Theoretical Population Biology 76 (2009) 52–58

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The positive effects of negative interactions: Can avoidance of competitors or predators increase resource sampling by prey? Adrian V. Bell a,∗ , Russell B. Rader a , Steven L. Peck a , Andrew Sih b a

Department of Biology, Brigham Young University, Provo, UT, USA

b

Department of Environmental Science and Policy, University of California Davis, Davis, CA, USA

article

info

Article history: Received 2 May 2008 Available online 14 April 2009 Keywords: Spatial dynamics Cellular automata

abstract Spatial overlap between predators and prey is key to predicting their interaction strength and population dynamics. We constructed a spatially-explicit simulation model to explore how predator and prey behavioral traits and patterns of resource distribution influence spatial overlap between predators, prey, and prey resources. Predator and prey spatial association primarily followed the ideal free distribution. Departures from this model were intriguing, especially from the interactions of predator and prey behavior. When prey weakly avoided conspecifics, they associated more highly with resources when predators were present. Predators increased the rate of prey movement between patches, which increased their ability to sample their environment and aggregate in patches with high resources. When prey strongly avoided each other, predators decreased prey association with resources. That is, an increased rate of prey movement increased the probability that prey would interact and avoid each other without regard to the distribution of resources. More generally, a more highly clumped distribution of resources acted as a spatial anchor that generally increased prey, predator, and resource association. Prey tended to congregate with resources and predators generally congregated with prey. © 2009 Elsevier Inc. All rights reserved.

1. Introduction The spatial distribution of organisms is an important aspect of ecological systems. At any given point in time, the roles of competition, foraging efforts, abiotic stress, and predation risk are diminished or magnified by the spatial context in which an organism is found (Silvertown et al., 1992). Theory shows how mechanisms such as dispersal and competition may strongly influence how a population can ‘‘track’’ spatial variation in carrying capacity (Roughgarden, 1974). If alternative habitats are available, habitat selection between two competing species may have long-term evolutionary outcomes on coexistence and the extent of competition (Rosenzweig, 1981). Similarly, in predator–prey systems, final outcomes (coexistence or local extinction) are influenced by the strength of trophic interactions, which are strongly influenced by spatial relationships (i.e. encounter rates). Spatial patchiness that creates, for example, prey refugia, unequal attack rates and density-dependent immigration between patches, can stabilize predator–prey dynamics through the same abstract

∗ Corresponding address: Graduate Group in Ecology, Department of Environmental Science and Policy, One Shields Avenue, University of California Davis, CA 95616, USA. E-mail addresses: [email protected] (A.V. Bell), [email protected] (R.B. Rader), [email protected] (S.L. Peck), [email protected] (A. Sih). 0040-5809/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.tpb.2009.03.008

principle (indirect density dependence Murdoch et al., 2003). Behind these dynamics lie the behavioral mechanisms that produce predator and prey movement behavior and the resulting spatial patterns. Spatial associations between predators and prey result from behavioral responses by both predators and prey. Perhaps obviously, predators and prey exhibit conflicting responses. Predators tend to go to areas with more prey (Burger et al., 2004; Honer et al., 2005; Lima, 2002; Veit et al., 1993), whereas prey tend to avoid predators (Caudill, 2005; Diehl et al., 2000; Doebeli and Killingback, 2003; Peckarsky, 1996; Peckarsky and Dodson, 1980; Sih et al., 1992; Van de Meutter et al., 2005). However, prey must also balance predator avoidance with the need to find food. Although a large number of studies have examined how predators and prey balance conflicting demands, a recent review emphasized that we know surprisingly little about outcomes and behavioral dynamics when both predators and prey can freely move and respond to one another (Lima (2002) but see Flaxman and Lou (2009)). The attraction–avoidance behavior of predators and prey in space has been deemed a predator–prey space race (Sih, 1998). Analysis of this interplay has used game theory to examine how individual space-use decisions result in a positive or negative spatial association of prey and predator populations (Alonzo, 2002; Cressman et al., 2004; Hugie and Dill, 1994; Krivan et al., 2008; Rosenheim, 2004). These models typically begin with patchily

