajorityoftheoreticalmodelsofIGP,includingthosethatposethediscrepancywithobservation(e.g., HoltandPolis 1997 ),arenon-spatial.Non-spatialmodelsassumethatindividualsaremixedhomogeneouslyinspace,ignoringvariousformsofspatialstructurethatareubiquitousinnature(Figure 1{2 ).Forexample,exceptinlandscapesthathavebeenarticiallyhomogenizedbyhumans(lawns,cropelds),weobservedistinctspatialvegetationpatternsinallterrestrialcommunities. Figure1{2. SpatialdistributionofpredatorsPandpreyN.Intherightgure,eachspeciesiswellmixed.Intheleftgure,thetwospeciesarespatiallysegregated. Regardlessofwhetherspatialstructureisgeneratedexogenouslyorendogenously(e.g., Bolker 2003 ),itcanhavealargeimpactonspeciesinteractions.Ifspeciesarewellmixed(Figure 1{2 ,left),thecommunity'sdynamicscanbewellapproximatedbytraditionalnon-spatialmodels.However,ifthereisaspatialpatterninanimaldistributionandifindividualsinteractonlywithindividualsinalocal

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neighborhood,spatialmodelsdescribingthetruedynamicsaremoreaccurate( Bolkeretal. 2000 ; Iwasa 2000 ; SatoandIwasa 2000 ).InChapter2,IexaminetheroleofspatialstructureinasimpleIGPcommunity.TheanalysispresentedinChapter2showsthatthenutrientcontentofIGpreyisanimportantmodelparameter;Chapter3exploresthisparameterexperimentally. FryxellandLundberg 1998 ).WhiletraditionalmodelssuchastheLotka-Volterramodelanditsnumerousvariantsassumethatbehavior(e.g.,foragingactivity)ofindividualsisconstantandindependentofenvironmentalfactors,thereisconsiderableevidencefordynamicvariationinbehavior.Inparticular,thereisalargebodyofevidencethatanimalsaltertheirforagingactivitywithrespecttopredationrisk( Caro 2005 )(Figure 1{3 ). Figure1{3. Hypotheticalactivitydynamics.Thetopgureshowsthedynamicsofpredatordensity.Thebottomgureshowsthecorrespondingforagingactivityofadaptive(solid)andnon-adaptive(dashed)foragers.

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Communitymodelsthatincludebehaviortypicallyassumethatanimalsforageoptimally,balancingtherisksofstarvationandpredation( StephensandKrebs 1986 ).Asaconsequence,thesemodelspredictthatforagersdecreasetheirforagingeort(e.g.,searchrate)whenpredationriskishigh(Figure 1{3 ).Thistypeofbehavioralanalysishasbeendoneforavarietyoffoodwebmodules( Bolkeretal. 2003 ).However,theinclusionofanimalbehaviorincommunitydynamicshasoverlookedmanyimportantaspectsofbehavior.Forexample,theoreticalmodelsthatincludeadaptivebehaviorhavelargelyignoredintraspecicinteractions.Inotherwords,themodelshowninFigure 1{3 overlooksthepossibleeectsofchangesinforagerdensitiesovertime,despitethefactthatanimalsareknowntoaltertheirbehaviorbasedonthebehavioranddensityoftheirpeers( GiraldeauandCaraco 2000 ; Caro 2005 ).Inchapter4,Iexaminetheevolutionofadaptiveforagingbehaviorinasimplepredator-preymodelbasedonEvolutionarilyStableStrategy(ESS)analysis.ThismodelincorporatesaHollingtypeIIfunctionalresponse.Wheneventhisbasicecologicaldetailisincludedinanotherwisestandardmodelofbehavioralresponses,itresultsininnitelymanyESSsduetotheevolutionofintraspecicinteractioncausedbythepredator'shandlingconstraint.Thisresultcautionsusintheinterpretationofresultsfromexistingmodelsandsuggeststhatconsiderationofbehaviorinexistingmodelsmaybetoosimplistic.Empiricalecologists,however,donottendtofocusonindividualbehavior.Evenstudiesthatdirectlyexaminebehaviorhavecollectedbehavioraldataatthelevelofpopulationsratherthanmeasuringindividualresponses(e.g., Anholtetal. 2000 ).Instead,muchoftheeortofexamininganimalbehaviorhasfocusedonindirecteectsgeneratedbysuchbehavior.Forexample,trait-mediatedindirectinteractions(TMIIs)areinducedbychangesinatrait(behavior)ofan

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intermediatespecies.Inathree-speciesfoodweb,theintermediateconsumersmayreducetheirforagingeortinresponsetopredationrisk,reducingtheirconsumptionrateofresources(Figure 1{3 ).Thus,predatorshaveapositiveindirecteectonresource(Figure 1{4 ).Anotherclassofwell-recognizedindirectinteractions,density-mediatedindirectinteractions,aretransmittedviachangesindensityofinterveningspeciesratherthanviatraitchanges. Figure1{4. Indirectinteractions.Blackarrowsindicatetheconsumptionofonespeciesbyanother(lethal/directdensityeect).Thicknessofthearrowrepresentstherateofconsumption.Grayarrowindicatesanon-lethaleect(directtraiteect).Thecommunityin(a)includesonlyasingleconsumerspeciesandtheresource,andthushasnoindirectinteractions.In(b),thepredatorspeciesPconsumestheconsumerspeciesthusdecreasingthedensityofconsumers(depictedbythesmallfont).Becausetheconsumerdensityissmaller,theconsumerpopulationremovesfewerresources.In(c),althoughtheydonotconsumetheconsumer,predatorsinduceantipredatorbehaviorbytheconsumer,whichdecreasestheconsumptionofresourcebyconsumer. AlthoughTMIIandDMIIarewidelydescribed( WernerandPeacor 2003 ),quantifyingthemisnotstraightforward.InChapter5,Iexamineindicesofindirecteectsthatarecommonlyusedinecologicalexperiments,focusingonhowtheycanbeusedtofacilitatetheconnectionbetweenindirecteectsandcommunitydynamics.

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inthepresenceofspatiallystructuredinteractions,behaviormayhavelittleeectonthedynamicsofcommunityandviceversa(e.g.,iftheyinteractinanon-additivemanner). Figure1{5. SpatialdistributionofpredatorsPandpreyN.TwopreyindividualsarelabeledasAandB. Asimplescenarioillustratesthepossibleeectofspatialstructureonadaptivebehavior.InFigure 1{5 ,thedistributionofpredatorsisconcentratedintheupperleftcorner.Preyarerandomlydistributed.Inthisscenario,preyindividualAmayforagemuchlessthanpreyindividualBbecauseitsperceivedpredationriskishigher(Figure 1{5 ).Thusspatialstructureleadstoconsiderablespatialvariationinindividualbehavior,variationthatthecommonnon-spatialmodelsneglect( Abrams 2001 ).Chapter6examineshowadaptivebehaviorandspatiallystructuredspeciesinteractionscanproducequalitativelydierentoutcomesincommunitydynamics.

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naturalhistorytellusanythingfurtherabouthowtomodeltheircommunitydynamics.First,considerFigure 1{3 again.Althoughantipredatorbehavioriswidelydescribedandthusthequalitativepatternweobserveintheguremakessense,behavioraltrackingoftheenvironmentinthiswayimpliesthatindividualscanmaintaininformationonthechangingstateoftheenvironment.Ifenvironmentalcuesindicatingpredatordensityarenotcontinuouslypresentandpreycanrespondonlytodirectencounterswithpredators,foragingeortisunlikelytotrackpredatordensityascleanlyasshowninFigure 1{3 .Howindividualsexhibitantipredatorbehaviorintheabsenceofimmediatethreatsandhowtheirbehavioraectedbytheenvironmentalvariablessuchasthedensityofpredators?Iaddressthisquestioninaseriesofthreestudies,eachfocusingonaspecictimescale.InChapter7,Iexaminethebehaviorofjumpingspidersafteranencounterwithapredator,behaviorthathasthepotentialtoproducetrackingbehaviorsuchasthatshowninFigure 1{3 (shorttimescale).InChapter8,Iexaminehowtherestingmetabolicratesofjumpingspidersareaectedbytheirpreviousexperiencewithpredatorsorpreyduringthepreviousday(intermediatetimescale).Inchapter9,Iexaminethegeneralactivitylevelofjumpingspidersintheeld(longtimescale).Theseresults,combinedwiththeresultsfrompreviouschapters,suggestthatthecommonlyusedmodellingframeworkisinappropriateforstudyingthecommunitydynamicsofjumpingspiders.

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behavior,whichisnotspecicallyexaminedinthisproject(i.e.,ballooning,[Belletal., 2005 ]),becauseitstronglyaectsthespatialstructureofthemodelandexclusionofthebehaviormayresultinanunrealisticdegreeofspatialstructure.Takentogether,thismodeldemonstratesthattheactivitypatternsofjumpingspidersthataredescribedinthisprojectplaykeyrolesinallowingthetwospeciesofjumpingspidersthatexhibitIGPtocoexist.Thisresultincorporatesnaturalhistorycharacteristicsofspiderssuchasballooning,furtherstrengtheningthevalidityofthisconclusion.Withoutthesimultaneousconsiderationofspatialandbehavioralfactorstogether,itwouldnotbepossibletoderivethisconclusion.AlthoughthefocusofthestudyisIGP,myresultsabouttherelationshipbetweenbehaviorandcommunityecologyaremoregeneral.Basedonthendingsofthisproject,ageneraldiscussionaboutbehavioralmodellingincommunityecologyisalsoprovidedtofacilitatereexaminationsofrelationshipsbetweenbehaviorandcommunityecology.

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PolisandHolt 1992 ; ArimandMarquet 2004 ).Eectiveimplementationofbiologicalcontrol,forexample,musttakeIGPintoconsideration( HarmonandAndow 2004 ; KossandSnyder 2005 ).IGPhasalsoaectedthesuccessofconservationandwildlifemanagementprograms(e.g., PalomaresandCaro 1999 ; Longcore 2003 ).ItisnowwellestablishedthatIGPdynamicshavestrongimplicationsforbothbasicandappliedecology.TheoreticalstudiesofIGPsuggestthatthecoexistenceofspeciesinIGPfoodwebsisdicult.Duetothedoublepressureofcompetitionandpredationfromintraguildpredators(IGpredators),modelspredictthatintraguildprey(IGprey)willbeeliminatedinawiderangeofparameterspace.BecauseIGPisubiquitousinnature( ArimandMarquet 2004 ),thereisadiscrepancybetweentheoryandobservations.Thisdiscrepancycontinuestopuzzleecologists( HoltandPolis 1997 ; KrivanandDiehl 2005 ).Simplemodelsshowthat 1. IGpreymustbebetteratexploitingthebasalresourcethanIGpredatorsinordertocoexist. 2. Atlowproductivitylevels,IGpreycanoutcompeteIGpredators.Whenproductivityishigh,IGpredatorswilldriveIGpreytoextinction.Atintermediateproductivitylevels,thetwospeciesmaycoexist. 3. Asproductivityincreaseswithintherangethatallowscoexistence,theequilibriumIGpreydensitydecreaseswhiletheequilibriumofIGpredatordensityincreases. 11

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Allthesepredictionshavebeenempiricallyveriedinparasitoidsystems(predictions1and2, Amarasekare ( 2000 );prediction3, Borer,Briggs,Murdoch,andSwarbric ( 2003 )),andinmicrocosms(prediction1, Morin ( 1999 );allpredictions, DiehlandFeissel ( 2000 )).TheseempiricalstudiesconrmthatsimplemodelscapturesomequalitativepropertiesofIGPinteractions.However,therealchallengetotheoryliesnotinthequalitative(im)possibilityofIGPcoexistencebutinitspredictedimprobability.ThenarrowparameterspacethatmodelssuggestcouldallowcoexistencedoesnotseemtosupporttheubiquitousoccurrenceofIGPinecologicalcommunities.Severalecologicalfactors(e.g.,size-structureandadaptivebehavior)havebeenexaminedtoseewhethertheyallowanincreasedprobabilityofcoexistence( Myliusetal. 2001 ; KrivanandDiehl 2005 );thecoexistenceparameterregionmayormaynotexpanddependingonthedetailsofthemodels.Althoughthesefactorsareimportant,itislikelythatwestilllacksomeimportantecologicalcomponentsinIGPmodels.OnefactorthathasnotbeenexaminedinIGPmodelsisspatialstructure(butAmarasekare(2000a,b)concludedthatthecompetition-dispersaltradeoisnotimportantinthecoexistenceofaparasitoidcommunitythatincludesIGPinapatchyenvironment).Spatiallyexplicitmodelling(e.g.,distinguishinglocalandglobalinteractions)hasgeneratedanumberofnewhypotheses( Amarasekare 2003a ).Furthermore,spatialstructureisknowntostabilizesimplepredator-preymodels( Keelingetal. 2000 ).AsallorganismsinanIGPfoodwebinherentlyinteractwithotherspeciesinaspatiallystructuredmanner,thisisanimportantaswellasarealisticaxistoexamine.Inthisstudy,Iusepairapproximations( SatoandIwasa 2000 )andanindividualbasedmodel(IBM)toexamineasimpleIGPfoodwebinaspatiallystructuredenvironment.Theso-calledpairapproximationkeepstrackoflocaldynamicsaswellasglobaldynamics,whilethemeaneld

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approximationmakesnodistinction(non-spatial);pairapproximationreducestothemeaneldapproximationinitsnon-spatiallimit(discussedbelow).Thus,usingpairapproximationallowsonetoexaminetheeectoflocalinteractionsbycomparingtheresultswiththeanalogousmeaneldmodel.WithIBMs,Iexaminetheeectsofspatialheterogeneityinproductivity,which HoltandPolis ( 1997 )suggestedshouldbeimportantinIGPsystems.Thethreemainquestionsare(1)howthequalitativepredictionsofIGPmodelsareaectedbytakingspaceintoaccount,(2)whetherspatialstructureexpandsthepossibilityofcoexistence,andifso,underwhatconditions,and(3)howspatialheterogeneityinresourcedistributionaectsIGPdynamics.

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Eachlatticesitehaszneighbors(e.g.,hexagonalneighborhoodswouldhavez=6)andiseitheroccupiedbyPorN,orisvacant(E).Thetwospeciescanhavedistinctneighborhoodsizes(zNforIGpreyandzPforIGpredators);however,unlessotherwisestated,weassumethatIGpreyandIGpredatorshavethesameneighborhoodsizes(z=zN=zP).ThefractionofsitesinstateP,N,andEarecalledglobaldensitiesandaredesignatedP;NandE.Wedeneqi=jasthelocaldensityofsitesinstateiwithaneighborinstatej.Forexample,qP=NisthefractionofPsitesthatarenexttoanNsite(i.e.,theprobabilitythatarandomlychosenNsiteislocatednexttoaPsite).IGprey(N)andIGpredators(P)canreproduce(ataratedependingonbasalresourceconsumption)onlyiftheyareadjacenttoavacantsite,andtheirreproductiveratepervacantsiteisRbN=zandRbP=z,respectively.Therefore,RbNandRbParethemaximumratesofreproductioninanemptyneighborhood.Hence,thereproductionrateofarandomlychosenIGpreyisz(RbN=z)qE=N=RbNqE=N,theproductofthemaximumbirthrateandtheexpectedfractionofvacantsitesintheneighborhood.Forsimplicity,Iassumethatspatialmovementoccursonlybymeansofreproduction.IGpreydieduetopredationbyIGpredatorsatarate,whichtogetherwithconversioneciency(e)alsodeterminesthereproductionofIGpredators.WedeneasthemaximumpredationratewhichisattainedwhentheIGpredatoriscompletelysurroundedbyIGprey.Basedontheserules,theequationsfortheglobalpopulationdensitiesaredN (2{1)dP

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2{3 and 2{4 representaspecialcaseofthemodeldescribedinFigure4ofPolisetal.(1989)whentheintensitiesofinter-andintra-speciccompetitionarethesame.Althoughcompetitionisforspaceratherthanforaresourcewithexplicitwithin-celldynamics,thenon-spatialversionofthemodelmatchesamodelderivedwithresourcecompetitioninmind.

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Duetothesedependencies,wehaveonlyveindependentvariables,whichcanbechosenarbitrarily.WewillchooseN;P;qN=N;qP=PandqP=Nastheindependentvariablesandexpressalltheothersintermsofthesevevariablesbasedontheaboveconstraints.Inordertocalculatethedynamicsoflocaldensity,forexampleqN=N,werstderivethedynamicsofthedoubletdensityNN(i.e.,twocellsthatareadjacenttoeachotherarebothoccupiedbyIGprey).dNN SatoandIwasa ( 2000 )and Iwasaetal. ( 1998 )fordiscussions/details.Thesedoubletdensitiesaretransformedtoconditional

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probabilities.Forexample,thedynamicsofqP=Parefoundtobe,dqP=P 2{1 .Theoutcomewasclassiedintooneoffourcases:IGpreycaninvadeIGpredators,butIGpredatorscannotinvadeIGprey(IGpreywin),IGpredatorscaninvadeIGprey,butIGpreycannotinvadeIGpredators(IGpredatorswin),eachspeciesisabletoinvadetheother(coexistence),andneitherIGpreynorIGpredatorscaninvadetheother(bistability)( MurrellandLaw 2003 ).

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Reproductionanddeatheventswererandomlyorderedforeachindividualineachiteration;ananimalthatdiedinatimestepmayormaynothavereproducedbeforedeath.ThepotentialfecundityforeachindividualwassimulatedasaPoissondeviatewithmeanRbN(IGprey)orRbP(IGpredators)whereRistheproductivityofthecellwhereindividualsreside(i.e.,eitherRLorRH:seebelowforthedescription).Foreachpotentialospring,theprobabilityofactualreproductionwasfractionofvacantadjacentcells(e.g.,theprobabilityofconvertingonepotentialreproductiontoanactualospringis1/4ifonlyoneadjacentcellisempty).Reproductionofospringwasrealizedsequentially,allowingfordepletionoffreespaceintheneighborhood.IGpredatorupdatingincludespredation,whichresultedinaPoissonreproductionprocess(numberofospring)withmeane.A51-by-51squarelatticewithperiodicboundarieswasusedastheenvironment.Simulationsalwaysbeganwith200IGpreyand100IGpredators,bothrandomlydistributedintheenvironment.Persistencewasdenedasfractionofsimulationsoutof50resultinginP>0andN>0att=5000.ThemodelwasimplementedinNetlogo( Wilensky 1999 ).

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halfthecellswereassignedtohighorlow,theaverageenvironmentalproductivitywasalways(RH+RL)=2.ExamplesofpatchesofdierentscalesareshowninFigure 2{1 Figure2{1. Examplesofrandombinarylandscapesbasedondierentpatchscales.Patchscalereferstothenumberoftimestheprocedurediffusewasapplied(seetext). 2.3.1MeanFieldApproximationInadditiontothetrivialequilibriumwherenospeciescansurvive(whichoccurswhenRc1andR>c2),coexistence(whenRc2),andbistability(whenR>c1andR
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Figure2{2. ParameterregionsindicatingtheoutcomeofIGPinanon-spatialmodel.Parameters:mP=0:3;mN=0:2;bP=0:5;bN=0:8;e=0:4.WhenR<0:25,neitherspeciescansurvive. Forexample,IGpreywilloutcompeteIGpredatorswhentheproductivitylevelislowandIGpredatorswillwinwhentheproductivitylevelishighprovided>0:5(Figure 2{2 ).Atintermediateproductivity,bothspeciescancoexist.Whenbothspeciescoexist,theamountofresourceinvacantcells,RE,(analogoustothestandingstockofunusedbasalresourceinanexplicitresourcesmodel)isatanintermediateproportionbetweenthatwithIGpreyaloneandthatwithIGpredatoralone.Whenthespeciescoexist,increasingproductivitywilldecreasethedensityofIGpreywhileincreasingthatofIGpredators.Thecoexistenceconditionbasedontheproductivitylevelabove(i.e.,Rc2)canberewrittenasbN=mN>bP=mP.Thusforspecieswithequalbackgroundmortalityrates,coexistenceisonlypossiblewhenIGpreyisabettercompetitorforresourcethanIGpredators(i.e.,bN>bP).

