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## Material Information- Title:
- Maintenance of Intraguild Predation in Jumping Spiders
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Ecological modeling ( jstor ) Ecology ( jstor ) Foraging ( jstor ) Modeling ( jstor ) Predation ( jstor ) Predators ( jstor ) Productivity ( jstor ) Spatial models ( jstor ) Spiders ( jstor )
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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. |

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PAGE 3 Anumberofpeoplehelpedmetocompletethisproject,whichmakesthisdissertationsomuchmoremeaningfultome.Idon'tlistthemhere,butIhopemyappreciationiswellunderstood.StillImustacknowledgemycommittee(BenBolker,JaneBrockmann,CraigOsenberg,JimHobert,BobHolt,andStevePhelps)fortheirvaluablecriticismandencouragementthroughout,whichIvalueverymuch.IamextremelygratefulabouttheTeachingAssistantshipandResearchAssistantshipopportunitiesaswellastheCLASfellowshipfortheirsupportandexperiences.ComplexSystemsSummerSchooloftheSantaFeInstitutealsoprovidedanextremelyenjoyableenvironmentinwhichIwasabletoinitiateapartofthisdissertation.Lastly,interactionsIhavehadwithBenBolkerhavebeenmymostvaluableexperiencehereatUF.Benimprovednotonlymyprojectbutalsomywayofapproachingecologicalproblems.IfIaccomplishedanythingworthwhileinthefuture,itisbecauseIwasfortunateenoughtoworkwithhim. iii PAGE 4 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 PAGE 5 .............................. 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 PAGE 6 .................... 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 PAGE 7 .............................. 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 PAGE 8 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 PAGE 9 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 PAGE 10 ........................... 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 PAGE 11 ............. 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 PAGE 12 xii PAGE 13 xiii PAGE 14 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 PAGE 15 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 PAGE 16 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 PAGE 17 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 PAGE 18 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. PAGE 19 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 PAGE 20 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. PAGE 21 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. PAGE 22 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. PAGE 23 behavior,whichisnotspecicallyexaminedinthisproject(i.e.,ballooning,[Belletal., 2005 ]),becauseitstronglyaectsthespatialstructureofthemodelandexclusionofthebehaviormayresultinanunrealisticdegreeofspatialstructure.Takentogether,thismodeldemonstratesthattheactivitypatternsofjumpingspidersthataredescribedinthisprojectplaykeyrolesinallowingthetwospeciesofjumpingspidersthatexhibitIGPtocoexist.Thisresultincorporatesnaturalhistorycharacteristicsofspiderssuchasballooning,furtherstrengtheningthevalidityofthisconclusion.Withoutthesimultaneousconsiderationofspatialandbehavioralfactorstogether,itwouldnotbepossibletoderivethisconclusion.AlthoughthefocusofthestudyisIGP,myresultsabouttherelationshipbetweenbehaviorandcommunityecologyaremoregeneral.Basedonthendingsofthisproject,ageneraldiscussionaboutbehavioralmodellingincommunityecologyisalsoprovidedtofacilitatereexaminationsofrelationshipsbetweenbehaviorandcommunityecology. PAGE 24 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 PAGE 25 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 PAGE 26 approximationmakesnodistinction(non-spatial);pairapproximationreducestothemeaneldapproximationinitsnon-spatiallimit(discussedbelow).Thus,usingpairapproximationallowsonetoexaminetheeectoflocalinteractionsbycomparingtheresultswiththeanalogousmeaneldmodel.WithIBMs,Iexaminetheeectsofspatialheterogeneityinproductivity,which HoltandPolis ( 1997 )suggestedshouldbeimportantinIGPsystems.Thethreemainquestionsare(1)howthequalitativepredictionsofIGPmodelsareaectedbytakingspaceintoaccount,(2)whetherspatialstructureexpandsthepossibilityofcoexistence,andifso,underwhatconditions,and(3)howspatialheterogeneityinresourcedistributionaectsIGPdynamics. PAGE 27 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 PAGE 28 2{3 and 2{4 representaspecialcaseofthemodeldescribedinFigure4ofPolisetal.