A.V. Bell et al. / Theoretical Population Biology 76 (2009) 52–58

distributed resources used by prey but not predators. The predator ideal free distribution balances their tendency to aggregate where there are more prey against competition with other predators. The prey ideal free distribution balances feeding, competition and predator avoidance. Under a broad range of conditions, at the joint ESS, both predators and prey tend to be more abundant in patches with more resources; i.e., predators and prey are positively associated with each other and with the prey’s resources (Sih, 1998). The intuition is that the system is ultimately anchored by the spatial distribution of resources (Sih, 2005). Interestingly, however, empirical studies on the spatial coincidence of predators and prey do not always support the basic prediction that prey and predators should be positively associated (Hammond et al., 2007; Sih, 1998, 2005). Some field surveys find a positive predator–prey spatial association, but others show negative or random patterns of association (e.g. Rader and McArthur (1995). Similarly, experimental studies yielded patterns going in all possible directions. New, perhaps more realistic, models are needed to explain these different patterns of predator–prey spatial association. First, although patterns of spatial association in nature emerge from the movements of individuals, game-theory models of predator–prey spatial interactions typically ignore movement dynamics, and focus instead on evolutionarily-stable equilibrium distributions (Brown and Vincent, 1992; Hugie, 2003). Predators and prey, however, appear to often use relatively simple behavioral rules that result in adaptive, but not necessarily optimal behaviors (e.g. Hammond et al., 2007). Thus an alternative modeling approach examines the patterns of predator–prey spatial association that emerge from specific, plausible movement rules. Second, earlier analyses of the predator–prey race modeled habitat heterogeneity in very simple ways. These models typically considered environments with only two or a few types of patches (e.g., suitable versus unsuitable) embedded in a featureless background matrix. Natural landscapes are, of course, much more complex. We modeled a more complex landscape in which predator and prey space use depends on variation in patch quality, variation in the quality of the surrounding environment, and the scale-dependency of organism responses. In contrast to ideal free models where organisms are omniscient within the area of study, movement and patch-use decisions take place at small scales. Movements take place at much smaller scales than the full range of patches available to the individual. By additionally increasing the number of patches considered and by allowing patch quality to vary along a gradient, a more diverse array of individual behavioral responses and the resulting population-level phenomenon can be studied (Ives et al., 1993; Zinner et al., 2001). Our main theoretical objective is to outline scenarios when predators and prey strongly (or weakly) associate spatially with each other and with resources given landscapes of different complexities and simple rules that govern the movement of individuals. We modeled the movements of predators and prey in a spatially explicit simulation model. We manipulated types of movement behaviors, the patchiness of resource distributions, and densities of predators, prey, and resources, and analyzed their effect on patterns of spatial association (negative, positive, or random). A priori, we expect that (1) predators and prey are positively associated when prey are free to aggregate and do not avoid predators, (2) both predators and prey are strongly clumped when there is less conspecific avoidance, (3) predators associate more with prey resources as resources become patchy, and (4) prey and predator conspecific avoidance decreases their association with resources and prey, respectively. Though the outcome was complex, simple and counterintuitive patterns emerged which differ from existing models of predator–prey spatial association. Some conditions resulted in predators and prey following the predictions of the ideal free distribution, while others showed how behavior at small scales coupled with spatial heterogeneity can dramatically affect how a population tracks their resources.