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themeaneldapproximationmodel(Figure 2{3 ).Astheneighborhoodsizegrows(e.g.,Figure 2{3 ,z=9999),thedynamicsofthepairapproximationapproachesthelimitingcase,themeaneldapproximationmodel.Inabistabilityregion,onespecies(NorP)canwindependingontheinitialdensities(i.e.,foundercontrol). Figure2{3. Resultsofinvasionanalysisinthepairapproximationmodel.TheparametersusedarethesameasinFigure 2{2 .Whenneighborhoodsize(z)islarge,theresultsareindistinguishablefromthenon-spatialmodel(Figure 2{2 ). 2{4 ).BothreproductionandmortalityparametersfortheIGpreyandIGpredatorswerexedatthesamevalue(bN=bPandmN=mP;seethegurecaptionfortheactualvalues).ThisconditionpreventsIGpreyfrompersistinginthenon-spatialmodel(seeintroduction)orwhenbothIGpreyandIGpredatorshadthesameneighborhoodsize(i.e.,zN=zP=4resultedIGpredatordominanceinalltheparameterregionsinFigure 2{4 ).However,asIGprey'sneighborhoodsizebecame

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Figure2{4. Resultsofinvasionanalysisinthepairapproximationmodel.SpatialscaleofIGpredatorswasxedatzP=4whilethatofIGpreyvaried.WhenzN=4(i.e.,zN=zP),IGpredatorsdominateintheentireparameterspaceshown.Parameters:mN=mP=1;bN=bP=1;e=1. greaterthanthatofIGpredators,coexistencebetweenIGpreyandIGpredatorsbecamepossible. 2{3 ).Forexample,inFigure 2{4 ,when=0:6fortherangeofproductivityexamined,strongspatialstructure(z=2)predictscoexistenceisimpossiblewhilecoexistencemaybepossibleinthecaseofweakerspatialstructuresuchas(z=6).Intheotherwords,theintervalofproductivitylevelsthatallowsforcoexistencechangeswithz.Thus,tomakeaquantitativecomparisonbetweenspatialandnon-spatialmodels,wecomparedtherangeofproductivitylevelsthatallowscoexistenceinthetwomodels.LetIspatialandInon-spatialbethecoexistenceintervalinproductivityforspatialandnon-spatialmodel,respectively(Theparametervaluesusedto

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obtainIspatialandInon-spatialaredescribedinthecaptionofFigure 2{5 ).Theratioofintervals,Ispatial=Inon-spatial,wereexamined:valuesgreaterthan1indicatethatspatialstructureenhancedtheprobabilityofcoexistencewithrespecttothenon-spatialmodel. Figure2{5. Parameterintervalsresultinginexpansionandreductionofthecoexistenceinterval.ThelineindicatesthecontouratIspatial=Inon-spatial=1.Whenthisratioisgreaterthan1,spatialstructureincreasedthesizecoexistenceintervals.Parameters:mP=0:3;mN=0:2;bP=0:3;bN=0:6;R2(0:1;10);z=4. Dependingontheparametervalues,spatialstructurecaneitherdecreaseorincreasethecoexistenceinterval(Figure 2{5 ).HighconversioneciencyeandahighattackrateofIGpredatorsmeantthatspatialstructureincreasedtheprobabilityofcoexistence(Figure 2{5 ).Althoughthecomparisonbetweenthespatialmodelwithz=4andthenon-spatialmodelisshown,theresultsforotherneighborhoodsizes(e.g.,z=6;z=8)aresimilar.

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productivitygradient:IGpredatorsareeliminatedatlowproductivitylevelsandIGpreyareeliminatedathighproductivitylevels(Figure 2{6 ). Figure2{6. Persistenceprobability(squares)anddensityofIGprey(circles)andIGpredators(triangles).Parameters:mP=0:2;mN=0:2;bP=0:5;bN=0:8;=0:9;e=0:9;z=4. Inheterogeneousenvironments(i.e.,eachcellisassignedeitherRHorRL),whentheproductivityofalowresourcepatchissmall(e.g.,iftheenvironmentswerehomogeneousatthisproductivity,evenIGpreyalonecouldnotpersist),asmallpatchscalewasfavorabletoIGpreyandIGpredatorswentextinctquickly.Whenthepatchscalewaslarge,however,IGpreywereeliminated.Atintermediatepatchscales,bothspeciescoexist.Becauseaverageproductivityatdierentpatchscalesisthesame,thissuggeststhatthespatialcongurationofpatchesmaystronglyaecttheoutcomeofIGP.Thisrelationship,however,ippedastheproductivityoflow-resourcepatches(RL)increased.WhenRLwasrelativelyhigh,persistenceoftheIGPsystemwashigherwhenthepatchscalewaseitherloworhigh.Persistenceprobabilitywaslowestatananintermediatelevelofpatchscale.Spatialheterogeneityalsomodiedtheeectofproductivitylevelonnumericaldominance.Forexample,thenon-spatialmodelpredictsthatwhenIGpreyand

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Figure2{7. Persistenceprobability(squares)anddensityofIGprey(circles)andIGpredators(triangles).Densitiesofconsumersarepresentedasfractionoftotalcellsoccupiedbythespecies.Parameters:mP=0:2;mN=0:2;bP=0:5;bN=0:8;=0:9;e=0:9;z=4. IGpredatorscoexist,asproductivityincreases,IGpreywilldeclineindensity.However,whenspatialheterogeneityisintroduced,IGpreydensitymayremainconstantasproductivityincreases(Figure 2{8 ). 2.4.1EectsofSpatialStructureontheBasicResultsofNonspatialModelsThehomogeneousenvironmentmodel(i.e.,pairapproximation)maintainedthequalitativepredictionsofnon-spatialmodels.Assuggestedbythenon-spatialmodel,resourceutilizationabilityofIGpreyhadtobebegreaterthanthatofIGpredatorsinorderforthetwospeciestocoexistwhentheyhavethesameneighborhoodsizes.Nonetheless,thepairapproximationmodelpredictsthatIGpreyandIGpredatorscancoexistevenwhentheresourceutilizationcondition(i.e.,bN>bP)isnotmetaslongasthespatialscaleforIGpreyislargerthanthatofIGpredators(Figure 2{4 ). Amarasekare ( 2000 )considersthisphenomenonadispersal-colonizationtradeo.RecognizingthispotentialtradeoisimportantbecauselaboratorymeasurementofparameterssuchasbNandbPoverlooksthe

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Figure2{8. Persistenceprobability(squares)anddensityofIGprey(circles)andIGpredators(triangles).Densitiesofconsumersarepresentedasfractionoftotalcellsoccupiedbythespecies.Parameters:mP=0:2;mN=0:2;bP=0:5;bN=0:8;=0:9;e=0:9;z=4,andSpatialscale=7. dierencesinspatialscaleofforaging,whichmaybeessentialinordertoteaseapartmechanismsofcoexistenceinIGPcommunities( Amarasekare 2003b ).Explicitlyconsideringthesefactors,Amarasekare(2000a,b)concludedthatlocalresourceutilizationdierences(e.g.,bN>bP)weremoreimportantthandispersal-colonizationtradeo(e.g.,zN>zP)incoexistenceofaparasitoidcommunity.Spatialheterogeneitycanoverturnthesecondprediction(speciesdominanceshiftsfromIGpreytocoexistence,andthentoIGpredatorsasproductivityincreases).Forexample,evenwhenaverageproductivityislow(i.e.,thehomogeneousmodelmodelpredictsthatIGpreywillexcludeIGpredators),IGpredatorscanstilloutcompeteIGpreywhenresourcesaredistributedatparticularpatchscales(Figure 2{7 ).Thus,althoughthehomogeneousmodelpredictsadominanceshift(i.e,IGpreydominance!coexistence!IGpredatordominance),thispredictioncanbeviolatedinthepresenceofresourceheterogeneityifpatchscaleislargein

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lowproductivityenvironmentsandsmallinintermediateandhighproductivityenvironments.Inotherwords,withoutknowinghowthespatialscaleofresourcechanges,wecannotreliablypredictchangesinspeciesdominancewithincreasingaverageproductivity.Furthermore,spatialheterogeneityaectsthequalitativepredictionthatthedensityofIGpreywilldecreasewhilethatofIGpredatorincreasesasproductivitylevelincreases.Onthecontrary,weseethatdensityofIGpreymayremainroughlyconstantasproductivityincreasesinthecoexistenceregion(Figure 2{8 )inaheterogenousenvironment.Intheeld, Boreretal. ( 2003 )observedthesamephenomenonthatdensityofIGpreywasunaectedbytheresourcelevel.Thusthespatialmodelcanpotentiallyexplainunresolvedresultsobservedinnature.TheresultsshowninFigure 2{8 assumeaconstantpatchscale.Ifthepatchscalevaried(e.g.,dierentscalesforeachproductivityvalue),wecouldpotentiallyseemanydierenttrends. 2{3 ),butthemodelsgavedierentquantitativepredictionsofcoexistenceprobability.Asspatialstructurebecomesstronger(i.e.,zdecreases),thecoexistenceregionexpandswhileshiftingintheparameterspace.ForexampleinFigure 2{3 ,aszdecreased,lowervaluesof(attackrateofIGpredatorsonIGprey)thatpreviouslyallowedforcoexistence(e.g.,R=1;=0:8)insteadallowedIGpreydominance.IntheparameterregionofcoexistencenearIGpreydominance,IGpredatorsbecomelesseectiveinutilizingIGpreyaszdecreases.Atthesametime,theparameterregionthatallowedIGpredatordominancebutwasneartheboundaryofcoexistenceparameterregion(e.g.,R=1:5;=0:6)shiftedtocoexistencebecauseIGpreybecamelessvulnerabletointraguildpredation.Theparameterehasasimilarinuenceinthe

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modelbecausereproductionduetoIGPisrealizedonlyifIGpredatorscancaptureIGprey.Ingeneral,givenahighattackrateofIGpredators()andhighprotabilityofIGprey(e)(i.e.,intraguildpredationismorebenecialtotheIGpredatorsthanresourceconsumption),spatialstructurefavorsthepersistenceoftheIGPsystem.OnehypothesisfortheevolutionofIGPisbasedonstoichiometry( DennoandFagan 2003 );IGpredatorsconsumeIGpreybecauseIGpreyhavetherightbalanceofnutrients(i.e.,eislarge).Forexample, Matsumuraetal. ( 2004 )documentedthatwolfspidersgrowbetterifotherspiders(i.e.,IGprey)wereincludedintheirdietthanwhentheywereraisedonadietthatdidnotincludeotherspidersasdiet.Thus,theparametersofnaturalsystemsarelikelytolieintheregionwherespatialstructurefavorsIGPpersistence.ThisrelationshipbetweenthebenetofIGPandspatialstructuresuggeststhatunderstandingtheproximalconsequencesanddeterminantsofIGP(e.g., Matsumuraetal. 2004 ; RickersandScheu 2005 )andtherolesofspatialstructureshouldfacilitateourunderstandingoftheecologicalandevolutionarysignicanceofIGP.

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Accordingly,predictionsofIGPoutcomesinaheterogeneousenvironmentdepartfrompredictionswhentheenvironmentishomogeneous.Whenaverageproductivitylevelwaslow(e.g.,0.65inFigure 2{7 ),IGPdynamicschangedfromIGpreydominancetocoexistenceandthentoIGpredatordominanceaspatchscaleincreased.Inthisexample,thelowpatchproductivitywasverylow(RL=0:3),andneitherIGpreyorIGpredatorscouldpersistinalargescalelowpatch.Thehighresourcepatches,incontrast,wereveryproductive(RH=1)sothatIGpredatorsdominated(Figure 2{7 ).Therefore,aspatchscaleincreasedandpatchproductivitydiverged,neitherspeciescouldpersistinthelow-resourcepatcheswhileIGpredatorswoninthehigh-resourcepatches.Atintermediatepatchscales,patchheterogeneitycreatedanenvironmentthatallowedbothspeciestocoexist,creatingahumpshapedrelationshipinpersistence.Thishump-shapewasippedathighmeanproductivitylevels(e.g.,0.835inFigure 2{7 correspondingtoRL=0:67).Inhigh-resourcepatches(RH),IGpredatorsdominated,andinlow-resourcepatches(RL),IGpreydominated;habitatsegregationresulted,causinganincreaseinpersistence.Ifpatchscalewasfurtherincreased,persistencewouldeventuallyapproach1.Forhabitatsegregationtobeeective,eachpatchtypemustbesucientlylarge.Forexample,continuous\spillover"ofIGpredatorsfromthehighproductivitypatchcanwipeoutasmalllowproductivitypatchwithIGprey.Thisresultsuggeststhatdetailsoflandscapecongurationmaysignicantlyaltercharacteristicsofcommunitystability.Furtherinvestigationsonhowlandscapestructureaectsmovementofspeciesandcommunitydynamicsisneeded( vanDyckandBaguette 2005 ).Althoughspatiallystructuredspeciesinteractionandspatiallyheterogeneousenvironmentarewellrecognizedfactorsinecology,systematicexplorationofthisaxishasonlybegunrecently( Bolker 2003 ; Hiebeler 2004a b ),andwedonotyethavecleargeneralhypothesesabouttheeectsofspaceeveninsimplemodels.

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Furtherexplorationsoftherolesofbothspatialstructureandspatialheterogeneityareneeded.Furthermore,althoughspatialinteractionsmaybediculttoanalyze,somespatialdataarerelativelyeasytocollectonceweknowexactlywhattocollect.Infact,eldstudieshaveoftencollectedthesedataasauxiliaryinformationevenwhentheiranalysesignoredspace.Thedevelopmentofspatialtheorieswillcreatemoretestablehypothesesandincreaseourabilitytoutilizedatamoreeciently,whichmayresolvesomeofthediscrepanciesbetweentheoryanddata.

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JoernandBehmer 1997 );feedingactivityandgrowthrateofzooplanktonarestronglyinuencedbyphosphoruslevelsintheiralgaldiet( PlathandBoersma 2001 ).Theorysuggeststhatanimalsshouldforageselectivelytomaximizetheirnutrientrequirements(e.g., Simpsonetal. 2004 ).Theseconsiderationsareimportantnotonlyforunderstandingthebehavioralandphysiologicalmechanismsofforagingbutalsoforunderstandingthedynamicsofecologicalcommunities.Nutrientsdirectlyaectthefunctionalandnumericalresponsesofspeciesinteractions( Andersenetal. 2004 ).Inrecentyears,analysisofcommunitymodelswithnutrientspecicinteractionshasbecamecommon(e.g.,usuallycalledecologicalstoichiometry( Loladzeetal. 2004 )ornutrienthomeostasis( Loganetal. 2004 ))andthesestudieshavehelpedtounderstandpreviouslyunexplainedpatternsinnature(reviewdin Moeetal. 2005 ).Intraguildpredation(IGP),predationwithinaguild(i.e.betweenmembersofdierentspeciesatthesamelevelinafoodchain),hasbeensuggestedtobearesponsetothemismatchintheratioofcarbontonitrogen(C:N)betweenpredatorsandherbivorousprey( DennoandFagan 2003 ).C:Ndecreasesastrophiclevelincreases( Faganetal. 2002 ).Thistypeofpredation(IGP,oromnivorymorebroadly)benetstheconsumersbecauseconsumingothernitrogen-richpredators(andthusdecreasingtheC:Nimbalance)helpssatisfytheirnutrientrequirements 31

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andfacilitatesgrowth( FaganandDenno 2004 ).IfIGPfacilitatesthegrowthofintraguildpredators,itwilldirectlyaectthesize-structureofindividualswithinthecommunitybecauseoccurrenceofIGPissize-dependent( Polis 1988 ).However,directexaminationofthishypothesisisrare(butsee Matsumuraetal. 2004 ).Inthisstudy,Iexaminedwhetherthenitrogencontentofpreyaectsthegrowthofjumpingspiders.Specically,Ishowedthatspidergrowthrateisfacilitatedbynitrogencontentofprey. MayntzandToft 2001 ).Inthecontrolgroup,fruitieswereraisedonDrosophilamedium(CarolinaBiologicalSupply).IntheN-richgroup,bloodmeal(PenningtonEnterprizes,Inc)wasaddedtothemedium(3:1=medium:bloodmeal).Inordertoexaminepotentialconfoundingfactorsofthetreatment(i.e.,treatmentmaycreatedierenceinaspectsofpreyinadditiontoNlevel),theenergeticcontentofpreywasalsoquantiedbasedonawhole-animalassaywithadichromateoxidationmethoddescribedin McEdwardandCarson ( 1987 ).

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3{1 ). Table3{1. Nutrientandenergycontentsofprey(standarderrorsinparentheses).Meanenergycontentsofpreywerenotsignicantlydierent(ANOVA,F2;12=0:8808;p=0:4396).Nitrogencontent(%N)ofieswashigherintheN-rich(blood)treatment(Welchtwosamplet-test,t1:183=9:2783;p=0:048). PreyN%C%Energy(J) ies(control)7.64(0.08)50.38(0.26)2.20(0.48)ies(blood)10.36(0.28)49.59(0.94)1.71(0.36)spiderlings10.06(0.57)35.88(1.27)1.50(0.26) Figure 3{1 showsthecarapacewidthsofspidersforeachtreatment.TheN-richgrouphadwidercarapacesonaverageforalltheinstarsexamined,butthedierenceswerenotstatisticallysignicant(Figure 3{1 ,t-test:p>0:05inall

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cases).Thesizedierenceswereinsignicantevenwhenthecumulativedierenceingrowthwasconsidered(i.e.,thechangesinsizefromthesecondinstartothefthinstar). Figure3{1. Growthincarapacewidthofspiders.nthinstardataindicatethedierenceinsizebetween(n+1)thandnthinstar.Nosignicantdierenceswerefoundbetweentreatments,foranyinstar.Treatments:N-rich(N)andcontrol(c).Topandbottomlinesofboxindicatethe75%quartileand25%quartileofsample,respectively.Thehorizontalbarintheboxindicatesthemedian.Topandbottombarsaroundtheboxindicate90%quartileand10%quartile,respectively.Theupperandlowernotchescorrespondstotheupperandlower95%CIaboutthemedian. ThemeandurationsofinstarswerealwaysshorterfortheN-richgroup(Figure 3{2 );thus,spidersgrewfasterwhilemoultingbetweeninstarsatthesamesizes.Thedierencesindurationwerestatisticallysignicantforthe2ndinstar(t-test:p=0:0006)and4thinstar(p=0:0003),butnotforthe3rdinstar(p=0:839).

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Figure3{2. Durationofeachinstar.Durationsof2ndinstarand4thinstarweresignicantlysmallerfortheN-richtreatment(indicatedby*).Treatments:N-rich(N)andcontrol(c). deRoosetal. 2003 ).BecauseoccurrenceofIGPdependsonthesize-structure,thegrowthconsequencesofstoichiometrymayhavestrongimplicationsforIGPcommunitydynamics.Futureworkshouldconsidertheprey'snutrientprolemorecarefully. Matsumuraetal. ( 2004 )havedoneexperimentssimilartothisstudyexaminingtheeectsofpreytypeonthegrowthlevelofwolfspiders(genusPardosa),nding

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thatspidersthatfedonintraguildprey(i.e.,otherspiders)alonedidnotenhancetheirgrowthrate.Yet,theyfoundthatamixeddiet(i.e.,additionofintraguildpreytoherbivorousprey)facilitatedthegrowthofwolfspiders.Researchershavefoundsimilarresults(i.e.,advantagesofmixeddiets)inotherecologicalsystems( Agrawaletal. 1999 ; Cruz-RiveraandHay 2000 ).Westilldonotclearlyunderstandtheoptimalnutrientrequirementsforthesecarnivores,norhowthosenutrientsaredistributedamongpreyintheeld.NordoweknowwhethersimplifyingthedescriptionofstoichiometrytoasingleC:Nratio,orC:N:P( Loganetal. 2004 ),isadequateforunderstandingcommunitydynamics.Forexample, Greenstone ( 1979 )foundthatwolfspidersforageselectivelytooptimizeaminoacidmakeup,whichsuggeststhatmorecomplexstoichiometricdescriptionsmaybenecessaryifwehopetostudystoichiometriccommunityecology.Inthisexperiment,thefood(i.e.,fruitymedium)ofpreyitemswasvariedtomanipulatethenitrogencontentofprey( MayntzandToft 2001 ).Wedonotknowwhetherthelevelofnitrogendierencebetweentreatmentandcontrolgroupswascreatedasaresultofnitrogenassimilationintoytissuesorbloodmealintheirgutcontent.Thisdierenceisnotcrucialtotheinterpretationofthisstudyasspidersnonethelessconsumednitrogenrichpreyandincreasedtheirgrowthrate(Figure 3{2 ).However,theresultshaveotherimplications.Forexample,nitrogencontentvariesgreatlyamongplants( Mattson 1980 ).Anthropogenicenvironmentalchanges(e.g.,increasedCO2andsoilpollution)alternutrientlevelsofplants( Newmanetal. 2003 ).Thebloodmealusedinthisstudyisacommonagriculturalfertilizer.Ifherbivoresthatconsumedierentplantsofdierentqualitiesinuencepredatorsasshowninthisstudy,theeectofstoichiometricinteractiononsystemswithIGPcouldoccuratverylargetemporalandspatialscales.Bycarefullyexaminingthenutrientrequirementoforganismsaswellastheowofnutrients,wemayobtaindeeperinsightsnotonlyintoaspecicecological

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communitywithIGPbutalsointogeneralpropertiesofthepersistenceofcomplexfoodwebs.