(1989)whentheintensitiesofinter-andintra-speciccompetitionarethesame.Althoughcompetitionisforspaceratherthanforaresourcewithexplicitwithin-celldynamics,thenon-spatialversionofthemodelmatchesamodelderivedwithresourcecompetitioninmind. PAGE 29 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 PAGE 30 probabilities.Forexample,thedynamicsofqP=Parefoundtobe,dqP=P 2{1 .Theoutcomewasclassiedintooneoffourcases:IGpreycaninvadeIGpredators,butIGpredatorscannotinvadeIGprey(IGpreywin),IGpredatorscaninvadeIGprey,butIGpreycannotinvadeIGpredators(IGpredatorswin),eachspeciesisabletoinvadetheother(coexistence),andneitherIGpreynorIGpredatorscaninvadetheother(bistability)( MurrellandLaw 2003 ). PAGE 31 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 ). PAGE 32 halfthecellswereassignedtohighorlow,theaverageenvironmentalproductivitywasalways(RH+RL)=2.ExamplesofpatchesofdierentscalesareshowninFigure 2{1 Figure2{1. Examplesofrandombinarylandscapesbasedondierentpatchscales.Patchscalereferstothenumberoftimestheprocedurediffusewasapplied(seetext). 2.3.1MeanFieldApproximationInadditiontothetrivialequilibriumwherenospeciescansurvive(whichoccurswhenR PAGE 33 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.,R PAGE 34 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 PAGE 35 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 PAGE 36 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. PAGE 37 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 PAGE 38 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 PAGE 39 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 PAGE 40 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 PAGE 41 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. PAGE 42 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. PAGE 43 Furtherexplorationsoftherolesofbothspatialstructureandspatialheterogeneityareneeded.Furthermore,althoughspatialinteractionsmaybediculttoanalyze,somespatialdataarerelativelyeasytocollectonceweknowexactlywhattocollect.Infact,eldstudieshaveoftencollectedthesedataasauxiliaryinformationevenwhentheiranalysesignoredspace.Thedevelopmentofspatialtheorieswillcreatemoretestablehypothesesandincreaseourabilitytoutilizedatamoreeciently,whichmayresolvesomeofthediscrepanciesbetweentheoryanddata. PAGE 44 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 PAGE 45 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 ). PAGE 46 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 PAGE 47 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). PAGE 48 Figure3{2. Durationofeachinstar.Durationsof2ndinstarand4thinstarweresignicantlysmallerfortheN-richtreatment(indicatedby*).Treatments:N-rich(N)andcontrol(c). deRoosetal. 2003 ).BecauseoccurrenceofIGPdependsonthesize-structure,thegrowthconsequencesofstoichiometrymayhavestrongimplicationsforIGPcommunitydynamics.Futureworkshouldconsidertheprey'snutrientprolemorecarefully. Matsumuraetal. ( 2004 )havedoneexperimentssimilartothisstudyexaminingtheeectsofpreytypeonthegrowthlevelofwolfspiders(genusPardosa),nding PAGE 49 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 PAGE 50 communitywithIGPbutalsointogeneralpropertiesofthepersistenceofcomplexfoodwebs. PAGE 51 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 PAGE 52 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 PAGE 53 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 PAGE 54 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 PAGE 55 4{2 :WN=bAaA2P PAGE 56 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 PAGE 57 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. PAGE 58 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. PAGE 59 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 PAGE 60 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). PAGE 61 1. Ifb PAGE 62 Table4{1. Equilibriumanalysis.ThecspeciedisthechoiceofESSinRegionIIIthatguaranteestheexistenceanonzeroequilibrium. h PAGE 63 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. PAGE 64 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). PAGE 