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Kondoh 2003 ; Yoshidaetal. 2003 ).Antipredatorbehavior(e.g.,activitylevel)isoneofthemostwellstudiedclassesofadaptivetraitsbothempiricallyandtheoretically;itiswidelyobservedinnature( WernerandPeacor 2003 ; Benerd 2004 ; Preisseretal. 2005 ; LuttbegandKerby 2005 )anditscommunitylevelconsequencescanbesignicant( FryxellandLundberg 1998 ; Bolkeretal. 2003 ).Oneoftheearliestapproachestothestudyofevolutionaryadaptation( MaynardSmithandPrice 1973 )goesunderthegeneralnameofevolutionarygametheory.Thisapproachseekstoidentifythesetofallstrategies(traitvalues)thatareevolutionarilystablebyapplyinganESS(EvolutionarilyStableStrategy)criterion.Astrategyiscalledevolutionarilystableifapopulationofindividualsadoptingthisstrategycannotbeinvadedbyamutantstrategy.Theusualindicatorformeasuringinvadabilityisthetness(contributiontothenextgeneration'sgenepool)oftheindividual( Roughgarden 1996 ).Thus,astrategyiscalledanESSifwhenitisadoptedbyalmostallmembersofapopulation,thenanymutantindividualwillhavealessertnessthanthatofanindividualofthegeneralpopulation.OneshortcomingofthisapproachtostudyingecologicaldynamicsisthatwhiletheESScriterionmakesgoodintuitivesense,itisbasedonastaticanalysisofthepopulationanddoesnotindicatehowthepopulationmayhave 38

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cometoevolvetosuchanESS.Infact,inhasbeenshownthatthenon-invadabilityofatraitvalue(aparticularESS)doesnotimplythatapopulationwithanearbydierenttraitvaluewillevolvetotheESSovertime( Taylor 1989 ; Christiansen 1991 ; TakadaandKigami 1991 ).Inotherwords,ifweregardevolutionasadynamicprocess,therecanexiststrategiesthatareevolutionarilystableaccordingtotheESScriterion,butthatarenotattainableinthedynamicsofevolution.ThedynamicalapproachtoevolutionofatraitCiscommonlymodelledbyincludingthefollowingequation: dt=g@W(~C;C) 4{1 havebeenmotivatedbythegeneralprinciplethatregardsevolutionasagradient-climbingprocessonanadaptivelandscape( Gavrilets 2004 ),andbysimilarprinciples( BrownandVincent 1987 1992 ; Rosenzweigetal. 1987 ; TakadaandKigami 1991 ; Vincent 1990 ; Abrams 1992 ; Abramsetal. 1993 ).Ithasalsobeenshownthatonecanobtainanequationsimilartothedynamics(Eq. 4{1 )asalimitingcaseofresultsfromquantitativegenetics( Lande 1976 ; Abrams 2001 ).InthederivationofEq. 4{1 byquantitativegeneticsitisassumedthatthetraitinquestionisdeterminedbyalargenumberofgeneticloci,eachcontributingasmalladditiveeect.Inthissetting,therateofevolutiongmaybeinterpretedastheratioofadditivegeneticvariancetopopulationmeantness( Iwasaetal. 1991 ; Abrams 2001 ).Whenthefocusofstudyisecologicaldynamics,weassumethatforagersbehaveoptimally(withanevolutionarilystablestrategy)andwestudytheconsequencesofthisbehaviortocommunitydynamics(e.g, Abrams 1992 ; Krivan

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1996 ; KrivanandSirot 2004 ).Thus,insteadofincorporatingtheevolutionarytraitequation,Eq. 4{1 ,anoptimalsolution(ESS)forthetraitCiscalculatedbytheESScriterionandthensubstitutedintotheecologicaldynamicsequations.Inotherwords,weassumethatevolutionhasalreadytakenplacetoshapetheadaptivebehavior,andthatanevolutionarilystablevalueforCisinplace.Anunderstandingoftherelationshipbetweengenesandbehaviorisnotnecessarywhenusingthisapproach.However,asweshallsee,eveninasimplemodel,suchanoptimalbehaviormaybeverycomplex.Inthispaper,usingasimpleLotka-Volterratypepredator-preymodelwithatypeIIfunctionalresponse( Holling 1959 ; Royama 1971 ; Jeschkeetal. 2002 )inwhichthepreyhaveadensity-dependentforagingeort,weanalyticallyderivetheESSofpreyactivity,asdenedbytheESScriterion.Specically,weshowthatatparticulardensitiesofpredatorsandprey,therearemultipleESSs.ToexaminetherelationshipbetweentheESSsandtraitevolution,wealsoexaminethecommondynamicalmodelofevolution(i.e.,Eq. 4{1 ).Toexamineecologicalimplicationsofadaptivebehavior,weexplorethedierencesthatmayariseincommunitydynamicsbetweentheevolutionarydynamicalapproachandthesituationwhereanyoneofthemultipleESSsofpreybehaviorisxedinthebaseecologicalmodel. dt=Nbcac2P dt=Pac2N

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themodelreducestothestandardLotka-VolterramodelwithatypeIIfunctionalresponse.Theparametersb;;h;a;mN;mPweregardtobeconstants,buttheforagingeortcweregardtobeafunctionc(N;P),sothatcismodelledhereasanadaptivebehaviorofthepreythatisdependentonthedensitiesNandP.Weinterpretbasthemaximumrateofbenet(reproduction)ofpreywhentheyforagemaximally.Handlingtime,h,isthetimerequiredforpredatorstoconsumeaprey.Thedensity-independentdeathratesofpreyandpredatorsaredenotedbymNandmP,respectively.Thesearcheciency,a,isacharacteristicofthepredatorsthatmeasurestheirsuccessrateofndingprey.Thesearcheciencyaofthepredatorsismodiedbythevulnerability(c)oftheprey.Theusualassumptionisthatvulnerabilityofthepreyincreaseswiththeirforagingeortasaconvexfunctionofc.Thereasonforchoosingalinearlyincreasingfunctionofctomodifythepreybenetratebandaconvexincreasingfunctiontomodifypredationeciencyaissothattheriskofpredationdoesnotoutweighthebenetofenhancedreproductionwhenincreasingforagingeortfromc=0inthepresenceofalargepredatorpopulation.NomatterwhatthedensitiesNandPare,therewillalwaysbesomepositivevalueofcwhichisbetterforthepreythanc=0.Forsimplicityweusethevulnerabilityfunction(c)=c2.Forexample,ifthepreydecreasetheireortfromc=1toc=0:5,thentheeectivesearcheciencyofthepredatorsdecreasesfromato0:25awhileeectivebenetdecreasesfrombto0:5b.Foodassimilationeciencyinconvertingingestedpreyintonewpredatorsisdenotedby.Noteagainthatweassumetheforagingeorthasboundedvalues.Sinceweassumec(N;P)2[0;1],cmaybeinterpretedasafractionofthemaximumforagingeort.AmethodforcalculatingallpossibleESSsforanadaptivebehaviorisbasedonthefollowing

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4{2 :WN=bAaA2P

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WeremarkagainthatweregardsuchstrategiesEandxtobedensity-dependentfunctionsE(N;P)andx(N;P). 4.3.1EvolutionarilyStableStrategy(ESS)ofForagingEortThederivationofevolutionarilystablestrategies(ESSs)forthetraitcisshowninAppendixA.TherearethreepossiblefunctionaltypesofESS:c=y1;c=y2andc=1,wherey1=1 a!;y2=1 a!EachofthethreeESSsisvalidonlyinacertainregionoftheNP-plane,asshowninFigure 4{1 .Whenpredatordensityisrelativelylow(RegionI,Figure 4{1 ),theonlyESSisforpreytoforagewiththemaximaleortofc=1.Whenpredatordensityishighrelativetothepreydensity(RegionII,Figure 4{1 ),theonlyESSisforpreytoforagewitheortofc(N;P)=y1(N;P).Thevalueofy1throughoutmostofRegionIIisgenerallylow(1),althoughc=y1agreeswithc=1attheboundarybetweenRegionIandRegionII.AtallpointsintheNP-planeofintermediatepredatordensity(RegionIII,Figure 4{1 ),eachofc=y1;c=y2andc=1isanESS.WhenwerefertoaparticularESS,c(N;P),itiswiththeunderstandingthatateachpoint(N;P)oftheNP-planechasawelldenedvaluethatisamongthepossiblevaluesgivenabove.TheexistenceofmultipleESSvaluesinRegionIIIimpliesthattheredoexistcomplicatedESS,becausethecriteriondoesnotrequirethatjustoneofc=y1;c=y2andc=1mustapplyuniformlytoallpointsinRegionIII.Onesuchcomplicated,andperhapsunlikely,strategyisdepictedinFigure 4{2 ,whereRegionIIIisdividedintomanysubregions,witheachsubregionassociatedtooneofthethreepossibleESSs.Forsimplicityinthesubsequent

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Figure4{1. SolutionsfortheESSforcineachofthreeregionsofthenondimensionalizedNP-plane(s=ahNandr=a bP).InbothRegionIandRegionIIthereispreciselyoneESSfunction.InRegionIIItherearethreepossibilitiesforanESS.Fortheexpressionsfory1andy2intermsofrands,seeAppendixA. analysiswewillconsiderthreebasicESSs,oneforwhichc=y1ischosenuniformlyforallpointsinRegionIII,andsimilarlythoseforwhichc=y2andc=1arechosenuniformlyinRegionIII.NomatterwhichofthethreepossibleESSsischosenuniformlyforRegionIII,therewillbeadiscontinuityoftheESSfunction.Ifc=1ischosenforRegionIII,thenthereisadiscontinuityatallpointsontheboundarybetweenRegionsIIIandII(Figure 4{3 );ifc=y1ischosenforRegionIII,thenthereisadiscontinuityatallpointsontheboundarybetweenRegionsIIIandI;ifc=y2ischosenthenthereisadiscontinuityatallpointsonbothboundariesofRegionIII.Wenotethatthestrategyc=y2(inRegionIII)isastrategythatiscountertointuitioninthesensethatforxedpreydensityN,asPincreasestheny2increasesinvalue,sothatapreyindividualthathasadoptedthestrategyofc=y2inRegionIIIwouldincreaseitsforagingeortasthepredatordensityincreases.

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Apreyindividualthathasadoptedthestrategyofc=y1inRegionIIIwoulddecreaseitsforagingeortasthepredatordensityincreases.Foraxedpredatordensity,similarcharacteristicsofthethreeESSfunctionsareobservedaspreydensityincreasesfromN=0(Figure 4{3 ). Figure4{2. AcomplicatedESSfunction,whereRegionIIIissplitintomanysubregions,witheachsubregionassociatedwithoneofthethreepossiblebasicESSs. Thefunctionalresponseofpredatorswillbeverydierent,dependingonthestrategythatpreyemployinRegionIII.Consideringeachofy1;y2;and1asastrategyemployeduniformlyinRegionIIIbytheprey,thetypeIIfunctionalresponsesappearasshowninFigure 4{4 .Thechoiceofc=1whileinRegionIII,naturallyyieldsaresponsethatisequivalenttoastandardtypeIIresponse.Thechoiceofc=y2howeveryieldsafunctionalresponsethatisoppositeintrendtothestandardresponse-increasingpreydensityresultsindecreasedkillrateforthepredatorswhileinRegionIII.

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Figure4{3. ThethreebasicESSfunctionsdeterminedbywhichofthethreestrategiesischosenuniformlyinRegionIII.Top(c=1),bottomleft(c=y1),bottomright(c=y2).PlotsoftheESSfunctionsareshownonthenondimensionalizedNP-plane(s=ahNandr=a bP). 4{2 inderivingtheESScriterion.Inthissectionwetakethebasicecologicalmodel(Eqs. 4{2 and 4{3 )andreplacetheforagingeortcintheseequationswithoneofthethreebasicESSstrategies.Thustheright-handsidesofequations( 4{2 )and( 4{3 )arenowformulatedasthree-partfunctions,sincetherearethreefunctionalformsforabasicESSforc,dependingonwhichofthreeregionsoftheNP-planethepoint(N;P)liesin.InRegionIIIwechoosejustoneofthethreepossibleESSstoincorporateintothesystem,andwelookateachofthesethreechoicesinturntocomparethe

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Figure4{4. ApparentfunctionalresponsesofpredatorswhenP=10,a=1;h=1;mN=0:1;mP=0:1.Inthisparameterregion,therearethreeESSs(Figure 4{1 ).ThesefunctionalresponseswereplottedassumingthatthethreeESSvaluesaredistinctstrategies.Left:c=y1.Middle:c=y2.Right:c=1. eectsontheecologicaldynamicswiththesechoices.Conceptually,wearenowlookingatcommunitydynamicswiththeassumptionthatevolutionhasalreadytakenplaceandhasarrivedatoneofthethreebasicESSs.Theanalysisinthissectionincludesequilibriumandstabilityresultsofthecommunitydynamics.Foranyxedpositivevaluesoftheparametersa;h;mNandmP,ifbothbandaresucientlylarge,thenthereisexactlyonenonzeroequilibrium(N;P)possibleandthisequilibriumisguaranteedtooccurforoneofthethreechoicesofESSinRegionIII.Thisnonzeroequilibriumislocallystableonlyifbothbandarefurthersucientlylarge.Inparticular,withbandsucientlylarge,alocallystableequilibriumwilloccurifandonlyifc=y1istheESSinRegionIII.Intheresultslistedbelow,forgivenstrategyc(N;P),theecologicalequilibrium(N;P)isrecordedalongwiththeeortcevaluatedatthisequilibrium,i.e.c=c(N;P).

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1. IfbmP,thenanequilibriumisguaranteedtoexistifc=1ischosenforRegionIII.ThisequilibriumisgivenbyN=mP 3. Ifb>2mNandmP< h<2mP,thenanequilibriumisguaranteedtoexistifc=y2ischosenforRegionIII.ThelocationofthisequilibriumandthecorrespondingvalueoftheESSatequilibriumisgivenbyN=mPb2 4. Ifb>2mNand h>2mP,thenanequilibriumisguaranteedtoexistifc=y1ischosenforRegionIII.ThelocationofthisequilibriumandthecorrespondingvalueoftheESSatthisequilibriumisgivenbyN=mPb2

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Table4{1. Equilibriumanalysis.ThecspeciedisthechoiceofESSinRegionIIIthatguaranteestheexistenceanonzeroequilibrium. h2mP h>maxf2mP;mP+1 2mNg hmP+1 2mN,thentheequilibriumislocallystable.Thelocationofthisequilibrium(N;P)maylieineitherRegionIIorRegionIII,dependingonthevaluesofb;=h;mN;mP.Table1summarizestheequilibriumanalysisresults. 4{2 and 4{3 weaddthefollowingdierentialequation,whichisEq. 4{1 appliedtothetnessfunctionoftheprey.dc dt=gb2acP MatsudaandAbrams 1994 )assumesthatccantakeonanarbitrarilylargevalueandsothesystemofthreedierentialequationsalwayshaswelldenedsolutions,although

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solutionswillgenerallyhaveunboundedcvalues.Inourmodel,weassumethatthevalueofcisrestrictedtoc2[0;1],andsowerestrictthethirddierentialequationtolimitthegrowthofc. dt=8>>><>>>:gb2acP dt<0,wheredc dtisgivenbyEq. 4{4 .Inastabilityanalysisofthesystem(Eqs. 4{2 4{3 4{4 ),solvingthetheequationdc dt=0easilyconrmsthattheequilibriavaluescfortheQGsystemarepreciselythesameastheESSsderivedbytheESScriterioninsection3.1.Thuswend,asexpected,thattheequilibria(N;P;c)fortheQGdynamicalsystemarethesameasthosederivedinsection3.2fromanalyzingtheecologicaldynamicswithESSinserted.

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Figure4{5. Evolutionarydynamicsofforagingeort(dc=dtversusc)inRegionI(left),RegionIII(middle),andRegionII(right),undertheassumptionoffastrateofevolutiong. WhichofthethreeESSinRegionIIIisfavoredbyevolutioncanbedeterminedbyexaminingthephaseplaneforcwithxedNandP.WemayassumeNandPtobeessentiallyconstantascevolves,becausewehaveassumedafastrateofevolutiong.Aphaseplanediagram(plotofdc=dtversusc)withvalueof(N;P)inRegionIIIisshowninFigure 4{5 .ThediagramshowsthatinthedynamicsoftheQGmodel,c=y1isstable,c=y2isunstableandc=1isstablewhenthesystemisinRegionIII.(Notethatc=1isboundedabove,sothatithasnowheretoevolvebutdown,butthedynamicsofcwillcauseanysmallperturbationtoalesserc-valuetoquicklyreturntoc=1.)Inparticular,ifthesystemisinastatesuchthat(N;P)isinRegionIII,thenavalueofcthatisgreaterthany2(N;P)willquicklyevolvetoc=1whileavalueofcthatislessthany2(N;P)willquicklyevolvetoc=y1(N;P).

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WhichofthethreeESSsisevolvedtowhileinRegionIIIisdependentonthetrajectoryoftheecologicalsystem.IfthetrajectoryofthesystemintheNP-planeentersRegionIIIwithtraitvaluecgreaterthany2,thenthetraitwillconvergetotheESSc=1,whileifthetraitvalueisbelowy2whenenteringRegionIII,thencwillevolvetoc=y1.Itiswellknownthatthepredator-preydynamicsofaLotka-VolterrasysteminvolvescounterclockwisetrajectoriesintheNP-plane.Thesameistrueofthissystem.Unlesstheparametersaresuchthatoneorbothspeciesaredyingout,trajectoriesproceedinacounterclockwisefashionintheNP-plane.Thismeansthatiftheinitialstateofthesystemissuchthat(N;P)liesinRegionI,theonlywaythattheresultanttrajectoryofthesystemintheNP-planemayenterRegionIIIisbycrossingtheboundarybetweenRegionIandRegionIII.Ingeneral,spiraltrajectoriesthatpassthrougheachofthethreeregionsproceedincounterclockwisecyclicorderof(I,III,II).Asnotedbefore,anytrajectorythatpassesthroughRegionIIquicklyevolvestoc=y1whileinRegionII,andanytrajectorythatpassesthroughRegionIquicklyevolvestoc=1whileinRegionI.Thusforanyinitialstate(N;P;c),unlessthetheecologicaldynamicsaresuchthattheresultingtrajectoryconvergestoanecologicalequilibriumwithouteverenteringRegionI,itisnecessarilythecasethatthetrajectorywillenterRegionIIIwithitstraitvaluexedatc=1.Asageneralprinciple,wecansaythatintheQGmodelwithfastg,anytrajectorythatinvolvesaspiraloracyclepassingthroughRegionItakesonthevaluec=1whileinRegionIII.ItwasshowninSection3.2thatstabilityoftheecologicalsystemispossibleonlyforcertainvaluesoftheparameters,andonlyiftheESSbeingusedhasc=y1xedinRegionIII.Theaboveanalysisshowshoweverthateveniftheparametersarefavorabletoecologicalstability,iftherateofevolutionisfast(largeg),thenthepotentialecologicalstabilitymaynotberealized,becausethesystemevolvestoc=1wheneverRegionIIIisenteredonatrajectorythatpassesthroughRegionI.

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Ontheotherhand,itisalsopossiblethatthetrajectoryofastableecologicalsystemmaybecontainedentirelywithinRegionsIIandIII,inwhichcaseafastrateofevolutionwouldnotaectthestability.Figure 4{6 showstypicaltrajectoriesinasimulationofthebaseecologicalsystemwhenthethreedierentESSsinturnwerexedinRegionIII.(Thesimulationshownisforthebasesystemwithoutthedynamicsofcincorporated.)ForthesimulationinFigure6,wherec=y2isxedinRegionIII,thereareperiodicoutbreaksofprey.Wherec=1isxedinRegionIII,thesystemsystemisunstablewithoscillationswithveryhighmagnitude(preygrowthismuchmorerapid).Wherec=y1isusedinRegionIII,theresultisalimitcyclethatpassesthrougheachofthethreeregions.ThisstablelimitcycleispossibleifweregardevolutionashavingalreadyoccurredandfurtherconsiderthatevolutionhasterminatedwiththebasicESSthatxesc=y1inRegionIII.ButthissamecycleisnotpossibleinthedynamicsoftheQGmodelwithfastg,becausethecyclepassesthroughthethreeregionsincyclicorder(II,I,III),thusresultinginthexingofc=1inRegionIII.Anotherinterestingconsequenceoftheaboveanalysisoftheevolutionarydynamicsisthatwehaveidentiedthebasicstrategyc=y2asunstableinthedynamicsofevolution.Thuswehave,atrstglance,theseeminglyparadoxicalexistenceofanunstablestrategythatisevolutionarilystable.Thepossibleconfusionliesinthetwowaysthattheword\stable"isbeingusedinthissentence.Thestrategyc=y2isstablerelativetoinvasionbymutants.IfthegeneralpopulationadoptsthebasicESSstrategythatxesc=y2inRegionIII,thenthepopulationisnotinvadablebyasmallnumberofmutants.Howeverthisstrategyisnotstablerelativetosmallshiftsinthebehaviorofthegeneralpopulation.Ifbysomehappenstancetheentirepopulationexperiencedasmallshiftinbehavior,dueforexampletoenvironmentalchangeortoalargescalemutation,thentheentire

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Figure4{6. Simulationofthedynamicsofpredatorsandpreyplottedontheeortdiagram(Figure 4{1 ).s=ahNandr=(a=b)P.Thegraylineindicatesr=p populationwouldevolveawayfromthestrategyy2towardsoneofthedynamicallystablestrategieswithc=y1orc=1inRegionIII.Similarly,wecansaythatthebasicESSthatxesc=1inRegionIIIisstablerelativetosmallshiftsinthebehaviorofthepopulationwithfastg,butthisbehaviordoesnotallowecologicalstability.

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ecologicaldynamics.Toattempttocalculateallpossibleeectsofslowevolutiononthecommunitydynamicswouldbemoredicultforthismodel,becausewewouldneedtocharacterizethoseforagingfunctionsc(N;P)thatallowforanecologicalequilibrium,andthendetermineforwhichoftheseinitialforagingfunctionstheQGdynamicswillmaintainstabilityofthecommunityasevolutionofcoccurs.Wewillleavesuchcalculationsforafuturepaper,butnotethatitisnotimplausiblethatslowevolutioncouldleadtothedestructionofecologicalstability.ItisalsoplausiblethatslowevolutioncouldconvergetoanESSthatsupportsecologicalstabilityandthatismorecomplicatedthanoneofthethreebasicESSs(i.e.anESSwhichispiecewisedenedonseveralsubregionsofRegionIII,asinFigure 4{2 ) 4{1 ).Amongtheseweidentiedonesimplesuchstrategythatallowsastableecologicalequilibrium.However,asystemthatisinitiallyinecologicalequilibriumandemployingthisESS,maywellevolveunderthequantitativegeneticsmodeltoanESSthatdestabilizesthecommunitydynamics.Evenwithoutconsideringtheissueofecologicalequilibrium,itisapparentthatdependingonwhichofthemultipleESSisadopted,communitydynamicscanbeverydierent(Figure 4{6 ),andsoweneedtobecautiousabouttheimplicationsderivedfromresultsofthesemodels.

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MostcommunitymodelswithacomponentofadaptivebehaviordonotincludeintraspecicinteractionssuchastheoneimposedbyatypeIIfunctionalresponse.Thisstudyshowsthattheinclusionofoneofthemostcommonlyusedfunctionsincommunityecology(i.e.,typeIIfunctionalresponse)inducestheexistenceofmultipleevolutionarilystablestrategiesofadaptivebehavior.AnyoneoftheseESSisbydenition,astrategythatisstablerelativetoinvasionbymutants.However,notallESSsareattainableintheevolutionarydynamicsthatarederivedfromspecicassumptionsabouthowgenesinuencebehavior.TheTypeIIfunctionalresponseisgenerallyconsideredtobeadestabilizingfactorincommunityecology( Murdochetal. 2003 ).Indeed,thesameLotka-Volterrasystemasusedinthisstudy,butwithouttheadaptiveforagingbehaviorcincorporated,isknowntobeagloballyunstablesystemwithoutlimitcycles.TheresultofthestabilityanalysisofoursystemgivessomeevidenceforthenotionthatinclusionofadaptivebehavioralongwithaTypeIIfunctionalresponsemayhaveastabilizingeectoncommunitydynamics.TheanalysisoftheESSsforthismodelwasconnedtowhatwereferredtoasthethree\basic"ESSs,thatisstrategiesthatuniformlyxoneofthethreefunctionalformsforanESSthroughoutRegionIII.However,theexistenceofthesethreebasicESSsimpliesthattherearetheoreticallyinnitelymanyESSsforthemodel(e.g.,Figure 4{2 ).Theanalysisofapparentfunctionalresponse(Figure 4{4 )wasdonejustforthethreebasicESSs.However,acomplicatedESSsuchasisshowninFigure 4{2 isapossiblebehavior,andifpreyweretoadoptsuchastrategy,thefunctionalresponsewouldappearveryerratic.Thecommunitydynamicsassociatedwithsuchanirregular,non-basicESSwouldalsobeveryhardtopredictoranalyze.Thebiologicalfeasibilityofastrategysetisdiculttoassesswithcondence.However,itisimportanttonotethatsimpleconsiderations(i.e.,typeIIfunctionalresponse)inasimplepredator-preymodelledtopotentially

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innitelymanyESSs,suggestingthatanalysisofbehavioraldataonactivitylevelcanbeverydicult.Thepresentstudyalsohasimplicationsforfunctionalresponsestudies.IfmultipleESSsarepossible,theapparentfunctionalresponsecanlookverydierentdependingonwhichESSisemployedbytheprey(Figure 4{4 ),eventhoughtheunderlyingmechanismofpredatoractivityisineachcasethetypeIIfunctionalresponse( Holling 1959 ; Royama 1971 ).WhenfacedwithexperimentalfunctionalresponsedatathatdiersfromtheclassicalTypeIIcurve,atypeIIfunctionalresponsecanstillbetted,providedthatinformationabouttheaboutactivitybehaviorofthepreycanbeincorporatedintothemodel.Furthermore,althoughcertainfunctionalresponsesmaylookunfamiliar(e.g.,Figure 4{4 ,middle),withoutcarefullyexaminingtheintraspecicbehaviorofpreyintheeld,weshouldnotdismisssuchresponsesaspossibilities.Forexample,jumpingspidersareknowntostayintheirretreatevenwhentheyarestarved(Okuyama,unpublishedmanuscript).Although,themechanismbehindthisbehaviorisstillunknown,ifweweretoestimateafunctionalresponseofpredatorsofthesejumpingspidersbyincludingtheinactiveindividualsintheanalysis,wemayseearelationshipthatisverydierentfromthecasewhenonlyactiveindividualsareusedintheanalysis.Inlaboratoryexperiments,suchinactivebehaviorsofpreyareoftennotrecoveredduetotheuseofasmallarena,andsosuchanalysesmayarticiallyleadtotheusualtypeIIfunctionalresponse.Itisimportanttoexaminehowactivitylevelisreallyexpressedinanaturalenvironment.Inmanytheoreticalinvestigationsofadaptivebehavior,noupperboundisimposedonthetraitvalue(e.g. MatsudaandAbrams 1994 ; Abrams 1992 ).Inthemodelstudiedhere,iftheforagingeortcwereallowedtotakeonarbitrarilylargevalues,thentherewouldbenoESSforcpossibleinRegionI.Atallotherpoints(inRegionIIandIII),bothofthestrategiesy1andy2wouldbeESSs,

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andy2wouldtakeonverylargevalueinRegionII.ThequantitativegeneticsmodelwithnoboundoncwouldalsoyielddrasticinstabilityinRegionI,sinceanytrajectorythatenteredRegionIwouldresultincevolvingtohigherandhigher,unboundedvalues.Althoughunboundedtraitvalueisacommonlyusedassumptionforitssimplicity,itcanhaveastronginuenceontheconclusionsthataremadeaboutcommunitydynamics.Inourmodel,thetraitcisadimensionlessquantityinterpretedasforagingeortandsotheonlyrealisticinterpretationiswithctakingvaluesbetween0and1. KrivanandSirot ( 2004 )investigatecommunitydynamicswithtypeIIpredatorfunctionalresponseundertheassumptionthatpreyeortisdeterminedbymaximizingpopulationtnessratherthanbyanESScriterionforindividualtness.ThedependenceofpreyeortonthedensitiesofNandPisaverydierentrelationwhenthecriterionismaximizingpopulationtnesscomparedtowhenthecriterionisESS.(Figure 4{7 ).Whenanalyzingadaptivebehaviorina Figure4{7. ForagingeortasafunctionofNandPforanEvolutionarystablestrategy(left)andforastrategythatmaximizespopulationtness(right).a=1;h=1;b=1. communityecologicalcontext,forexamplewhenconsideringtraitmediatedindirectinteractions,thecriterionfor\optimalbehavior"needstobeconsideredcarefullyandstatedexplicitly.Foraxedpreydensity,anincreasingpredatordensitywould

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increasetheeectsizeofthetrait-mediatedindirectinteractioninthecaseofanESS,butmaydecreaseitinanon-ESSsolution(Figure 4{7 ).Asfarastheantheinterfaceofevolutionandecologyisconcerned,thecurrentstudyhighlightstheimportanceofunderstandingthegeneticbasisofbehavior.Whilemoreprogressinthiseldhasbeenmadeinrecentyears(e.g., Greenspan 2004 ),westillhavelittleinformationaboutthemechanismsofbehaviorformosttraits,andthusabouthowbehaviorcomestoxation.Ithasnotbeenwellestablishedthattheassumptionsofthequantitativegeneticsapproach( Abrams 2001 )areappropriateforthestudyofadaptivebehavior.Untilthegeneticbasisofbehaviorismorewellgrounded,evolutionaryecologicalmodellingremainshighlyphenomenological,evenifthemodelisbasedonthemechanisticgeneticsargument.Whileresearchinbehavioralgeneticsisalreadyrecognizedasanexcitingresearchfront,theeldofcommunityecologyalsoawaitsitsexcitingprogress.

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1+sy2:ThentheESSproblemisequivalentto:FindE2[0;1]suchthatF(x;E)0forallx2[0;1].ItisclearforanyE,thatF(E;E)=0,sotheproblemisequivalentto:FindE2[0;1]suchthatminx2[0;1]F(x;E)=0.Foranyxedy-valueE,thefunctionF(x;E)isquadraticinxsoitiseasytolocatethevaluex0thatyieldstheminvalueforF(x;E).Thatvalueisx0=(1+sE2) 2r:Therearetwocasestoconsider,becausethevalueofx0mayormaynotlieintheinterval[0;1].Inparticular,x02[0;1]()E2(2r1)=s. 1. IfE2(2r1)=s,thentheminvalueofF(x;E)occursatx=x0,andsotheproblemrequiresinthiscasethatF(x0;E)=1 4r(1+sE2)((1+sE2)2rE)2=0:IfEistobeanESSinthiscase,thenEmustsatisfysE22rE+1=0: IfE2(2r1)=s,thentheminvalueofF(x;E)occursatx=1,andsotheproblemrequiresinthiscasethatF(1;E)=(E1)((1+sE2)r(1+E)) (1+sE2)=0:Itisnothardtoshow,withtheassumptionE2(2r1)=s,thattheonlysolutionofthisequationforEintheinterval[0;1]isE=1.

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Insummary,supposeEisanESS.Thenthetwocasestoconsiderare: 1. IfE2(2r1)=s,thenEmustsatisfysE22rE+1=0: IfE2(2r1)=s,thenEmustsatisfyE=1:NextconsiderthepolynomialequationgiveninCondition(1).Thesolutionsoftheequationsy22ry+1=0are:y=rp s:ThusanESSforcondition(1)ispossibleonlyifr2s.Furthermore,ifwelety1=rp sandy2=r+p sthenbothy1;y2arepositiveandwehavethefollowingrequirementsfory1andy2tolieintheinterval[0;1]: 2,theny1istheonlyESS.(y2>1inthiscase.) 2,theny1
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2,theny1istheonlyESS.(y1<1andy2>1inthiscase.) 2,theny=1istheonlyESS.(Bothy1;y2areeither>1ornotrealinthiscase.)

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WernerandPeacor 2003 ; Bolkeretal. 2003 ).Twowell-knownphenomenaincommunityecology,trophiccascadesandkeystonepredation,illustratetheimportanceofbothtraitanddensityeects( Schmitz 1997 ; Wissingeretal. 1999 ; Schmitzetal. 2004 );TMIIcanalsopromotecoexistenceinecologicalcommunities(e.g., Damiani 2005 ).EcologistshavequantiedthestrengthsofTMIIandDMIIinavarietyofsystems( WernerandPeacor 2003 ; Preisseretal. 2005 ),typicallyconcentratingontherelativestrengthsofthetwotypesofindirectinteractions,andtheireectsonlong-termcommunitydynamics( KrivanandSchmitz 2004 ; vanVeenetal. 2005 ).Inorderforustomakeprogressinthisarea,however,wemustquantifyindirectinteractionsinwaysthatareaccurate,consistentamongstudies,andconsistentwiththeunderlyingcommunitydynamics.Here,wepointoutthatthemethodsusedinpreviousstudieshavebeeninconsistentandmayinaccuratelyestimatetherelativestrengthoftraitanddensityeects,oneofthemaingoalsofthesestudies.Weexplorethestrengthsandweaknessesofdierentmetricsusingtheexampleofathree-specieslinearfoodchain(predators-foragers-resources).Predatorsbothkillforagers(densityeects)andinduceantipredatorbehaviorinforagers(traiteects),inbothcasesreducingtheabsoluterateatwhichtheforagerpopulationconsumesresourcesandthusincreasingthedensityofresources.Wendthatratio-basedmetricstypicallyquantifyTMIIandDMIImostconsistently,althoughothermetricsmay 63

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berequiredinspeciccaseswhereabsolutedierencesinresourcedensityareofinterestorwherethecommunityisobservedoveralongtimescale. 5.2.1StandardExperimentalDesignStudiesthataimtoquantifythestrengthofTMIIandDMIIaretypicallyshort-term,usuallymuchshorterthanagenerationtime,withnegligiblereproductionorregrowthofanyofthespeciesinthecommunity.Thusresearcherstypicallyquantifyindirecteectsbasedonthechangeinresourcedensitybetweenthebeginningandendoftheexperiment(whichisequivalenttothetotalresourceconsumedbyforagersiftheregrowthofresourceisnegligible).Previousattemptstoquantifythestrengthofindirecteectshaveusedsomeorallofthefollowingtreatments. 1. Thetruepredatortreatmentincludesunmanipulatedpredators,foragers,andresource,mimickingthenaturalsystem; 2. Thethreatpredatortreatmentincludespredators(orpredatorcues),inducingantipredatortraitsinforagers,butpreventspredatorsfromconsumingforagers(e.g.,predatorsaredisabledorcaged); 3. Thenopredatortreatmentcontainsonlyforagersandresources,andthuseliminatesindirecteects. 4. Thecullingtreatmentremovesforagersinawaythatmatchesthepredationrateinthetruepredatortreatmentintheabsenceofpredators.Whiletherstthreetreatmentsarestandard,cullingisrarer( PeacorandWerner 2001 ; GrinandThaler 2006 ).Wewilldiscusstheimportanceofcullingbelow;wesimplynoteherethattheaccuracyofthecullingtreatment(i.e.,thedegreetowhichitmimicsthenaturalremovalofforagersbypredators)isimportant( GrinandThaler 2006 ).Experimentersmustrecordthenumberofsurvivingforagersinthetruepredatortreatmentatfrequentintervalsandremoveforagersinano-predatortreatmenttomatchthepopulationtrajectoryinthepredatortreatment.

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JulianoandWilliams 1987 ));(2)variationinforagerstrategyovertime( Luttbegetal. 2003 );(3)dierentialmortalityduetocostsofantipredatorbehavior;and(4)intraspecicinterference.ThenwecandeneFandfastheper-forageruptakeintheabsenceandpresenceofpredators;sinceantipredatorbehaviorgenerallyreducesforagingeortoreciency,wesupposef
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Alternatively,wecanquantifytheindirecteectsbasedonproportionalchanges(ratios): TMIIr=nopred threat=F fTMIIr+=cull true=F fDMIIr=nopred cull=N nDMIIr+=threat true=N nTIIr=nopred true=FN fn(5{3)orsimilarly(asusedbyallexistingstudies): TMIIr2=1nopred threat=1F fTMIIr2+=1cull true=1F fDMIIr2=1nopred cull=1N nDMIIr2+=1threat true=1N nTIIr2=1nopred true=1FN fn(5{4)Allexistingstudiesthatusedratiobasedindiceshaveusedeq. 5{4 ratherthaneq. 5{3 .Usingtheratio-basedindices,thedierencebetween+anddisappears(e.g.,TMIIr=TMIIr=TMIIr+andDMIIr=DMIIr=DMIIr+).However,aswediscussbelow,thedierencebetweenindiceswithdierentsubscripts(e.g.,r+vs.r)canbecomeimportantinsomecircumstances.Existingstudiesvarywidely(Table 5{1 ),usingbothadditive(eq. 5{2 )andratio(eq. 5{4 )indices.Inaddition,somestudieshavecalculatedtheindicesofTMIIandDMIIdirectlyfromthecontrastsshownabove(directmethod),whileothershavequantiedTMIIusingthecontrastsbutderivedDMIIbysubtractingTMIIfromtheoverallsizeofindirecteects:wediscussthisindirectmethodfurtherbelow.

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Table5{1. ExistingstudiesthathaveexplicitlycomparedTMIIandDMII. StudyAdditive/RatioDirect/IndirectCulling HuangandSih ( 1991 )AdditiveIndirectNo WissingerandMcGrady ( 1993 )AdditiveIndirectNo PeacorandWerner ( 2001 )AdditiveDirectYes GrabowskiandKimbro ( 2005 )RatioIndirectNo WojdakandLuttbeg ( 2005 )RatioDirectNo GrinandThaler ( 2006 )RatioDirectYes metricsdenedaboveleadto TIIa=FNfn=DMIIa+TMIIa+=DMIIa++TMIIaTIIr=FN fn=DMIIrTMIIrTIIr2=1FN fn=1(1DMIIr2)(1TMIIr2):(5{5)Whileonecandecomposetotaleectsinanyofthethreeframeworksshownabove,theratioframeworkissimplest,andforsomepurposescanbesimpliedfurtherbytakinglogarithms:logTIIr=logDMIIr+logTMIIr.Furthermore,thedecompositionoftotaladditiveeectsintocomponentswithdierentsubscriptsisproblematic:wediscussthisfurtherbelow.Althoughsomestudieshaveusedratiomeasures( GrinandThaler 2006 ),thegeneralimportanceofassessingcontrastsonanappropriatescaledoesnotseemtohavebeenappreciatedasithasinthecloselyanalogousproblemofdetectingmulti-predatorinteractions( BillickandCase 1994 ; Wootton 1994 ).However,additiveindicesmaybepreferablewhenthegoalistoquantifytheabsolutechangeinresourcedepletioninsteadoftherelativesizeofTMIIandDMII.Forexample,inastudyofeutrophicationonemightwanttoknowtheabsolutechangeinphytoplanktoninalakeduetoTMIIorDMII;inthiscase,TMIIa+andTMIIawillquantifythechangeinresourcedepletionduetotheantipredatorbehaviorifwexedthethedensityofforagerstothatofthetruepredatorandnopredatortreatments,respectively.

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5{5 ).Forexample, HuangandSih ( 1991 )quantiedmetricssimilartoTMIIaandTIIaandestimatedDMII,whichcorrespondstoDMIIa+,bysubtractingtraiteectsfromthetotal.Toseetheproblem,supposethatpredatorsreducedboththeaveragedensityandtheaverageuptakerateofforagersbyaproportionp,inwhichcasewewouldprobablyliketoconcludethatthemagnitudesofDMIIandTMIIareequal.Carryingthroughtheequationsabovewithf=(1p)F,n=(1p)Nshowsthattraiteects(TMIIa)arealwaysestimatedtobe1=(1p)timeslargerthandensityestimatesinthiscase(DMIIa+).(TheproblemstillappliesifFandNarereducedbythesameabsoluteamounts|althoughitwouldbehardtointerpretthisscenarioinanycasesinceFandNhavedierentunits.)Similarly,ifonetriestouseadditivemetricswithouthavingrunacullingtreatment,onecanonlyestimateTMIIaandDMIIa+.Indirectmethodscanwork|forexampledividingTIIrbyTMIIrshouldgiveaconsistentestimateofDMIIr|butonlyinthecasewhereallthesimplifyingassumptionsstatedabove(nodepletion,nointraspeciccompetition,etc.)hold. 5.3.1BiologicalComplexities:Short-termWhatifbiologicalcomplexitiessuchasdepletionofresourcesorintraspecicinterferencedooccur?Restatingeq. 5{1 moregenerallyas

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highlightsourimplicitassumptionsabove.Forexample,byassumingthatf1=f2,weareassumingthatantipredatorbehaviorisindependentofpopulationdensity;byassumingthatF1=F2,weareassumingthatpercapitaforagingsuccessintheabsenceofpredatorcuesisindependentofforagerdensity( Luttbegetal. 2003 ).AssumingN1=N2issafeunlesssignicantnumbersofforagersdieduetothecostsofantipredatorbehavior(easilydetectedinanexperiment);assumingn1=n2maybereasonablesinceitisanexplicitgoalofthecullingtreatment.Inthestandardexperimentaldesignwithoutculling,wehavethreetreatmentswithwhichtotesttwocontrasts,andnoremaininginformationwithwhichtotestourassumptions.Thecullingtreatmentprovidesasecondpairofcontraststhatwereinitiallysupposed(eq. 5{3 )tobeequivalent.Continuinginthetraditionofthemultiple-predator-eectsliterature( BillickandCase 1994 ; Wootton 1994 ),wemaybeabletousethelog-ratioindicesandinterpretnon-additivityorinteractiontermsasevidenceforadditionalecologicalmechanisms.Forexample,wecanthinkofpreyrelaxingantipredatorbehaviorunderhighconspecicdensityasaninteractionbetweendensityandtraiteects,inboththeecologicalandstatisticalsense:thisphenomenoncouldbequantied(ifF1=F2)aslogf2=f1=logTMIIrlogTMIIr+.Unfortunately,as Peacor ( 2003 )suggested,conspecicdensitymayalsochangeforagerbehaviorevenintheabsenceofpredators,meaningF16=F2.Whiletheavailablecontrastsdonotprovideenoughinformationtodisentangleallofthepossibleeects,atleastthepresenceofaninteractiontellsusthatsomethinginterestingmaybehappening.Auxiliarymeasurementsofbehavioralproxiesforuptake,ormeasurementsofresourceuptakeatarangeofdierentforagerdensities,aremoredetailedpotentialsolutionstotheproblemofadditionalinteractions.Wehavealsoassumedsofarthattheabsoluterateofforagerconsumptionisindependentoftheamountofresourceavailable|givenenoughtime,foragers

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willreducetheresourcedensitylinearlytozero,whichmaybereasonableinsmallexperimentalarenas.Ifalternativelyforagersdepleteresourceexponentially(sowecanredeneFandfaspredationprobabilityofoneunitofresourceperforagerintheabsenceandpresenceofpredatorsrespectively),thenthechangeintheamountofresource(e.g.inthenopredatortreatmentisproportionalto(1(1F)N).Wecandeneyetanothersetofindicesinthiscaseas(e.g.) DMIIr3=log(threat) log(true)(5{7)where(threat)and(true)aretheproportionalreductionofresourceswithrespecttotheprevioustimestep.Wecallthese\log-log-ratiometrics",becausethedecompositionlogTIIr3=logTMIIr3+logDMIIr3involvestakingthelogarithmoftheresponsevariablestwice.Theequivalenceofthe+andindices,andthecleandecompositionofTIIintotraitanddensityeects,stillholdsinthiscase.Ecologicalsystemsarediverse,andwehavecertainlynotcoveredallofthepossiblescenarios.Forexample,stronglynonlineardynamics(e.g.self-competitionamongtheresource)could,likemoststronglynonlinearinteractions,leadtopeculiarresults|forexample,resourcedensitiesdroppingasforagerdensitiesorforagingeortsdecreased( Abrams 1992 ).Ifstrong,suchdynamicsshouldbeobviousfromunusualsignsormagnitudesoftheindices(e.g.F=f<1);ifweak,theycouldthrowointerpretationsofdata.Theonlypreventivemeasureswecansuggestarecommonsense(avoidusingresourceswithpotentialforsuchstrongself-suppression)andauxiliaryobservations(behaviorproxies)orexperiments(rangesofforagerdensities).