65 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. PAGE 66 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 PAGE 67 Figure4{6. Simulationofthedynamicsofpredatorsandpreyplottedontheeortdiagram(Figure 4{1 ).s=ahNandr=(a=b)P.Thegraylineindicatesr=p populationwouldevolveawayfromthestrategyy2towardsoneofthedynamicallystablestrategieswithc=y1orc=1inRegionIII.Similarly,wecansaythatthebasicESSthatxesc=1inRegionIIIisstablerelativetosmallshiftsinthebehaviorofthepopulationwithfastg,butthisbehaviordoesnotallowecologicalstability. PAGE 68 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. PAGE 69 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 PAGE 70 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, PAGE 71 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 PAGE 72 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. PAGE 73 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. PAGE 74 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 PAGE 75 2,theny1istheonlyESS.(y1<1andy2>1inthiscase.) 2,theny=1istheonlyESS.(Bothy1;y2areeither>1ornotrealinthiscase.) PAGE 76 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 PAGE 77 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. PAGE 78 JulianoandWilliams 1987 ));(2)variationinforagerstrategyovertime( Luttbegetal. 2003 );(3)dierentialmortalityduetocostsofantipredatorbehavior;and(4)intraspecicinterference.ThenwecandeneFandfastheper-forageruptakeintheabsenceandpresenceofpredators;sinceantipredatorbehaviorgenerallyreducesforagingeortoreciency,wesupposef PAGE 79 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. PAGE 80 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. PAGE 81 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 PAGE 82 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 PAGE 83 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). PAGE 84 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 PAGE 85 necessaryinordertocomparetrait-anddensity-mediatedeectsaccurately. GrinandThaler ( 2006 )foundlargedierencesbetweenTMII+randTMIIraswellasbetweenDMII+randDMIIrina3-dayexperiment;whiledierencesbetweentheTMIIindicescouldbecausedbyintraspecicinteractionsassuggestedabove,dierencesinDMIIaremoreconstrainedandmayreecttheeectsofvariationindensityandbehaviorovertime.Afewpossiblesolutionstothesedicultiesareto: vanVeenetal. 2005 ).Hereevenalittlebitofperiod-by-perioddata,evenifthesamplingfrequencyistooslowtocapturethedetailsofthedynamics,canbeenormouslyusefulforvalidatingthefunctionalformsincorporatedinthemodel. PAGE 86 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. PAGE 87 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 PAGE 88 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. PAGE 89 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 PAGE 90 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. PAGE 91 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): PAGE 92 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. PAGE 93 Thenumberofpredatorswasvariedfrom5to50inincrementsof5.Predatorsareconsideredencounterediftheyarefoundinthecellsthatweresearched(Figure 6{1 )inaccordancewiththeforagingsolution.Foragersweresettotheirinitiallocationthroughouttheseason(i.e.,theyforagedaroundarandomxedlocation)whereaspredatorsrelocateddaily.PredatorsdispersedrandomlytoanemptycellwithinaradiusofD.Foreachpossibleparameterset(Table 6{1 ),30simulationswereconducted.Attheendofeachsimulation,thenumberofsurvivingforagers,fecundityofthesurvivors,andthedepletionofresourcebytheforagerpopulationwererecorded.Thesurvivalandfecundityrepresentthedirecteectofpredatorsonforagers,whileresourceuptakeisusedtoquantifyindirecteects(discussedbelow). PAGE 94 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 PAGE 95 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 PAGE 96 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. PAGE 97 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 PAGE 98 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 PAGE 99 thesespecicitiesindeedseemtoactdistinctivelyinrealecosystems( Preisseretal. 