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However,indirectinteractionsclearlyactoverlongertimescalesaswell.Luttbegetal.(2003)havepointedoutthatforagerstrategiesmayvaryevenoverthecourseofafairlyshort-termexperimentwheredensitiesareheldconstant,andofcoursethedensitiesofpredators,foragers,andresourcemayallvaryoverlongertimescales.Ifwearetotrytounderstandthelonger-termdynamicsofecologicalcommunities,whetherempiricallyortheoretically,wewilleventuallyneedtothinkabouthowtoquantifyindirectinteractionsthatrunoverlongenoughtimescalesthatpopulationdensityandbehaviorvarysignicantly.IfwerunanexperimentoverTtimestepsandsimplyaddtogetherthelog-ratioindicesfrom(eq. 5{5 ),wedopreservethedecompositionofindirecteects: (threat)| {z }period-by-period6=Pt(nopred) {z }overall:(seee.g. EarnandJohnstone ( 1997 )forotherbiologicalimplicationsofthefactthatsumsofratiosarenotequaltotheratiosofsums).ThisdierencecancausealargedierenceintherelativesizesofTMIIandDMIIevenoverafairlyshortexperiment.InthiscasetheproductofTMIIrandDMIIrcomputedfromtheendpointdata(thedierencebetweenbeginningandendingresourcelevels)willnolongersatisfythedecompositiongivenineq. 5{5 ,andthe+andratioindiceswillnolongerbeequivalent(TMIIr+6=TMIIr,DMIIr+6=DMIIr).Anotherconsequenceisthatacullingtreatmentwillbe

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necessaryinordertocomparetrait-anddensity-mediatedeectsaccurately. GrinandThaler ( 2006 )foundlargedierencesbetweenTMII+randTMIIraswellasbetweenDMII+randDMIIrina3-dayexperiment;whiledierencesbetweentheTMIIindicescouldbecausedbyintraspecicinteractionsassuggestedabove,dierencesinDMIIaremoreconstrainedandmayreecttheeectsofvariationindensityandbehaviorovertime.Afewpossiblesolutionstothesedicultiesareto: vanVeenetal. 2005 ).Hereevenalittlebitofperiod-by-perioddata,evenifthesamplingfrequencyistooslowtocapturethedetailsofthedynamics,canbeenormouslyusefulforvalidatingthefunctionalformsincorporatedinthemodel.

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moreclearlyandconsistentlydecomposetotalindirecteectsintotrait-anddensity-mediatedcomponents|itisalsoclearthatsignicantcomplexitieslurkoncewegobeyondshort-term,highlycontrolledexperimentsinsmallarenas.However,thesecomplexitiesareactuallythesignatureofinterestingecologicaldynamics,representingthenextstagebeyondthenow-familiarquestionsof\aretrait-mediatedeectsdetectable?"and\whatistherelativemagnitudeoftrait-vsdensity-mediatedeects?"( WernerandPeacor 2003 ; Preisseretal. 2005 ).Wesuggestthat,asinstudiesofmultiplepredatoreects,ratio-basedindicesshouldprobablybethedefault,butthatempiricistsinterestedinquantifyingindirecteectsshould(1)considermetricsthataremostappropriatefortheirparticularsystemandquestion(e.g.additivevs.log-ratiovs.log-log-ratio,linearvs.geometricresourceconsumption);(2)report\raw"measures(e.g.resourcedensitiesorconsumptionrates)toallowreaderstocalculatedierentindicesfromthedata;(3)incorporatecullingtreatmentsintheirexperimentsandusetheadditionalcontraststotestforandinterpretinteractionsbetweentraitanddensityeects;and(4)considerrunninglongerexperiments,despitethepotentialaddedcomplexities,togaininformationonalargerandrichersetofecologicalphenomena.

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StephensandKrebs 1986 )aswellasinunderstandinghowthosebehaviorsaectcommunitydynamics(reviewedin Bolkeretal. 2003 ).Inthescenariowhereforagersadjusttheirforagingactivitylevelbasedontheirperceptionoftheenvironment,thesimplestcaseassumesasinglehomogeneousforagingarenawithaknowndensityofpredatorandprey(e.g., Abrams 1992 ).Inthesemodels,foragersareassumedtoreacttotheaveragepredationriskoftheenvironment.Thisbehaviorintroducestraitinteractionsintothecommunity( Abrams 1995 ),whichinuencethedynamicsofthecommunityinimportantmanner( WernerandPeacor 2003 ).However,predationriskcanvaryspatiallybasedonexogenousfactors(e.g.,microhabitats)( Schmitz 1998 ; Bakkeretal. 2005 )andendogenousfactors( Keelingetal. 2000 ; Liebholdetal. 2004 ).Thus,modelsassumingthatapopulationofforagersrespondingtoanaverage(i.e.,spatialaverage)riskofpredationmaygiveinaccurateresultsifanimalsrespondtospatiallyvariablelocalcues(e.g.,encounterwithapredator)( Jennionsetal. 2003 ; Hemmi 2005b ; Dacieretal. 2006 ).Spatialpropertiesofforagersandpredatorscaninuencetheresultingspeciesinteractions.Forexample,whilevisualforagerscandetectpredatorsthatarelocatedwithintheirperceptualrangeatanymoment( Cronin 2005 ),chemosensoryforagers( Cooper 2003 ; GreenstoneandDickens 2005 )maydetectthepresenceofpredatorsbasedoncuesthatmayormaynotbecloselyassociatedwithpredator'sactuallocationdependingonhowthepredator'schemicalcues 74

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traveltheenvironmentandhowlongthechemicalcuespersist.Thus,ecologicalcommunitieswithdierentspatialpropertiesmayexhibitdierentoutcomesinspeciesinteractions.Forexample,ameta-analysisbyPreisseretal.( 2005 )showedthattrait-mediatedeectsarestrongerinaquaticthaninterrestrialsystems.Whetherornotthisdierencecanbeattributedtothespatialcharacteristics(e.g.,physicalpropertiesofthepredatorcues)discussedhereisnotclear,butmostcommunityecologicalstudiesthatexaminedtrait-eectofchemicalforagersarebasedonaquaticsystems( WernerandPeacor 2003 ),suggestingthepossibilitythattheobservedtrendisinuencedbythespatialproperties.Inthispaper,Iexaminedhowspatialconsiderationmayaectthestrengthofspeciesinteractionsbyconstructingtwotypesofforagersinasimplethreespecieslinearfoodchain(resource{forager{predator).Thersttypeofforagers,GlobalInformationForagers(GIFs),representthecommonlyusedmodellingframework(e.g., Krivan 2000 )whereforagersdetecttheaveragepredationriskoftheenvironmentregardlessoftheircurrentactivity(e.g.,evenwhenforagersarehiding)ortheactuallocationsofthepredators.Thisscenariomaybeappropriateifpredatorcues(e.g.,chemical)diuserapidlyintheenvironment.Forexample,aquaticchemicalforagerscandetectpredatordensitybasedontheconcentrationofdiusingchemicalcues(i.e.,actualpresenceofpredatorsisnotrequiredtoinduceantipredatorbehavior)( HolkerandStief 2005 ).Thesecondtypeofforagers,LocalInformationForagers,onlydetectlocalpredatorcuesthatareassociatedwiththeactualpredators.LIFsdeveloptheirperceptionofpredatordensitybasedontheirexperienceofencounterswithpredators(Chapters7and8).ThedierencebetweenGIFsandLIFsisnotonlythespatialrangeoverwhichtheyestimatethepredatordensitybutalsohowtheyobtaintheinformation.WhileGIFscandetectthepredatordensitypassivelyevenwhentheystayinarefuge,LIFsmustleavetheirrefugeandsampletheenvironmenttogaininformationaboutpredators.

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Iexaminedtwoimportantdeterminantsofthefateofecologicalcommunities;directinteractions(i.e.,theperformanceofforagers)andindirectinteractions(e.g.,interactionsbetweenpredatorsandresources).Toquantifydirectandindirectspeciesinteractions,IsolvedfortheforagingeortforGIFsandLIFsthatmaximizestheirtnessundertheirrespectivebiologicalandphysicalconstraintsbyusingdynamicstatevariablemodels( ClarkandMangel 2000 ).Thesolutionswerethensimulatedinaspatiallyexplicitlatticeenvironment.Thisprocedureallowedmetoexaminetheperformance(i.e.survivalandreproduction)offoragerswithdierentsensoryproperties.Ialsoexaminedhowthesetwodierentforagingstrategiesaectindirectspeciesinteractions.Specicallydensity-andtrait-mediatedindirectinteractions(DMIIandTMII,respectively)ofpredatorsontheresourcepopulationwereexamined.DMIIistheindirecteectofpredatorsontheforagers'resourcethroughreductionsinforagerdensity,whileTMIIistheeectofpredatorsontheresourcethroughreductionsinforageractivity(i.e.,duetoantipredatorbehavior)( WernerandPeacor 2003 ). 2000 )dynamicoptimizationmodelwasdevelopedforaKKsquarelatticespacewithperiodicboundaryconditions(i.e.,edgesoftheenvironmentareconnectedtotheoppositeedges).Themodelisathreespecieslinearfoodchainwherepredatorsconsumeforagers,whileforagersconsumeresources.Eachcellisoccupiedbyapredatororaforagerorisempty.Thus,predatorsandforagershaveexplicitspatiallocations.Resourcesarerandomlydistributedacrossspace,andareinstantaneouslyrenewed{hencetheyareonlyrepresentedimplicitlyinthemodel.Predatorsandforagersreproduceattheendofone40-dayforagingseason.ThefollowingfecundityrulefromLuttbeg

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andSchmitz(2000)wasusedfortheforagers;Numberofospring=x0:75 LuttbegandSchmitz 2000 ).TheforagingeortCofforagersisdescribedbythenumberoflatticecellssearchedeachday.Ifaforagersearchesmorecells,itismorelikelytondresources,butitalsobecomesmorevulnerabletopredators.TherearesixpossiblelevelsofforagingeortCrangingfrom0to80(Figure 6{1 ). Figure6{1. Schematicrepresentationofforagingeorts.Theblackcentersquareistheforager'slocation.Thegraysquaresindicatecellsinwhichtheforagerwillseekfoodi.e.,C=0,4,12,and28fromlefttorightrespectively.ForagingeortofC=48and80canbesimilarlycharacterized(notshown). Givenaprobabilityofndingaresourceinasinglecell,andthatresourcesareassumedtobeindependentbetweencells,theprobabilityofndingaresourceforagivenlevelofeortis=1(1)C.Foragersexpendenergyonmetabolismatarateofaperday;ifaforagerndsaresource,itincreasesitsenergystatebyY.Themaximumenergystateobtainablewassetto40.Foragersstarvetodeathiftheirenergystatefallsbelow1.

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GIFs'perceptionoftheprobabilityofencounteringapredatorbysearchingonecellisProbG(predator)=P K2wherePistheactualnumberofpredatorsintheentirelatticespace(e.g.,averagerisk).LIFsbasetheirestimateofpredatorprobabilityonpastexperience:basedonthenumberofpredatorsencountered(p)whileforaginginkcellsoverthepastmtimesteps,foragerspredicttheencounterprobabilitybasedonabinomialdistributionpBinomial(k;ProbL(predator)):whereProbL(predator)istheperceptionofforagerabouttheencounterprobability.LIFshaveapriorknowledgeaboutthisencounterprobability,whichissetasBeta(;)whereandconstituteinnateknowledgeoftheforagers(i.e.,priors)abouttheenvironment.Ichoseaweakpriorthatcorrespondstoanintermediatepredatordensity(=0:01;=0:99).Thispriorisweakandisequivalenttoasinglepriorobservationinabinomialprocesswith1%ofprobabilityofencounteringapredatorforagivencell(e.g.,25predatorsintheenvironment).ThesespecicationsleadtotheposteriordistributionforProbL(predator),Beta(+p;+kp),whichisusedbyLIFstodeterminetheiroptimalstrategies.Theperceptionoftheprobabilityofsurvivingagivenforagingeort,C,forGIFsisapproximatedbyProbG(survive)=(1dProbG(predator))Cwheredistheprobabilityofbeingkilledgivenanencounterwithapredator.LIFs'perceptionofthisprobabilityisapproximatedbyProbL(survive)=(1dU)C;UBeta(+p;+kp):

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ThetnessfunctionsF(x;t)(forGIFs)andF(x;p;k;t)(forLIFs)aredenedasthemaximumexpectedreproductivesuccessbetweendaytandtheendoftheforager'slifegiventhatitscurrentenergystateisxandthatithasencounteredppredatorswhilesearchingkcellsinpastmtimesteps.ThedynamicoptimizationrulescanbedescribedbyF(x;t)=ProbG(survive)fF(x+Ya;t+1)+(1)F(xa;t+1)gF(x;p;k;t)=Z10ProbL(survive)Beta(u;;;p;k)fF(x+Ya;t+1)+(1)F(xa;t+1)gduThenwecansolvefortheoptimalforagingeortCbyusingthebackwarditerationprocedure( ClarkandMangel 2000 ).Table1showstheparametersusedforthebackwardsolutions. Table6{1. Parametervaluesusedforthesimulations.Forthedescriptionofparameters,seethetext. ParameterNotationValue LatticeK51P(foodjcell)0.05Memorym3Predationd0.5ResourcevalueY3DispersalD1,2,3,4,5Metabolisma1 6.2.1.1Directeects:performanceofforagersAfterthebehavioralsolutionsforGIFsandLIFswerefound,spatiallyexplicitsimulationswereconductedwith100foragerswithaninitialenergystateof5units.Predatorsandforagerswererandomlydistributedoverthelatticespaceatthebeginningofthesimulation.

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Thenumberofpredatorswasvariedfrom5to50inincrementsof5.Predatorsareconsideredencounterediftheyarefoundinthecellsthatweresearched(Figure 6{1 )inaccordancewiththeforagingsolution.Foragersweresettotheirinitiallocationthroughouttheseason(i.e.,theyforagedaroundarandomxedlocation)whereaspredatorsrelocateddaily.PredatorsdispersedrandomlytoanemptycellwithinaradiusofD.Foreachpossibleparameterset(Table 6{1 ),30simulationswereconducted.Attheendofeachsimulation,thenumberofsurvivingforagers,fecundityofthesurvivors,andthedepletionofresourcebytheforagerpopulationwererecorded.Thesurvivalandfecundityrepresentthedirecteectofpredatorsonforagers,whileresourceuptakeisusedtoquantifyindirecteects(discussedbelow).

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bytheforagerpopulationattheendoftheseason.Thus,thesemeasurementsareapproximationstotheactualTMIIandDMIIthatoccurinthesystemthroughouttheseason(Chapter5). 6.3.1DirectEects:PerformanceofGIFsandLIFsWhenpredatordensitywashigh(e.g.,50predators),GIFssurvivedbetterthanLIFs(Figure 6{2 ).Ontheotherhand,theaveragefecundityofsurvivingLIFswasalwayshigherthanthatofGIFs.Fitness(i.e.,theproductofsurvivalandfecundity)ofLIFswasuniformlyhigherwhenpredators'movementrangewassmall,butasthedispersalrangeDofpredatorsincreased,theiradvantageoverGIFsdiminished(Figure 6{2 ). 6{3 ).DMIIwasuniformlylargerthanTMIIinLIFs.InGIFs,therelativestrengthofTMIIandDMIIchangeddependingonpredatordispersalandresourcelevel(Figure 6{3 ).ThestrengthofDMIIdecreasedwithincreasingresourcelevelwhileitincreasedwithincreasingpredatordispersalDanddensityP.ThestrengthofTMIIforGIFswasgreaterwhenthepredatordensitywashighthanwhenthepredatordensitywaslow,butwasrelativelyunaectedbypredators'dispersalrange. KareivaandOdell 1987 ; SchellhornandAndow 2005 )),thespatial

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Figure6{2. Proportionofpreysurviving,averagenumberofospring,andtnessofGIFs(G)andLIFs(L).Numberofpredators=50. eectsseenwereduetothesamplingerror(i.e.,randomvariability)ofpredationriskinthespatialenvironment.Forexample,iftheenvironmentcontainsasinglepredator,thenalocationnearthepredatorandanotherlocationfarfromthepredatorhaveverydierentactualpredationrisk.Thisdierencediminishesaspredatordensityincreasesbecauseeverylocationbecomesclosertoapredator.Thus,samplingerrorislargestwhenthedensityofpredatorsislow.Whenpredatordensityislow,evenwhenpredatordispersalishigh,LIFshavehighertnessthanGIFs(resultsnotshown).Dispersalofpredatorsalsoactstohomogenizethepredationriskintheenvironment.Ifdispersalisunlimited,themodellosesitsspatialcharacteristics.Limiteddispersalofpredatorsenhancedthe

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Figure6{3. EectsizeforTMII(T)andDMII(D)withvariablenumberofpredators(P). samplingerrorsofpredationriskintheenvironmentandgaveanadvantagetoLIFs(Figure 6{2 ).Inaspatiallystructuredenvironment(e.g.,withlimitedpredatordispersal),GIFssurvivedbetterbutsacricedfecunditycomparedtoLIFs(Figure 6{2 ).Becausesurvivalandfecundityrepresentdirectdensityandtraiteectsofpredatorsonforagersrespectively,wecaninterpretthatthedierentmechanisms(i.e.,GIFvs.LIF)resultinthetradeobetweendirecttraitanddensityeects.ThisresultmaybeconsistentwithPreisseretal's( 2005 )meta-analysis,whichfoundthattrait-mediatedeectsarestrongerinaquaticsystemthaninterrestrialsystems.Inaquaticsystems,predatorcuesmaydiuseintheenvironmentmorereadilyand/orpersistlongerandthusforagerscannotrespondtotheactuallocationofpredators;theymustactlikeGIFs.Consequently,aquaticchemicalforagersmayexhibithighlevelsofantipredatorbehaviorevenwhenactualpredationriskislow.

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Theeectofthespatialstructureonindirectinteractionswaslarge(Figure 6{3 ).Likedirecttraiteects,trait-mediatedindirectinteractionsweregenerallystrongerinGIFsthaninLIFs.Inparticular,TMIIofLIFsisalmostnegligiblethroughouttheparameterspace,indicatingantipredatorbehavioralonedoesnotproducemucheect.Thisisbecausepredationprobabilityusedinthesimulationwasrelativelyhigh(d=0:5).Underthehighlyecientpredators,LIFsbecomemoreopportunisticandthevalueofantipredatorbehaviorbecomessmall.Antipredatorbehaviorinducedthroughexperience,asinLIFs,hasvalueonlywhenforagershavesucientlygoodchanceofsurvivingtheencounter( Sih 1992 ).Whentheprobabilityofsurvivinganencounterissmall,thereislittlechanceoflearningfromtheexperience.Ifthepredationriskislowered(d=0:25),theeectsizeofTMIIincreases,butthegeneralcharacteristicdiscussedhereisnotaectedbythischange.InLIFs,DMIIwasalwaysstrongerthanTMII.Ontheotherhand,inLIF,therelativemagnitudeofTMIIandDMIIweresensitivetothepredatordispersal,theresourceavailability,andnumberofpredators.GIFschangetheirbehaviorbasedonthenumberofpredatorsintheenvironment,notwherepredatorsarelocated,thuspredatordispersaldoesnotaecttraitexpression.Ontheotherhand,predatorswithahighdispersalabilitycanmoreeectivelydepleteforagersintheenvironment.Therefore,whenpredatordispersalishigh,parameterregionwhereDMIIisgreaterthanTMIIbecomeswide.Previously,therelativesizeofindirecteectswerediscussedpotentiallyasanimportantindexthathelpsdeterminecommunitystability( WernerandPeacor 2003 ).Iftrue,thisresultindicatesthatweshouldconsiderspatialstructureaswellastheperceptionofforagersinthesemodelsastheycanqualitativelyaltersucharelationship.Theresultsfromdirectandindirecteectsalsohaveimplicationsforexperimentaldesignsthatarecommonlyusedtoquantifytraiteects( Werner

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andPeacor 2003 ).Inexperimentswhereforagersdetectpredatorsbasedonthecuesthatmayspreadfarfrompredators(e.g.,GIFs),articialarenasmaycausepreytoexaggeratetheirtraitexpression.Forexample,inaquaticsystemwithchemosensoryforagers,antipredatorbehaviorsareoftenstudiedbyintroducingwaterthatheldpredatorspeciesbecauseitcontainschemicalcuesusedforidentifyingtheexistenceofpredatorsbyforagers( HolkerandStief 2005 )orintroducingcagedpredators( AnholtandWerner 1998 ).However,nostudyhasexaminedhowthechemicalcuediusesinwaterorhowrapidlyitdecays.Thusalthoughthereisevidencethatwaterthatcontainedmorepredatorsismoreeectiveininducingantipredatorbehavior( HolkerandStief 2005 ),emergingspatialinteractionswillbestronglyaectedbysuchunknownphysicaldetails.Forexample,chemicalforagersinaquaticandterrestrialenvironmentwouldmediateverydierenttraiteectsbecauseofthedierencesbetweenthephysicalpropertiesofwaterandair.Ifthecueisquicklyhomogenizedintheenvironment,thesystembecomessimilartoGIFsexaminedinthispaper.Relativelysmallarenasusedinexperimentsmaypotentiallycreateabiasbecauseitprohibitsforagersfrommovingtoareaswherethechemicalcueisabsent(e.g.,eventuallypredatorcuesmayllupthearena).Todate,mostcommunitymodelswithadaptiveforagingbehaviorshavenotincorporatedspatialstructure( Abrams 1993 ; FryxellandLundberg 1998 ; Krivan 2000 ; Abrams 2001 ).Thus,wedonotunderstandhowtheseadaptivebehaviorsresultincommunitydynamicsinaspatiallyexplicitenvironmentorthepossibleroleofthephysicalenvironment.Furthermore,becauseconventionalnon-spatialmodelsgiveresultssimilartoGIFs,itispossiblecurrentgeneralunderstandingabouttheeectoftraitchangeoncommunitydynamics( Bolkeretal. 2003 )mayapplyonlytospecicscenarios.Behavioristshavelongknownthatphysicalenvironmentaectsbehaviorthroughsensorymechanisms( Endler 1992 ),and

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thesespecicitiesindeedseemtoactdistinctivelyinrealecosystems( Preisseretal. 2005 ).Althoughmorestudiesareneeded,investigationofadaptivebehaviorthroughsensoryconstraintsmaybeafruitfulwaytofurtheradvancetheinterfaceofadaptivebehaviorandcommunitydynamics.

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LimaandDill 1990 ; Eisneretal. 2000 ; Mappesetal. 2005 ; Caro 2005 ).Onecommonantipredatortraitisvigilancebehavior,whereanimalsincreasetheirabilitytodetectpredatorsatthecostofreducedresourceintake(e.g., Bertram 1980 ; Beko 1995 ; BednekoandLima 2002 ; Randler 2005 ).Communityecologistshavebeenincreasinglyinterestedinthistypeofbehaviorbecauseitisknowntoaectthedynamicsofecologicalcommunities( WernerandPeacor 2003 ; Bolkeretal. 2003 ).Mostcommunitymodelswithadaptivebehaviorincludeavariabledescribingthelevelofforagingeort.Thenatureofthisvariablevariesamongstudies.Somestudiesarevagueaboutforagingeort( MatsudaandAbrams 1994 ; Abrams 1992 ; LuttbegandSchmitz 2000 );someidentifyitwithameasureofforagingintensitysuchassearchspeed( LeonardssonandJohansson 1997 );andothersdeneitasfrequency,thefractionoftotaltimeavailablethatforagersspendforaging( Abrams 1990 ).Empiricalstudiesindicatethatforagerscanrespondtoenvironmentalcuesbychangingbothintensityandfrequencyofforaging( JohanssonandLeonardsson 1998 ; Anholtetal. 2000 ).Thecommunityimplicationsofpredator-inducedchangesinforagingeortdependonwhetherforagerschangetheirintensityorfrequencyofforaging.Indeed,oneofthecentralfociofcommunitymodelsaretheindirecteectsthatariseontheprey'sresourcesinresponsetochangesintheforager'sbehaviorinducedbythepredator. 87

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ConsideraforagerwhoseresourceintakeisdescribedbyaHolling'stypeIIfunctionalresponse( Holling 1959 ; Royama 1971 ; Jeschkeetal. 2002 ).Inthesimplestcase,therateofconsumptionandthereforetherateofresourcedepletionbyaforagingpopulationisNaR=(1+ahR)whereaandhareattackrateandhandlingtime,andNandRaredensitiesofforagersandresource,respectively.Letcbeaparameterthatrepresentstheeectofapredatorontheforagingeort.Ifpredatorsaect\intensity",thenthepredatoreectisrepresentedbyareductionintheencouragerate: 7{1 and 7{2 )canbesubstantial.Forexample,Figure 7{1 showsasmuchasa17%dierenceinforagingeortdependingonwhetherfrequencyorintensityisassumed.Thiscanalterthecommunitydynamicsqualitatively(Okuyama,unpublishedmanuscript).Mostcommunitymodelsthatallowforadaptiveforaginguseamathematicalformulationbasedonintensityratherthanfrequency(althoughthedistinctionisirrelevantinmodelswithtypeIfunctionalresponse).Thus,weneedempiricalstudiesthathelpusdierentiatebetweenthesemodelsandthusbetterguidetheoreticalapproaches.Antipredatorbehavioralsoincludesbothescapebehaviorandavoidancebehavior( Sih 1985 ).Ecologistshaverarelydistinguishedbetweenescapeandavoidance,andtheyareoftenusedinterchangeably( Hemmi 2005a ).Here,Iconsiderescapebehaviortobeadirectresponseofpreytoanencounterwithapredator(e.g.,runningaway)whileavoidanceispreybehaviorthatdecreasesthe

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Figure7{1. ResourcedepletionrateunderaTypeIIfunctionalresponsebasedonthetwodierentassumptionsofforagingeort:intensityofeort(Wintsolidline)orfrequencyofeort(Wfreq,dashedline).ParametersusedwereobtainedfromthedamseyIschnuraelegansfeedingonDaphnia(a=1.38;h=0.032; Thompson ( 1975 )).Theverticalline(atforagingeortC=0.56)correspondstoareductioninforaginginresponsetoashpredatorforthedamselfyIschnuraverticalisfeedingonDaph-nia( PeacorandWerner 2004 ).N=60;P(predatordensity)=15. probabilityofencounterwith,ordetectionbyapredator.Justaswithfrequencyandintensityofforaging,avoidanceandescapebehaviorresultinqualitativelydierentmodelpredictions.Forexample,ineqn. 7{1 ,escapebehaviorreducestherealizedattackratebypushingcbelow1.Incontrast,avoidancebehaviorwouldinducedensity-dependenteectsbecauseinorderforaforagertoescapefrombeingdetectedbypredators,itneedstoreduceitsactivitylevelaspredatordensityincreases.Mostmathematicalmodelsassumeforagerscloselytrackthedensityofpredatorsandexhibitavoidancebehavior.Onewaytoidentifythetypeofactivityexpressions(e.g.,intensityvs.frequency)andtovalidatetheexistingmodelsistoextrapolateprey'sproximateresponses(e.g.,encounterwithapredator).Decisiontheoreticalmodels( Dalletal. 2005 )describehowanimalsuseproximateinformationtoinferthedensityofpredatorsanddeterminebehavior.Atypicalmodelpredictionfromsuchamodelisthataforagerthathadencounteredapredatortwicewouldincrease

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vigilance,foraginglessintenselyinreturnforincreasedpredatordetectionwhencomparedwithaforagerthathadencounteredoneorzeropredators(e.g., LuttbegandSchmitz 2000 ).Thus,foragersinanenvironmentwithahighpredatordensitywillexhibitlowerforagingintensitythanforagersinanenvironmentwithalowpredatordensitybecauseonaveragetheyencountermorepredators.Theseproximatecue-basedmodelshavetwoadvantages.First,themechanisticnatureofproximatemodelsallowsthemtomakecontextdependentpredictionsthatotherphenomenologicalmodelscannot.Forexample,agiving-updensitymodelthatincorporatesaforager'sproximatecues(e.g.,encounterfrequency)forassessingtheremainingresourcedensitycanpredictchangesinaforager'sstrategyinresponsetothestatisticaldistributionofresourcesamongpatches( Iwasaetal. 1981 ).Inthiscase,aphenomenologicalmodelrepresentsaspeciccaseofamechanisticmodel.Second,predictionsfromproximatecuemodelscanbeusedtostudyhowanimalstranslatetheirexperienceintobehavior.Despitethesefeaturesofmechanisticmodelsandnumerousantipredatorbehaviorstudies( Caro 2005 ),weknowlittleabouthowanimalsperceivepredationriskandadjusttheirbehavior( LimaandSteury 2005 ).Asdiscussedabove,detailsofbehavior(i.e.,intensityvs.frequency,oravoidancevs.escape)cancauselargedierencesincommunitydynamics.Ecologistsneedtounderstandbehaviorinordertopredicthowtheywillscaleuptocommunitydynamics,becausecommunitydynamicsareinherentlysummariesofbehavioralprocessessuchaspredationandreproduction.Thisstudyexamineshowaninitialencounterwithapredatorinuencestheavoidancebehavior(i.e.,timetocomeoutofarefuge)ofjumpingspidersovertime.Thestudyalsoexamineshowtheirbehaviortocomeoutoftherefugeisinuencebyaneutralstimulus.Thedurationoftheiravoidancebehaviorgivessomeinformationaboutthetypeofdynamicstheymayproduce(eqn. 7{1 vs. 7{2 ).Theirresponsetotheneutralstimulusafteranencounterwithapredatorwillgive

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ussomeinformationabouttheirpotentialdecisionmakingprocessesthatcanbediscussedinadecision-theoreticalcontext( Dalletal. 2005 ). Gardner 1964 ).Theexperimentwasdoneina22factorialdesign.Oneofthefactorswaspredator(encounterorno-encounter).Thistreatmentwasestablishedtoexaminewhetherencounterwithapredatorhadaneectonthespiders'subsequentresponsetoanambiguousvisualstimulus(theotherfactor).Jumpingspiders,likeothervisualpredators( Cronin 2005 ),detectanobjectandsubsequentlyidentifyit(asaprey,predator,etc).Thus,theresponseofspiderstoanambiguousstimuluscanprovidenerinformationonthedurationofbehavioralantipredatorresponsesthantheirresponsetoanactualpreyvisualcue.Thisisbecauseanabsenceofresponsetoanactualpreyitemdoesnotnecessarilyindicatethatthateectofpredatorhasdiminished.Theexperimentalarenawasmadeoftwo2cmlongcircularrubbertubes(Figure 7{2 ).Asmallergreensemi-transparenttube(4mmininteriordiameter,1mmthickness)wasplacedinsidealarger(10mmininteriordiameter,1.5mmthickness)cleartube,withanemptyspacebetweentheinnerandoutertubes.Thisspacewasclosedatbothendswithpiecesofrubber.Oneendofthesmallertube

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Figure7{2. Schematicdiagramoftheexperimentalarena.Thearenaconsistsoftwophysicallyseparatedopenspaces:aninnerspaceandanouterspace.Theopeningsofbothspacesareclosed(coloredgray)exceptforthefrontopeningoftheinnerspace.Numbersrepresentdierenttimeperiods(steps).Step1representsthetimespidersspentatlocation1beforeapproachingtheexit,Step2isthetimespiderstookfromleavinglocation1toreachinglocation3,andStep3isthetimespidersspentatlocation3beforeleavingthetube. wasclosedwithaspongesothatonlyoneendwasopen.Ineachexperiment,aspiderwasplacedinsidetheinnertube. 7.2.1.1PredatortreatmentEachtrialstartedbyintroducingajumpingspiderintotheinnertube.Whenthespidercametothetipoftheentrance/exithole,avisualpredatorstimulus(adeadadultPhidippusaudaxfemale)wasshownbyplacingitattheexit.Intraguildpredationandcannibalism,wherelargerindividualsconsumesmallerindividuals,arecommoninjumpingspiders.Becausespiderspossessaninnateperceptiontowardsotherspiders( Land 1972 ),presentationofthedeadadultindividualresultedinspidersrunningbacktotheinnerendofthetube,whichwasthereferencepoint(location1inFigure 7{2 )forsubsequentmovement(seeBehavioralmeasuresbelow).Fortheno-predatortreatment,itwasimpossibletomakeaspiderretreatinsidethetubewithoutastimulus.Evenarticialvisualstimulisuchashuman

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ngers,forceps,andmatchstickswereinsucienttomakespidersretreatbackintothetube.Thus,tomakespidersmovetotheendoftheinnertubesothatthebehavioralassaycouldberuninacomparablemanner,amatchstickwasusedtonudgespiderstothebackofthetube. Land 1972 ).Spidersrstdetectmovementofanobjectwiththeirposteriorlateraleyes(PLE),thenorienttowardstheobjectandidentifyitwiththeiranteriormedianeyes(AME).Thus,theinitialdetectionofanobjectdoesnotallowspiderstoidentifytheobject(i.e.,thevisualstimuluswillbeambiguous)unlesstheyhavetheobjectinclearviewoftheAME.Tointroducesuchvisualstimuli,veadultourbeetles,Triboliumconfusum,werecontainedinthespacebetweenthetwotubes(outerspace)andthespacewasclosedsuchthatthebeetlescouldonlymovearoundwithinthespace(Figure 7{2 ).Becausetheinnertubewasnotcompletelyopaque,aspiderinsidetheinnerspacecouldvisuallysensethemovementoftheourbeetlesbut(arguably)notidentifythem. 7{2 ).Step1isthedurationfromthetimespiderstouchedthebackwallorwerenudgedtotheinnerendofthetubetothetimetheybegantomovetotheexit;step2isthetimethatspiderstooktomovefromtheinnerendofthetubetotheexit;andstep3isthetimeintervalfromthespiders'arrivalattheexitofthetubetowhentheyexitedthetubecompletely.Anobservercouldseethelocationofaspiderthroughtheinnerandoutertubes,althoughtheinnertubemadeitimpossibletoidentifytheorientationofanindividual.

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7{1 ,Table 7{1 ).Theambiguousvisualstimulididnotaectthetimetocomeoutwhenspidershadnotencounteredapredator,butiftheyhadencounteredapredator,theambiguousstimulicausedspiderstostayinthetubesignicantlylonger(Table 7{1 ).

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Figure7{3. Timespentinthetubeineachtreatment. Foreachtimestep,spidersshowedasimilarbehaviorwheretheyexhibiteddierentialresponsetotheambiguousstimulionlywhentheyencounteredapredatorstimulus(Figure 7{4 ).Theinteractionsbetweenpredatortreatmentandstepwerealwaysnegativeexceptforthestep2inthewithoutambiguousstimulitreatment(Table 7{2 ).Theseinteractionsweresignicantforstep3inbothcases(i.e.,withandwithoutambiguousstimuli),butonlysignicantforthetreatmentwithoutambiguousstimuliinstep2.Adetailedbiologicalinterpretationisgivenindiscussion. Table7{1. Estimatedparametersfromtherandomeectmodel.SE(standarderror)andRE(randomeect). ValueSEREt-valuep-value Intercept13.611.233.2811.04<0:01Predator10.832.258.664.79<0:01Ambiguousstimuli-0.990.148-0.670.51Interaction9.122.104.33<0:01Residual5.36

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Table7{2. Statisticalresultsforwithandwithoutambiguousstimuli.REindicatestherandomeectdescribingthestandarddeviations.ParameterswithmissingREvaluesarexedeects. Intercept6.110.720.968.45<0:01Predator8.881.384.936.42<0:01Step23.141.193.402.64<0:01Step30.581.112.560.520.60PredatorStep2-3.281.39-2.350.02PredatorStep3-4.871.39-3.49<0:01Residual3.56 Intercept6.860.941.807.28<0:01Predator13.391.605.168.38<0:01Step20.801.635.370.490.62Step30.491.373.060.350.72PredatorStep20.721.740.410.68PredatorStep39.381.745.36<0:01Residual4.46

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Figure7{4. Boxplotforthedurationofeachstepforeachtreatment. 7{1 ).Similarpredatoreecthasbeenobservedinotheranimals( Jennionsetal. 2003 ; Hugie 2004 )suggestingthatahidingbehaviorisatypicalantipredatorbehaviorinwidevarietyoftaxa.Theobservedbehaviorqualitativelyagreeswiththepredictionsofdecision-theoreticalmodels(e.g. LuttbegandSchmitz 2000 ).Inthesemodels,individualsformulatetheirperceptionabouttheenvironmentinlightofbothpriorinformation(e.g.,innatedecisionrules)andexperience(e.g.,encounters).Thisnewperception

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abouttheenvironmentiscalledtheposteriordistribution,andanimalsbasetheirsubsequentdecisionsonthisposteriordistribution( Dalletal. 2005 ).Inthecurrentstudy,ifweconsiderthepredatortreatmentasmanipulatingpriorinformation(spiders'priorestimateoftheirpredatorencounterprobabilityishigherwhentheyaregivenapredatorstimulus)andtheambiguousstimulustreatmentastheirexperience,wecaninterpretthedierenceinthespiders'responsetotheambiguousstimulusasresultingfromtheirdierenceinposteriordistributions(i.e.,spidersaremorelikelytointerprettheambiguousstimulusasapredatoraftertheencounter).Thisiswhythisstudyisdierentfromotherstudiesthatinvestigatedtheeectofpredatorsonly.Forexample, Jennionsetal. ( 2003 )examinedhidingbehavior(i.e.,timetostayinanest)ofddlercrabsafterexposingthemtoapredatorvisualcueandfoundthesignicantpredatoreectsimilartothejumpingspiders.However,withoutexaminingtheresponsetoanambiguousstimulus,itisnotpossibletoexaminehowtheperceptionwithrespecttothepredationriskmayoperateinthedecision-theoreticframeworkasdiscussedabove.Similarly,theuseofapreystimulusinthisstudyinsteadoftheambiguousstimuliwouldhavebeenineectivebecausethepreystimulusmaynotcontributeanyinformationtowardsthepriorinformationofthepredationriskunlessanimalspossesssomeinnateperceptionabouttheoccurrenceofpredatorsandprey.Theadaptivebehavioralmodelsusedincommunityecologyassumethatforagerscankeeptrackofthedensityofrelevantstimuli(e.g.,conspecics,predators,resources)inordertoadjusttheirbehavioraccordingly.Thedecision-theoreticalmechanismdescribedabovecanpotentiallyleadtosuchbehavior,buttheshorttemporaleectshownherewouldnotbestrongenoughtocreateamagnitudeoftraiteectthatisexhibitedbytypicalmodels.InChapter6,Ihaveshownthatevenwithamoderatelypersistentpredatoreect(morepersistenteectthantheresultofthisexperiment),whenanencounterwithapredatorisinfrequent,the

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modelexhibitsmuchweakertraiteectthantypicalmodels.However,thisstudyonlyexaminedthepredatoreectonthetimetocomeoutofrefuge,anditispossiblethatthepredatoreectismorepersistentonotheraspectsofbehavior(Chapter8).Furtherstudiesontheeectofpredatorsonasuiteoftraitexpressionareneededtoexaminethevalidityoftheexistingmodelsofbehaviorincommunityecology.Theinterpretationofthetwotreatments(i.e.,predatortreatmentandambiguousstimulitreatment)requiressomecaution.First,itisuncertainwhetherspiderswereunabletoidentifytheambiguousvisualstimuliasthetreatmentintended.However,suchidenticationofobjectsisunlikelybecausejumpingspidersrequireaclearcontrastofanobjectwithitsbackgroundforidentication( Land 1972 ),whichwaslikelyremovedbythesemi-transparentinnertubes.Second,theno-predatortreatmentdidnotconsistofno-stimulus,butspiderswerepushedtothereferencepointwithamatchstick.However,spidersdidnotrespondtothearticialobject(i.e.,matchstick)verystrongly,whiletheirresponsetoapredatorstimuluswasverydistinct(personalobservation).Thus,theno-predatortreatmenteectivelyproducedtheresponsesimilartothescenariowherenostimuluswasgiventospiders.Otherwise,theno-predatortreatmentwouldhavemadethepredatoreectconservativebecause,ifanything,themanipulationwilllikelytomakespidersmorecautious,butverylargetreatmenteectswereobtained.Whenthestatisticalanalysisincorporateddierentsteps,theinteraction(betweenpredatorandstep2)wassignicantonlyinthetreatmentwithouttheambiguousstimulus(Table 7{2 ).Biologically,thisnegativeinteractionbetweenstep2andthepredatorfactor(3.28,Table 7{2 )maybeinterpretedasawaningofthepredatoreect(8.88,Table 7{2 )overtime.Thus,thepredatoreectdiminishedbyabout37%inthetreatmentwhentheambiguousvisualstimuli

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wasabsent.Thiseectwasnotstatisticallydierentfromzerowhenspiderswereexposedtotheambiguousvisualstimulus.Furtherstudiesareneededtounderstandlatencyofpredatoreectasitisavitalassumptionintheoreticalmodelsthatallowforagerstotrackdynamicenvironments( LuttbegandSchmitz 2000 ).Communityecologicalmodelsincorporatingadaptiveindividualbehaviorarenowcommon,butbehavioraldetails,whichmayhavesignicanteectsoncommunitydynamics(Figure 7{1 ),haverarelybeenexamined.Thoughcommonmodelsappearsimpleandtransparent,theymakeimplicitassumptionsaboutbehavioralexpression.Wedonotknowhowwellanyspecicmodelstructurecanapproximateavarietyofbehavioralexpressions( Caro 2005 ).Torespondtothisissue,moremechanisticframeworkssuchasdecision-theoreticalmodelshavebeendeveloped( Dalletal. 2005 ).Thisstudyexaminedthetemporalcarryovereectofthepredatorencounter,whichisacentralassumptioninthosemechanisticmodels.Inordertofurtherunderstandandvalidateexistingcommunitymodelswithadaptivebehavioralcomponents,weneedtounderstandhowindividualstranslatetheirexperiencetobehavioralexpressionratherthansimplydemonstratingageneralpatternsuchasreductionsinforaginginhigh-riskenvironments( LimaandSteury 2005 ).Understandingthedetailsofbehavioralexpressionwillallowustoexaminemoreappropriatelyhowbehaviorscalesuptocommunitydynamics