2005 ).Althoughmorestudiesareneeded,investigationofadaptivebehaviorthroughsensoryconstraintsmaybeafruitfulwaytofurtheradvancetheinterfaceofadaptivebehaviorandcommunitydynamics. PAGE 100 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 PAGE 101 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 PAGE 102 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 PAGE 103 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 PAGE 104 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 PAGE 105 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 PAGE 106 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. PAGE 107 7{1 ,Table 7{1 ).Theambiguousvisualstimulididnotaectthetimetocomeoutwhenspidershadnotencounteredapredator,butiftheyhadencounteredapredator,theambiguousstimulicausedspiderstostayinthetubesignicantlylonger(Table 7{1 ). PAGE 108 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 PAGE 109 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 PAGE 110 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 PAGE 111 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 PAGE 112 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 PAGE 113 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 PAGE 114 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 PAGE 115 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. PAGE 116 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 PAGE 117 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 PAGE 118 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). PAGE 119 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. PAGE 120 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 PAGE 121 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 PAGE 122 changesinresponsetoexperiencewillenhanceourunderstandingofthelinkagesbetweenbehavior,physiology,andecologicaldynamics. PAGE 123 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 PAGE 124 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 PAGE 125 nightandwhennotactivelyforaging( HoeerandJakob 2006 ).Wecategorizedindividualsthatwerefoundintheirretreatasinactiveandindividualsthatwerefoundoutsidearetreatasactivebasedonourdaytimeobservations.Thisclassicationisonlyapproximate,becauseindividualsthatwerefoundintheretreatmayhavebeeninactivefordierentamountsoftime(e.g.,theymayhavebeenactivejustpriortotheobservation).Toobtainsomeideaabouttheirpreviousactivity,\fooddeprivationdegree"{equivalenttothetimesinceaspiderhadlastfedtosatiation{wasalsoestimated. BildeandToft ( 1998 )usedabehavioralassaytoquantifythefooddeprivationlevelof PAGE 126 sheet-webspiders;theycreatedapopulation-levelreferencecurveinthelaboratorythattranslatesanumberofpreyconsumedinashorttimeperiodtoafooddeprivationlevelindaysandthencomparedbehaviorofeld-collectedspiderstothereferencecurve.However,jumpingspidersexhibitalargevariationintheirpredationbehavior(thiswasalsoapparentinthelaboratoryexperimentdescribedbelow).Thus,webasedanindexoffooddeprivationdegreeonbodymassinsteadofbehavior.Fooddeprivationlevelofanindividualwasestimatedbyreferencingitsmassintheeldtotheindividualmasslossprole.Forexample,ifspideri'smassintheeldwaswianditsmassatsatiationwasfi,thefooddeprivationdegreeTiwasestimatedastherstdaysincethesatiationthatsatisedwi PAGE 127 Figure9{1. Boxplotsforthefooddeprivationdegreesofthespidersintheeld.Spiderswereclassiedbasedonsexandthelocationwheretheywerefound:outsideretreat(active)orinsideretreat(inactive). Gardner 1964 ; WalkerandRypstra 2003 ).Ifspiderseatmorewhentheyaremoredeprivedoffood,wewouldhaveseentheoppositetrendintheresults|inactivespiderswouldhavebeenmoresatiated. 9{1 )isthatspidershaveanactivephaseandaninactivephase,andthattheyswitchbetweenthetwophasesbasedonaphysiologicalstatevariablethatisrelatedtothedegreeoffooddeprivation.Thephysiologicalvariable(e.g.,bodymass) PAGE 128 decreasesastheyaredeprivedoffood,andinactivespiderswillbecomeactiveoncetheirphysiologicalstatedropsbelowathresholdLI!A;activeindividualsbecomeinactiveoncetheycaptureenoughpreytoraisetheirphysiologicalstateabovea(possiblydierent)thresholdLA!I.IfLI!A PAGE 129 Immediatelypriortotheexperiment,thespiderswerefedadlibitumfor24hrsfollowedbysixdaysoffooddeprivation.Theirwatersupplywasmaintaineddailywithwater-soakedspongesduringthefasting. PAGE 130 