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WernerandPeacor 2003 ).Inpredator-preycommunities,activitydynamicsmaycausequalitativechangesinthecommunitydynamics.Forexample,theymayallowpersistence( Kondoh 2003 )orinducepopulationcycles( AbramsandMatsuda 1997 ).Anumberoftheoreticalstudiesexaminingtherelationshipbetweenactivityandpopulationdynamics( Bolkeretal. 2003 )havesuggestedthatthepredictionsofcommunitymodelsaresensitivetothedetailsofbehavioralmodels.Vigilancebehaviorisoneofthebest-studiedmechanismsofactivitydynamics( LimaandDill 1990 ; Caro 2005 ).Typicalmathematicalmodelsofvigilancebehaviorassumethatanimalsadjusttheirvigilanceinresponsetothedensitiesofpredatorsandresources(e.g., Abrams 1992 1995 );manyempiricalstudiessupportthisassumption( StephensandKrebs 1986 ; KagataandOhgushi 2002 ; Caro 2005 ).Forexample,larvalfrogsreducetheiractivitylevelinresponsetoincreasesinresourceorpredatordensities( Anholtetal. 2000 ).However,thesestudiesquantifychangesinactivitylevelundertheinuenceofapersistentstimulus.Forexample,Anholtetal.'s( 2000 )studyusedcagedpredatorstosupplyachemicalcuethatvariedinconcentrationamongtreatment,butpersistedintime.Theactivitylevelsofpreyunderthesetypicalexperimentalconditions( AnholtandWerner 1998 ; PeacorandWerner 2001 ; HolkerandStief 2005 ),however,onlyreectthe 101

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behaviorofpreythatarepersistentlyexposedtopredatorcues(Chapter5).Suchascenarioisunlikelyforpreythatdetectpredatorsonlyintermittently(commoninpreythatdetectpredatorsvisually).Mostmodelsusedtopredictthelinkeddynamicsofbehaviorandpopulationsimplicitlyassumethatpreycanmodifytheirbehaviorinconcertwithchangesinpredatorpopulationdensities;preymustshowsustainedresponseswithrespecttopredatorcuesinordertotrackpredatordensitieswhenpredatorcuesareintermittent.Thisstudyexamineshowjumpingspiders,whichareexposedtopredatorandpreyvisualcuesduringtheday,changetheirmetabolicratesonsubsequentnights.Spidersmaintainonlyhalftherestingmetabolicrateofotherpoikilothermicpredators( Anderson 1970 1996 ),loweringmetabolismstillfurtherwhentheyexperiencepreyshortages( Anderson 1974 ).Althoughtheeectsofpredatorandpreystimulionimmediateresponses(e.g.,timetocomeoutofarefuge;Chapter7)arewellstudiedinthesespidersandotherorganisms(e.g., Jennionsetal. 2003 ),littleisknownabouttheeectsofsuchexperiencesduringthedayonlonger-termresponsessuchaschangesinmetabolismatnight.Inthisstudy,Ihypothesizedthatbecausejumpingspidersvisuallydetectobjects,visualstimulialonemaybesucienttomaintainrestingmetabolicrateofstarvingspidersabovetheexpectedlevelintheabsenceofsuchstimuli.Totestthisidea,Iexaminedhowvisualstimuliofpredatororpreyalone,intheabsenceofaccesstoprey,aecttherestingmetabolicratesofstarvingjumpingspiders.

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chamber(light:dark=12:12,theshiftsoccurringat0700and1900,temperature=25C)duringthesecondinstarandapartoftheirthirdinstarbeforebeingusedintheexperiment.Duringthistime,spiderswerefedonefruity(Drosophilamelanogaster)everythirdday.Forthethreedaysbeforetheexperimentstarted,spiderswerefedoneydaily.Allindividualswerefedonthesamescheduletoensuresimilarstarvationlevelsamongtestsubjectsatthebeginningoftheexperiment. Anderson 1974 ).Spidersinthesecondtreatment(preytreatment)wereshownprey(i.e.,receivedvisualpreystimuli)butwerenotallowedtoconsumethem.Spidersinthethirdtreatment(predatortreatment)wereshownapredatorstimulusandwerenotgivenprey(i.e.,neitherconsumptionnorvisualpreystimulus).Spidersinthelasttreatment(fedtreatment),weregivenasinglepreydailytoconsume(consumptionandvisualpreystimulus).Groupsofvespiders,representingasinglesample(seeOxygenmeasure-mentbelow),werekeptinanexperimentalarenaconsistingoftwonestedclearplasticcontainers.Asmallercontainer(62mmindiameterand43mminheight,approximatelycylindrical)wasplacedinsidethelargercontainer(107mmindiameterand78mminheight,approximatelycylindrical)sothatthewallsoftheinnercontainerphysicallyseparatedaninnerfromanouterspace,butobjectsineachspacewerevisiblefromtheotherspace.Subjectspiderswere

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keptinsidetheinnerspace.Inthenonetreatment,boththeinnerandouoterspacewereleftemptyexceptforthetreatmentsubjects.Inthefedtreatment,veD.melanogasterpreywereintroducedintotheinnerspace;althoughallpreywerealwaysconsumed,itisuncertainhowthepreyweresharedamongthevesubjects.Inthepreytreatment,veD.melanogasterwereplacedintheouterspace,visiblebutinaccessibletothesubjects.Inthepredatortreatment,oneadultfemalejumpingspiderPlexippuspaykulliwasplacedintheouterspace,visiblebutinaccessible.PlexippluspaykulliandPhidippusaudaxliveinsamehabitatsandintraguildpredationissize-dependent;thus,intheeldPlexippuspaykulliadultspreyonPhidippusaudaxjuveniles(personalobservation).Subjectspidersandthepredatorstimulusspidersreactedtoeachotheralthoughseparatedbythewallofsmallercontainer.Inparticular,subjectspidersexhibitedtypicalvigilancebehaviortowardsthepredatorstimulus(e.g.,haltingmotion,makingslowbackwardmovements,andrunningaway).Incontrast,subjectspidersoccasionallyattemptedtojumponpreythatwereseparatedfromthemintheouterspace,indicatingthatsubjectswereabletodistinguishthetwovisualstimuliinthearena.Thefourtreatmentsdescribedabovewereappliedfrom1000-1600daily.At1600,spiders(ingroupsofveindividuals)weregentlytransferredtoaplasticsyringefortheiroxygenconsumptionmeasurement(followedby2hoursofacclimationperiod,furtherdescribedbelow).Theoxygenmeasurementprocedurelasteduntilthenextmorningat0700.Thus,eachtreatmentalternatedbetweentreatmentsandoxygenmeasurement.Thisregimenwascontinuedforfourdays. Lee ( 1995 ).Therespirationmeasurementswereconductedinaroomwithacontrolledtemperatureof25.762:2C(meansd).Syringes(60ml)connectedtoaglasspipetwereusedastherespirometricchambers.Thesyringeplungerswerepushed

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insothata10mlspacewasactuallyavailabletospiders.Becauseofthesmallsizeandlowrespirationrateofindividualspiders,veindividualsweregroupedformeasurements;eachgroupofvespiderswascontainedinaseparatesyringeandwasacclimatedforapproximately2hourseachday.Althoughspidersarecannibalistic,cannibalismisrareatthisearlyinstarandwasneverobservedinthisexperiment.Afteracclimation,thepipetopeningsofthesyringeswereclosedwith15%KOHsolutionandleftforstabilization(approximately30min).Measurementsstartedat1900andendedat0700nextday.RespiredCO2intheclosedairisabsorbedbyKOHsolution,reducingthepressureinsidethechamberandcausingtheKOHsolutiontomoveintothepipet,andthusconvertedtothevolumeofoxygenconsumedbythespiders( Lee 1995 ).Therecordedvalueswerecalibratedfromthemeasurementsfromtwoemptysyringes.Alltreatmentsofallsampleswerecollectedsimultaneouslytominimizeerrorscausedbypotentialtemperatureandhumidityuctuations.Estimatesofsizeandweightsweremadefromonerandomsamplefromeachsamplegroupof5spiders(hence7samplesfromeachtreatment).Thepooledcarapacewidthsandbodyweightwere0.980.0004mmand2.260.09mg,respectively(meanse).Nostatisticaldierencesineithervariableswereobservedamongthetreatments(ANOVA,p>0:1forbothsizeandweight).

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respectively.Thenthefourmodelswere:(1)visualstimuli(eitherpreyorpredator)donotaectmetabolicrate(i.e.,None=Prey=Predator);(2)onlypreyvisualstimuliaectmetabolism(i.e.,Prey6=None=Predator);(3)onlypredatorvisualstimuliaectsmetabolism(i.e.,Predator6=Prey=None);and(4)preyandpredatorvisualstimuliaectmetabolismdierently(i.e.,None6=Prey6=Predator).ThebestdescribedmodelwaschosenbasedonAIC(=2l+2p)wherelandparethelog-likelihoodandnumberofparametersofthemodel.Allmodelswerebasedonnormaldistributionswithequalvariances. 8{1 ).Inthestarvationtreatments,individualsdecreasedtheirmetabolismsignicantlyregardlessofthepresenceorabsenceofvisualstimuli(t-tests,p<0:001forallcomparisons). Figure8{1. Averagechange(SE)inoxygenconsumptionsofspiders(l/hr/individual).Averageweightofindividualspiderswas2.26mg. Model2(onlypreyvisualstimuliaectmetabolism)wasselectedondays1and2.Onday3,thetreatmenteectdisappeared|model1(allmetabolicratesequal)wasselected.

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Table8{1. AICforeachmodel.p=numberofparameters.No.ModelpAIC(day1)AIC(day2)AIC(day3) 1None=Prey=Predator226.1330.7136.662Prey6=None=Predator321.0430.1038.493Predator6=Prey=None323.7431.9237.194None6=Prey6=Predator421.9132.0839.14 8{1 ).Thusvisualstimuliaswellasphysiologicalstate(degreeofstarvation)inuencemetabolicrate,althoughstarvationisaprimarydriverofmetabolicrate(ashasbeenshowninotherspiders: Anderson 1974 ; TanakaandIto 1982 ).Starvationrapidlyinducedlowermetabolicrates(Figure 8{1 ),consistentwithresultsfoundinwolfspiders( Anderson 1974 ).Consideringthatspiders'labilemetabolicrateisoftencitedasanadaptationtounpredictablepreyshortages( Anderson 1974 ),itissurprisingthattheirmetaboliccontrolissosensitivetobriefperiodsofstarvation|temporarypreyshortagescanoccurbychanceevenwhenoverallpreyavailabilityishigh,inwhichcaseshuttingdownmetabolismwouldbeinappropriate.Visualstimulimayaectmetabolicratesindirectlythroughchangesinactivitylevel.Forexample,spidersinthepreytreatmentmayhavebeenmoreactivethanthoseinthepredatorandnonetreatmentintryingtocaptureprey.Similarly,spidersinthepredatortreatmentmaybeminimizingtheirmotionformostofthetimeinordertoavoidpredation.Iftheseactivityshiftsoccur,spiders'proximateresponsetostimuliistoloweractivitylevels,andloweredmetabolicratesmaysimplybeabyproductofloweredactivity.However,earlierexperimentshavesuggestedaweakcorrelationbetweenactivityandmetabolicrateinotherspiders:wolfspidersexhibitdramaticchangesinmetabolicrateinresponsetofasting

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evenwhenactivitylevelwasmaintainedduringthefastingperiod( Anderson 1974 ).Jumpingspiderscanalsorecovertheiroxygenconsumptionrateaswellasremovinglactateaccumulatedduetoactivitywitharelativelyshortperiod(1hour)( Prestwich 1983 ).Thus,Isuggestthatspidersadjusttheirmetabolicratesindirectresponsetovisualstimuli,ratherthanasanindirectconsequenceofchangesinforagingbehavior.Toteaseapartthecauseandeectrelationshipbetweenmetabolismandactivityorotherfactors,amoreelaborateexperimentaldesignisrequired.Whethertheobservedresultsweredirectresponsesorbyproductsofchangesinactivity,thisstudyindicatesthepossibilitythatchangesinpreydensitiesandpredatordensitiescandrivechangesinspiderbehaviorandhencethatbehaviorandpopulationdynamicscaninteract.Forexample,evenwhenspidersfailtocaptureprey,ahighdensityofpreymayinduceahigherdegreeofforagingactivitythatmaintainahighmetabolicrates.Whileindividualswithhighmetabolicratesmustfeedmoretomeettheirmetabolicneeds,ahighmetabolicratemayalsoincreasetheprobabilityofpreycapturebyallowingenergeticallydemandingactivity( Speakmanetal. 2004 ).Thus,spidersthatmaintainahighermetabolicratemaybesuccessfulnotonlyincapturingpreybutalsoinreproducingorescapingpredation( AnillettaandSears 2000 ).Ingeneral,maintenanceofmetabolicratehasgreaterconsequencesinectothermsthaninendotherms.Thus,spidersalteringtheirmetabolicratesindependentlyofactualpreyconsumptionisanotheravenueforadaptivechangeinbehaviorasafunctionoftheenvironment( Bolkeretal. 2003 ).Thisresultsuggeststhatafailuretoobserveanobviousadaptivebehavior(e.g.,immediateresponsetopredatorencounter)doesnotprovetheabsenceofadaptivedynamicspredictedbycommonmodels.Thesustainedeectofexperience(i.e.,seeingprey)mayinduceavarietyoftraiteects.Furtherunderstandingtheecologicalconsequencesofmetabolic

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changesinresponsetoexperiencewillenhanceourunderstandingofthelinkagesbetweenbehavior,physiology,andecologicaldynamics.

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FryxellandLundberg 1998 ; Bolkeretal. 2003 ; WernerandPeacor 2003 ).Behavioraectscommunitydynamicsbyalteringthestrengthofspeciesinteractions( Abrams 1991 ; Berlowetal. 2004 ),whichinturnmodifybothfunctionalandnumericalresponsesofthespeciesinvolved.Forexample,foragersthatexpressantipredatorbehavior(e.g.byreducingtheiractivitylevel)willlowertherealizedattackrateoftheirpredators( Anholtetal. 2000 ).Ifforagerschangetheirbehavioradaptively,thisattackratemaybecomedensity-dependentandcanproducerichdynamicseveninasimplemodel(e.g., Abrams 1992 ).Oneconclusionfromthesestudiesisthatchangesinactivitylevel(e.g.,foragingintensity,Chapter7)ofanimalscanaectcommunitydynamicsprofoundly.Tomodelactivitydynamics,mosttheoreticalmodelsassumethatactivitylevelisdeterminedsolelybythetrade-obetweenresourceuptakeandcurrentpredationrisk(e.g., Abrams 1993 ; Bolkeretal. 2003 ),orbytherelativeabundancesofresourcespeciesifthefocusisontheswitchingbehavior( Krivan 2000 ; vanBaalenetal. 2001 ; Kondoh 2003 ).However,thisapproximationissensitivetoavarietyofdetails.Forexample,whenanimalsexhibitantipredatorbehaviorinthefrequencyratherthantheintensityoftheiractivity(Chapter7),orwhenspatialconstraintsareimposedonthespeciesinteractions(Chapter6),theapproximationdepartsfromthetruedynamics.Onereasonwhytheapproximationfailsinthesesituationsisbecauseactivitychangesonafastertimescalethanthemodelscapture.For 110

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example,commonmodelsusuallypredictasingleoptimallevelofforagingactivitywithrespecttoeachdensityofpredators(e.g., Abrams 1993 ; KrivanandSirot 2004 ).However,innature,individualschangetheirbehavioratamuchfastertimescaleinresponsetoenvironmentalcues(e.g.,Chapter7).Apreyindividualmayexhibitavarietyoflevelsofactivityinonehouroreveninaminutewhilethedensityofpredatorsmaystayrelativelyconstantforlongperiods(e.g.,weeks).Nevertheless,animplicitassumptionofthemodelsisthatbehavioraldynamicsthatmattertothemodelareonlyasfastasthedynamicsofthecommunity;thecommonmodelsignorebehavioralvariationthatoccursatafasttimescale(oddly,sincethesemodelswereinspiredbytheneedtoincludeprocessesthatactedatafasttimescale).Thesefastbehavioraldynamics,however,maysignicantlyaectcommunitydynamics(Chapter10)andneedtobedescribedinamannerthatcanbeextrapolatedtolong-termcommunitydynamics.Thispaperinvestigatedjumpingspidersintheeldtogainabetterunderstandingoftheiractivitydynamics.Becausedemographicdynamicsofjumpingspidersoccuronamuchslowerscalethanthisstudyaddresses( Edwards 1980 ),thedetailsofactivityIdescribeinthisstudyareexamplesofthedetailsthatareignoredbytheconventionalframeworkofcommunitymodelswithadaptivebehavior.Inparticular,Iwilldemonstratethatthesespidersemployabiphasicactivitypatternwhereindividualsalternatebetweenperiodsofactivityandinactivity.Inaddition,Ishowthatthisactivitypatternisnotsimplydescribedbytheirfooddeprivationlevel.Asimplebehavioralmodelthatpredictstheobservedpatternintheeldwasexaminedwithlaboratoryexperimentsandcomputersimulations. Land 1972 ; Hill 1979 ; Lietal. 2003 ).Theyactivelyforage(theymovetondpreyratherthanbuildingwebstocaptureprey)butdoproducesilkinordertoconstructretreatsforshelterat

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nightandwhennotactivelyforaging( HoeerandJakob 2006 ).Wecategorizedindividualsthatwerefoundintheirretreatasinactiveandindividualsthatwerefoundoutsidearetreatasactivebasedonourdaytimeobservations.Thisclassicationisonlyapproximate,becauseindividualsthatwerefoundintheretreatmayhavebeeninactivefordierentamountsoftime(e.g.,theymayhavebeenactivejustpriortotheobservation).Toobtainsomeideaabouttheirpreviousactivity,\fooddeprivationdegree"{equivalenttothetimesinceaspiderhadlastfedtosatiation{wasalsoestimated. BildeandToft ( 1998 )usedabehavioralassaytoquantifythefooddeprivationlevelof

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sheet-webspiders;theycreatedapopulation-levelreferencecurveinthelaboratorythattranslatesanumberofpreyconsumedinashorttimeperiodtoafooddeprivationlevelindaysandthencomparedbehaviorofeld-collectedspiderstothereferencecurve.However,jumpingspidersexhibitalargevariationintheirpredationbehavior(thiswasalsoapparentinthelaboratoryexperimentdescribedbelow).Thus,webasedanindexoffooddeprivationdegreeonbodymassinsteadofbehavior.Fooddeprivationlevelofanindividualwasestimatedbyreferencingitsmassintheeldtotheindividualmasslossprole.Forexample,ifspideri'smassintheeldwaswianditsmassatsatiationwasfi,thefooddeprivationdegreeTiwasestimatedastherstdaysincethesatiationthatsatisedwi
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Figure9{1. Boxplotsforthefooddeprivationdegreesofthespidersintheeld.Spiderswereclassiedbasedonsexandthelocationwheretheywerefound:outsideretreat(active)orinsideretreat(inactive). Gardner 1964 ; WalkerandRypstra 2003 ).Ifspiderseatmorewhentheyaremoredeprivedoffood,wewouldhaveseentheoppositetrendintheresults|inactivespiderswouldhavebeenmoresatiated. 9{1 )isthatspidershaveanactivephaseandaninactivephase,andthattheyswitchbetweenthetwophasesbasedonaphysiologicalstatevariablethatisrelatedtothedegreeoffooddeprivation.Thephysiologicalvariable(e.g.,bodymass)

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decreasesastheyaredeprivedoffood,andinactivespiderswillbecomeactiveoncetheirphysiologicalstatedropsbelowathresholdLI!A;activeindividualsbecomeinactiveoncetheycaptureenoughpreytoraisetheirphysiologicalstateabovea(possiblydierent)thresholdLA!I.IfLI!A
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Immediatelypriortotheexperiment,thespiderswerefedadlibitumfor24hrsfollowedbysixdaysoffooddeprivation.Theirwatersupplywasmaintaineddailywithwater-soakedspongesduringthefasting.

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y,thepreywasremovedfromthecupbeforethespidercouldconsumeit,sothatthespiderdidnotgainanynutritionalvaluefromtheprey.Thisprocedurewasnecessarytoexaminewhetherthetransitionbacktoanactivephaseisinuencedbythedegreeoffooddeprivationifthespidersbecameinactive(i.e.,didnotattackprey).Exceptonthetreatmentday(Day0),theobservationwasconductedat1100daily.Observationsonthetreatmentdaywereconductedat1800. Spiegelhalteretal. ( 2002 ))wasusedtoselectthebestmodel.

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Figure9{2. Proportionofindividualsthatattackeday.Eachtreatmenthad11individuals. Becausethesespiderswerenotsatiated(exceptpossiblyforthe5-preytreatment,furtherdiscussedbelow),asignicanttreatmenteectwouldsuggestthatspidersthataredeprivedoffooddonotnecessarilyforage(i.e.,spidersmayremaininanactiveoraninactivephasepartlyindependentlyoftheirfooddeprivationstatus)buttheirtendencytoattackispartiallygovernedbytheirsatiation(i.e.,treatmenteect).Furthermore,persistenceofthetreatmenteectwillprovideinformationontherelationshipbetweenLI!AandLA!I.Forexample,datashowingthatthetreatmenteectpersistsformorethan6days(i.e.,theinitialfooddeprivationdegreepriortotheexperiment)wouldsupportthehypothesisthatLI!A
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Table9{1. Estimatedparametersofthereducedmodel.Themodelisanhierarchicallogisticregressionmodelwiththeintercepta1+a2jandslopeb1+b2j,wherejisthedayoftheexperiment. Parametersmean2:5%97:5% experiment.Theeectofdayvarieddependingonthetreatment:increasingaswellasdecreasingtrendswereobserved(Figure 9{2 ).Thereducedmodelwasselectedbasedonthemodelselectioncriteria(Fullmodel:DIC=222.291,Reducedmodel:DIC=206.284).Table 9{1 showstheestimatedparameters.Thenegativevalueofa2indicatesthatspiderstendtoforagelessastheystarve(approximately30%reductioninattackratebetweentherstandthelastdayoftheexperiment).Thatfooddeprivedindividualsforagelessiscounterintuitive;wediscussthisfurtherbelow.Thetreatmenteectbecomesweakerasspidersbecomedeprivedoffoodbecausepositiveb1andnegativeb2cancelout.Forexample,betweenthe0-preyand5-preytreatments,spidersinthe5-preytreatmentexhibitedapproximately70%reductionintheirattackrateontherstday,butthisreductionwasincreasedto30%attheendoftheexperiment.Byday6,spidersinthe5-preytreatmentareroughlyequivalentintermsoffooddeprivationleveltoallgroupsjustpriortotheexperiment(becausethedurationoffooddeprivationconditioningafteradlibitumfeedingwas6days).However,onday6oftheexperimentaltrial,manyindividualsdidnotattackay,whileallindividualsattackedapreyatthebeginning,suggestingthefooddeprivationisnotthesolefactorthatdeterminestheactivityofspiders.Furthermore,inthe5-preytreatment,althoughallofthespiderscaptureday

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Figure9{3. Treatmenteectparameterestimates.Solidlineandtwodashedlinesindicatethemeanand95%credibleregionsofthereducedmodel.Squaresindicatethemeansforthefullmodel. within10minonday0,only5individualscapturedaywithin10minatday10.Thisdierenceissignicant(Fisher'sexacttest,p=0:01238).TheseresultsareconsistentwiththehypothesisLI!A
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ThusthedynamicsofmasscanbedescribedasSi(t+1)=8>><>>:Si(t)mw;A;ifAi(t)=0Si(t)mw;A+Y;ifAi(t)=1Ifanindividualisinactive(i.e.,Ai=0),itsmassdecreasesbymw;Aduetometaboliccost,whichdependsonitsmassandactivityphase.Ifanindividualisactive(i.e.,Ai=1),itincreasesitsmassbyYfrompreyconsumption.However,Ywillbearandomvariable,incorporatingtheprobabilityofcapturingaprey.Activitytransitionsaregovernedbythefollowingrules:Ai(t+1)=8>>>>>>>>>><>>>>>>>>>>:0;ifAi(t)=0andSi(t)>LI!A1;ifAi(t)=0andSi(t)LA!I1;ifAi(t)=1andSi(t)
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Figure9{4. Relationshipbetweenmassandfooddeprivationdegree.Theestimatedfunctioniswt=0:145(t+1)0:063. Themetaboliclossrateforinactiveindividuals(i.e,mw;A=0)canbederivedfromthisrelationship(w=a(t+1)b)andrecognizingthat(t+1)b1=(w=a)11=b, dt=ab(t+1)b1=bw 9{1 (i.e.,mw;A=1=ba1=bw11=bu).Otherparametersofthemodelsweresetarbitrarily.Forexample,thethresholdswereset(LI!A=0:116;LA!I=0:14)suchthattheconditionLI!A
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Figure9{5. Simulationofanindividual.Circlesandsquaresarerealizationsofthesimulationcorrespondingwithactiveandinactivephase,respectively.SolidlinetracesthedeterministicpredictionofthestochasticIBM.HorizontallinesareLA!I(top)andLI!A(bottom). mean0.01wasusedtoreectthevariationofpreysizeintheeld.Theadditionalmetaboliccostforactiveindividualswassetasu=0:008. 9{5 ).Whenindividualsareinactive,massdecreasesslowlybasedonthemetaboliclossrelationship.Oncespidersbecomeactive,theycanquicklyattainenoughpreytobecomeinactiveagain.Duetothelowmetabolicrateofspiders(Chapter8),thelossisslowandthustheperiodofinactivityislongerthantheactiveperiod.Althoughthequantitativepredictiondiersfromtheactualobservation,thisactivitypatternoftheIBMproducedpatternssimilartothoseobservedinthedata(Figure 9{6 ).Inparticular,onaverageinactiveindividualsaremoredeprivedoffoodthanactiveindividuals.Inactiveindividualsalsovarymoreintheirfooddeprivationdegreethanactiveindividuals.

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Figure9{6. ResultsoftheIBMbasedon1000individuals.125individualsareactive. BildeandToft ( 1998 ),althoughtheoverallfooddeprivationdegreeofjumpingspidersappearstobelargerthanthatofsheetwebspiders.Thisdierencemaybeduetotheirforagingtactics:whilejumpingspiderspracticeactiveforaging,sheetwebspidersarespecializedinasit-and-waitbehaviorintheirretreat.Becausesheet-webspiderswouldhavenocleardistinctionbetweenstayinginretreatandforaging,theymaynotenterastateofinactivity,andhencemayexperiencelessvariationinnutritionalstate.Theobservedtrends(Figure 9{1 )maybeexplainedbasedonthemechanisticrulesoftheIBM.First,moreindividualswereobservedinactivethanactiveinthe

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eld.Thismaybeduetothelowmetabolicrateofthespidersandthebiphasicactivitypattern;thelowmetabolicratemakestheperiodofinactivitylongbecausethetimetoreachthelowerthreshold(LI!A)becomeslonger(Figure 9{5 ).Thislossinmassmaybeeasilycompensatedbyafewsuccessfulpreycaptures,whichmakesactivephasesshorterthaninactivephasesonaverageprovidedpreyarenotscarce.Anotherimportanttrend,thatinactiveindividualsaremoredeprivedoffoodthanactiveindividuals,maybeexplainedbythenonlinearmasschangeininactiveandactivephases.Ininactivephase,themassdecreasesexponentiallysuchthatonaverage,aspider'smassislow(e.g.,closetoLA!I).Ontheotherhand,activeindividuals'masseswillbeonaveragehigherbecausethelowerthemassofanindividual,thehighertheprobabilityitwillcaptureaprey.Therefore,althoughasdiscussedabove,theresultthatmorestarvedindividualsforagemoreintensely( Gardner 1964 )appearstocontradictstheobservedtrend,itactuallycomplementsthemodeltoproducetheobservedpatternifsucharelationshipbetweenfooddeprivationandforagingbehaviorisexpressedonlywhenspidersareactive.Thedeclineoftreatmenteect(thedierenceinforagingactivityamongdierentlyfooddeprivedspiders)overtimewasnotonlycausedbyanincreasingforagingintensityofinitiallyinactivespiderswithfooddeprivation.Infact,insometreatments,theproportionof(attacking)individualsdeclinedovertimeeventhoughindividualsbecamemoredeprivedoffood.Ihypothesizethatthiseectwasduetothequalityofspiders'retreats.Thespidersbuiltathinretreatduringtherstdayoftheexperiment,andinthesubsequentdaystheycontinuedtobuilduponitandcreatedmorerobustretreats.Robustretreatsarethickandlesstransparent.Existenceofthisretreatisprobablyessentialforspiderstobeinactive.Forexample,inactivespidersthatwerecollectedintheeldatemanypreyinthelaboratorybecausetheywereseparatedfromtheirretreat,whichforced

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themtobecomeactiveinthelaboratory.Therefore,althoughthetreatmenteectbecameinsignicantonday9,itmaytakelongerthan9daysforspidersintheeldtobecomeactiveagain(i.e.,LI!Amaybeoverestimated).Ignoringsuchdetailsofnaturalhistorymayleadtomisleadingresults:forexample,ifthesamelaboratoryexperimentwereconductedinacommonexperimentalarena(i.e.,allspiderswereremovedfromtheirretreats),wemighthavefoundeectsneitheroftreatmentnoroftime.Thesebehavioraldynamicsmayhavelargeimplicationsforthedynamicsofactivityandpopulationdensities,whichareignoredbythemostcommonlyusedmodelsofactivitydynamics(Chapter10).Althoughtheparticularbehaviordescribedheremaybespecictospidersthathavehighfooddeprivationtolerance,muchsmallerdeviationsfromthestandardmodelcanhavemajoreectsonmodelpredictions(Chapters6and7).Althoughthisstudyconsidersalongertimescalethanmanyotherbehavioralstudies(e.g.,Chapter6;Caro2005),itstilldealswithamuchfastertimescalethanisconsideredbytypicalcommunitymodels:onthetimescaleofthisexperiment,densitiesofpredators(e.g.,otherjumpingspiders)areunlikelytochangedramatically.Inotherwords,allofthedynamicsdescribedinthisstudyaredetailsthatareconsideredunimportantinthecommonframeworkofmathematicalmodels.Nevertheless,becausetheapproximationsmadebycommonmodelsaresensitivetovariationinbehaviorthatoccursatfasttimescales(Chapters6,7,and10),weneedtore-examinehowactivitydynamicsandcommunitydynamicsarerelatedtoeachother.Furthercarefulexaminationsofactivitydynamicsinavarietyoforganismswillhelptovalidatetheassumptionsoftheexistingmodelsandimprovetheirgeneralrobustness.Moregenerally,theywillidentifywhatbehavioralinformationwewillneedinordertoreliablyscalebehavioralmodelsuptothelevelofcommunitydynamics.

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127

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Iwillconcludethisdissertationwithashortgeneraldiscussionaboutbehavioralmodellingincommunityecology.Indirecteectsarealmostcertainlyimportantinecologicaldynamics( Wootton 2002 ; WernerandPeacor 2003 ).Thebehaviorsdescribedinthisdissertationwouldcreatecomplexindirectinteractionsthatmaystronglyinteractwithspatialstructure.However,asdiscussedinChapter5,wearenotyetatastagewherewecanconnectspecicmeasuresofindirecteectssuchastrait-anddensity-mediatedindirectinteractions(TMIIandDMII,respectively)tocommunitycharacteristics.Thisinabilityisinlargepartbecauseindirect-eectexperimentshavefocusedstronglyonquantifyingtheeectofattackratesonthesemeasuresandhaveneglectedotherinformationsuchasdirecteects(e.g.,costinreproductionduetothetraitchange),whichecologistshavelongknowntobekeycomponentsgoverningcommunitydynamics( Murdochetal. 2003 ).Thisisnottosaythereisnounderlyingbiologicalrelationshipthatconnectsshort-termTMIIandDMIItolong-termdynamics,butithasyettobediscovered.Thefurtherresearchthatisneededtobridgethesedisciplinesisbeyondthescopeofthisdissertation.Inthissynthesis,Iwillfocusontheeectofbehavioralandspatialfactorswithoutexplicitlyfocusingonindirecteects.

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5151patches).EachindependentpatchcanharboranunlimitednumberofIGpredatorsandIGpreyintheabsenceofbiologicalconstraints.

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variationinpreybenets(Chapter9),IassumedYisarandomdeviatefromaPoissondistributionwithmean5.

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Suter 1991 ; Foelix 1966 ; Belletal. 2005 ).Thus,intheabsenceofballooningbehavior,themodelmayinduceunrealisticallystrongspatialeects.Toreecttheballooningbehavior,IassumednewbornsdispersedgloballyratherthanbeingrestrictedtothelocalneighborhooddeterminedbyU.Eachsimulationwasinitiatedwithinitialnumbersof15IGpredatorsand30IGpreyrandomlydistributedintheenvironmentwithaninitialenergystateof15forallindividualsofbothspecies.Inthecaseofbiphasicactivity,individuals'

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initialactivitystateswererandomlydetermined.Eachsimulationwascontinueduntileitheroneofthetwospecieswereexcludedor1500timestepshadelapsed.Averagedurationofpersistencewasusedtoexploretheeectsofvariousecologicalfactorsonthepersistenceofthecommunity. 10{1 ).Undermonophasicactivity,persistencewashighestatanintermediatelevelofproductivity.Underbiphasicactivity,persistenceincreasedwithproductivitylevel(Figure 10{1 ).Ingeneral,persistencewaslongerunderbiphasicthanmonophasicactivity.Regardlessoftheactivitydynamics,increasingspatialstructure(i.e.,smallerU)increasedpersistence. Figure10{1. EectofproductivitylevelonthepersistenceofIGPcommunityundervariousdegreesofspatialstructure.ThesmallertheneighborhoodsizeU,thestrongerthespatialstructure.Localreproductionandnon-adaptivebehavior(i.e.,==0)areassumed. Theeectofadaptivebehavior(bothand)wasrelativelyweakcomparedtotheeectofbiphasicactivitydynamics.ThisbehaviorrepresentsIGprey'schangeinforagingeortCinresponsetotheirlocalenvironment.Withmonophasic

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activity,adaptivebehaviorshiftedthepeakinpersistencetohigherproductivitylevels(Figure 10{2 ). Figure10{2. Averagepersistencewithandwithoutadaptivebehavior.Adaptivebehaviorindicatesthatbothandarepositive(seetext).U=7.Localreproductionisassumed. Ballooningbehaviordecreasedpersistencedramaticallyundermonophasicactivitydynamics.However,withbiphasicactivity,persistencewasmaintainedevenwithballooningbehavior(Figure 10{3 ). Figure10{3. Averagepersistencewithandwithoutballooning.U=7.Adaptivebehaviorsareincluded.

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10{1 ).Furthermore,persistencebasedonbiphasicactivitywasrobusttoglobalreproduction(i.e.,ballooning);incontrast,globalreproductioncollapsedpersistenceundermonophasicactivity,suggestingtheactivitydynamicsmaybeanimportantmechanismthatallowsIGpreyandIGpredatorstocoexistinjumpingspidercommunities. Hastings 2000 ).Thisadvantageousscenario(forIGpredators,andthusdisadvantageousforIGprey)isfurtherenhancedwhenIGpreyarebenecialtoIGpredators(Chapter3)|whichappearstobethecaseforjumpingspiders,becausenitrogencontentofpreyhadasignicanteectonthegrowthrateoftheanimals(Chapter3).

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10{2 ).Atmost,adaptivebehaviorcausedaweakquantitativeeectundermonophasicactivity.InlightoftheresultsofChapter6,theweakeectofadaptivebehaviorisnotsurprisingbecausespatialeectsweakentraiteects.Injumpingspiders,althoughantipredatorbehaviorissustainedtemporarily(Chapters7and8),thedurationofthetraitexpressionwasrelativelyshortandwouldnotbestrongenoughtoproducethemagnitudeoftraiteectpredicted

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bycommonmodels(Chapter6).However,theseresultsdonotnecessarilyimplythatadaptivebehaviorisunimportant.Forexample,althoughbiphasicactivitywasmodelledsimply,spiders'loweringofmetabolicrateinresponsetopredatorencounters(Chapter8)mightbeassociatedwiththethresholdLA!I.Ifanencounterexperiencewithapredatorincreasesspiders'tendenciestobecomeinactive,thispredatorinducedtraiteectwouldbemuchstrongerthanwhatispredictedbythecurrentmodel.AsdiscussedinChapter9,mostempiricalstudieshaveexaminedtheeectofpredatorsonimmediatebehavioralresponses,suchastimetocomeoutofarefuge(e.g., Jennionsetal. 2003 );however,predatorencountercanaectpreyinavarietyofways.Morecomprehensivetreatmentsoftheeectsofpredatorthreatwillbeusefultofurtherappreciatetheroleofadaptivebehaviorincommunitydynamics. deRoosetal. 2003 ).Inthisscenario,conventionalmodelspredictthatthisindividualwouldkeeploweringitsforagingactivityonaveragethroughoutitslifetimebecausethedensityofIGpredatorsdecreases.Whetherornotthispredictionbasedonaverageactivitylevelistrueornotneedstobeinvestigated,whichwouldbeadiculttaskbecauseitrequiresmorethansimplydocumentinghighpredatordensityresultinginmoreencounterswithpredators(andthusinducingescapebehaviormorefrequently).Whetherornotthispredictionbasedonaverageactivitylevelistrueornotneedstobe

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investigated,whichwouldbeadiculttaskbecauseitrequiresmorethansimplydocumentinghighpredatordensityresultinginmoreencounterswithpredators(andthusinducingescapebehaviormorefrequently).However,itiseasytoshowthatthereislargevariancearoundtheaveragetraitexpression.Forexample,ifanindividualencountersandescapesapredator,itwillsubsequentlyloweritsactivity(Chapter7).Inotherwords,althoughtypicalmodelsassumethatanimalsrespondtoanaveragedensityofpredators(e.g., Abrams 1992 ),behavioralvarianceisinducedbyotherdetailssuchasspatialstructure(Chapters2and6)andindividualexperience(Chapters7and8).Furthermore,individualexperienceandspatialstructurearenotindependent(Chapter6)becausepreyintheregionwherethedensityofpredatorsishighwillexperiencepredationriskdierentlyfrompreyintheareawherethedensityofpredatorislow.Theeectoftheseignoredvariationsofbehaviorcanbesubstantial.Forexample,iftherelationshipbetweenforagingactivityandresourceintakeisconvex,includingbehavioralvariationwilllowertheactualresourcedepletionduetoJensen'sinequality.Thus,simplemodelsmayoverestimateresourcedepletion(cf.Chapter7).Althoughthemagnitudeofparticularbiasescanbeestimatedandcorrectedforifvariancesofbehaviorareknown,theeectsofsuchbiasesonthedynamicsofmulti-speciescommunityareunknownandneedtobeinvestigated.Typicalcommunitymodelswithadaptivebehaviorignoremorethanjustbehavioralvariation.Asillustratedbybiphasicactivityofjumpingspiders,somebehaviorsmaynotbeabletobesimplyapproximatedbyincludingvarianceinthemodel.Injumpingspiders,communitydynamicsunderbiphasicactivitycouldnotberecoveredbysimplyadjustingparametersofmonophasicactivity.Asdiscussedabove,thisdierenceprobablyoccursbecausesuchdynamicsnotonlylowertherateofIGPbutalsoinduceotherdynamicsthathaveyettobedescribed.Becausetheoreticalinvestigationsthatscalebehaviortocommunitydynamics

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aredominatedbyacommonrecipethatneglectstheeectofrapidvariationinbehavior,wehavelittleunderstandingoftheimportanceofempiricallyobservedbehaviors.Theintentofthisdiscussionistopointoutpersistingweaknessesintheconnectionbetweentheoryanddata.Whiletheoreticianshaveneglectedbehavioralvariation( Bolkeretal. 2003 ),empiricistshaveworkedhardtoquantifythem( WernerandPeacor 2003 ).While\allmodelsarewrong"( Box 1979 ; Sterman 2002 ),wemustneverforgettoexaminethereliabilityofvariousapproximations.Whilesimplemodelsareeasiertounderstand,modelsareuselessiftheydonotpredictthedynamicsofthetargetcommunity( Peters 1991 ).Myworkquestionstherobustnessofcommonapproximationsmadeinmodelslinkingbehaviorandcommunitydynamics.Goodexperimentsarenotoriouslydiculttoconduct.Theslowtimescaleofcommunitydynamicsmakesmanyexperimentslogisticallyinfeasible.Fortunately,inclusionofbehaviorallowsempiriciststoexaminethemodelsmorerigorously,whichmakesthisfusionagreatplacefortheoreticiansandempiriciststocollaborate.Inthisdissertation,Ishowedthatbehavioraldynamicsthatoccuratafastertimescalethanthedemographicdynamicscanhaveastrongimpactoncommunitydynamics.Thus,theoriesthatincorporatefastdynamicsintheconnectionbetweenbehavioralandcommunitydynamicswillfacilitatethiscollaborationdramatically.Withoutthiseort,wecannotidentifyhowmuchandorwhatkindofbehavioralvariationcanbeincludedintothemodelasvariation,asmechanisticdetails,orignored.Althoughsimplifyingbehavioraldynamicsisneededasitwillallowmoregeneralanalysis,ecologistshavepaidlittleattentiontothegoodnessoftheirapproximations.Consequently,wehavesurprisinglylittlegeneralunderstandinghowbehaviorscalesuptocommunitydynamicsotherthanstatingitisimportant.Forexample,despitetheincreasingpopularityof

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quantifyingTMIIandDMII( Preisseretal. 2005 ; WojdakandLuttbeg 2005 ),wehavelittleideawhattheyimplyinalongtermcommunitydynamics(Chapter5).IhopethattheinformationIhaveprovidedinthisdissertationwillmotivatebothempiricistsandtheoreticianstofacilitatetheconnectionbetweentheoryanddatatofurtherimproveourunderstandingabouttherolesofbehaviorinecologicalcommunities.

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