y,thepreywasremovedfromthecupbeforethespidercouldconsumeit,sothatthespiderdidnotgainanynutritionalvaluefromtheprey.Thisprocedurewasnecessarytoexaminewhetherthetransitionbacktoanactivephaseisinuencedbythedegreeoffooddeprivationifthespidersbecameinactive(i.e.,didnotattackprey).Exceptonthetreatmentday(Day0),theobservationwasconductedat1100daily.Observationsonthetreatmentdaywereconductedat1800. Spiegelhalteretal. ( 2002 ))wasusedtoselectthebestmodel. PAGE 131 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 PAGE 132 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 PAGE 133 Figure9{3. Treatmenteectparameterestimates.Solidlineandtwodashedlinesindicatethemeanand95%credibleregionsofthereducedmodel.Squaresindicatethemeansforthefullmodel. within10minonday0,only5individualscapturedaywithin10minatday10.Thisdierenceissignicant(Fisher'sexacttest,p=0:01238).TheseresultsareconsistentwiththehypothesisLI!A PAGE 134 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) PAGE 135 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 PAGE 136 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. PAGE 137 Figure9{6. ResultsoftheIBMbasedon1000individuals.125individualsareactive. BildeandToft ( 1998 ),althoughtheoverallfooddeprivationdegreeofjumpingspidersappearstobelargerthanthatofsheetwebspiders.Thisdierencemaybeduetotheirforagingtactics:whilejumpingspiderspracticeactiveforaging,sheetwebspidersarespecializedinasit-and-waitbehaviorintheirretreat.Becausesheet-webspiderswouldhavenocleardistinctionbetweenstayinginretreatandforaging,theymaynotenterastateofinactivity,andhencemayexperiencelessvariationinnutritionalstate.Theobservedtrends(Figure 9{1 )maybeexplainedbasedonthemechanisticrulesoftheIBM.First,moreindividualswereobservedinactivethanactiveinthe PAGE 138 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 PAGE 139 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. PAGE 140 127 PAGE 141 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. PAGE 142 5151patches).EachindependentpatchcanharboranunlimitednumberofIGpredatorsandIGpreyintheabsenceofbiologicalconstraints. PAGE 143 variationinpreybenets(Chapter9),IassumedYisarandomdeviatefromaPoissondistributionwithmean5. PAGE 144 Suter 1991 ; Foelix 1966 ; Belletal. 2005 ).Thus,intheabsenceofballooningbehavior,themodelmayinduceunrealisticallystrongspatialeects.Toreecttheballooningbehavior,IassumednewbornsdispersedgloballyratherthanbeingrestrictedtothelocalneighborhooddeterminedbyU.Eachsimulationwasinitiatedwithinitialnumbersof15IGpredatorsand30IGpreyrandomlydistributedintheenvironmentwithaninitialenergystateof15forallindividualsofbothspecies.Inthecaseofbiphasicactivity,individuals' PAGE 145 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 PAGE 146 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. PAGE 147 10{1 ).Furthermore,persistencebasedonbiphasicactivitywasrobusttoglobalreproduction(i.e.,ballooning);incontrast,globalreproductioncollapsedpersistenceundermonophasicactivity,suggestingtheactivitydynamicsmaybeanimportantmechanismthatallowsIGpreyandIGpredatorstocoexistinjumpingspidercommunities. Hastings 2000 ).Thisadvantageousscenario(forIGpredators,andthusdisadvantageousforIGprey)isfurtherenhancedwhenIGpreyarebenecialtoIGpredators(Chapter3)|whichappearstobethecaseforjumpingspiders,becausenitrogencontentofpreyhadasignicanteectonthegrowthrateoftheanimals(Chapter3). PAGE 148 10{2 ).Atmost,adaptivebehaviorcausedaweakquantitativeeectundermonophasicactivity.InlightoftheresultsofChapter6,theweakeectofadaptivebehaviorisnotsurprisingbecausespatialeectsweakentraiteects.Injumpingspiders,althoughantipredatorbehaviorissustainedtemporarily(Chapters7and8),thedurationofthetraitexpressionwasrelativelyshortandwouldnotbestrongenoughtoproducethemagnitudeoftraiteectpredicted PAGE 149 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 PAGE 150 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 PAGE 151 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 PAGE 152 quantifyingTMIIandDMII( Preisseretal. 2005 ; WojdakandLuttbeg 2005 ),wehavelittleideawhattheyimplyinalongtermcommunitydynamics(Chapter5).IhopethattheinformationIhaveprovidedinthisdissertationwillmotivatebothempiricistsandtheoreticianstofacilitatetheconnectionbetweentheoryanddatatofurtherimproveourunderstandingabouttherolesofbehaviorinecologicalcommunities. 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