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2. Methods The model is a 30 × 30 cellular automata populated by a single species of predator and a single species of prey. At each time step, individual predators and prey have a probability of leaving their current patch that depends on prey density, predator density, and resource levels within that patch (see equations below). If an individual leaves, it moves a distance of one cell, with an equal probability of moving in any available direction. Typically there are four possible directions except at the boundaries of the grid, where the number of directions decreases to two or three. As the study is concerned with the behavioral response race (spatial correlations), we assume that movement behaviors occur on a faster time scale than consumption. Therefore, reproduction and mortality are not incorporated in the model. Including population dynamics would add layers of complexity to answer questions outside the scope of this article (e.g. ecological stability), and are often ignored in behavioral models (e.g. evolutionary game theory and ideal free distribution). Next, we describe the rules that determined the movement of predators and prey. Per capita prey and predator emigration out of patch i(ENi , EPi ) were a function of the numbers of prey (Ni ), predators (Pi ), and resources (carrying capacity, Ki ) in that patch. For simplicity, we assumed that emigration rates were a linear combination of negative effects (what to avoid) scaled by a linear combination of positive effects (attraction). Formulated this way, emigration probabilities are exactly the cost to benefit ratio to remaining in a patch. ENi =

aNi + bPi Ki

EPi =

Pi cNi + dKi

(1)

Prey are more likely to emigrate from a patch if that patch has lower resources (lower K ), higher competition (higher N) or higher predation risk (higher P). Predators are more likely to emigrate if the patch has more competitors (higher P), fewer prey or less resources for prey. We manipulated the values of coefficients a, b, c, and d to define various movement rules used by predators and prey (Table 1). Larger coefficients resulted in a stronger effect of a given factor (predators, prey, or resources) on emigration rates from a given patch. These behavioral rules are similar to those tested by real data using information criteria (Burnham and Anderson, 2002; Hammond et al., 2007). In the cellular automata, we incorporated asynchronous movements of predators and prey, where movement responses of the whole population (e.g. predators) are based on a simultaneous analysis of all prey, predator, and resource densities. That is, all prey respond and move, then all predators respond and move, then prey again, and so on. All predators and prey exhibited a fixed behavioral response (a, b, c, and d are constant) for each treatment. Any values of Eq. (1) greater than unity were set to unity. To explore the effects of different resource patchiness, we spatially divided the total carrying capacity into uniform or aggregated distributions. In ‘‘uniform’’ distributions, each patch in the system had the same prey carrying capacity. In aggregated systems, we divided the system’s total carrying capacity into a number of clumps on the landscape. With each run of the simulation, we randomly chose points on the cellular automata, five for the ‘‘aggregated’’ resource treatment and one for the ‘‘highly aggregated’’ resource treatment, from where the carrying capacity would be symmetrically distributed. Carrying capacity decreased with distance from the chosen patch at a rate equal to exp(-sh), where s = 0.3 and h is the number of steps from the chosen step (cell order). Given the nature of the behavioral rules, we expected different densities of predators, prey, and resources to have an important effect on spatial associations. To systematically investigate density

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Table 1 Parameter values for the behavioral rules. The first letter describes the density-dependent response to conspecifics; the second letter describes the density-dependent response to the other trophic level. DD = density-dependent. Prey behavioral type

Parameter values

Predator behavioral type

Parameter values

HH

a = 1; b = 1

ll

c = 1; d = 1

a = 1; b = 0.1

hl

a = 0.1; b = 1

lh

a = 0.1; b = 0.1

hh

High DD conspecific avoidance, high DD response to predators High DD conspecific avoidance, low DD response to predators Low DD conspecific avoidance, high DD response to predators Low DD conspecific avoidance, low DD response to predators

HL LH LL

effects on spatial patterns given the free movement of predators and prey, we varied the relative densities of prey (N) and predators (P), total carrying capacity (K ), to the total number of patches (C , which is constant). This way, we tested for contrasting patterns of high versus low resource systems and high versus low densities of predators and prey. At the start of each simulation, predators and prey were randomly distributed. Our response variable was the spatial association between predators, prey, and resources at the end of each simulation. We used three statistical indices to quantify these associations, Peason’s r, Moran’s I, and Lee’s L (shown below in this order).

! i

r = " I =

i

j

& !! i

j

C L= 

vij

! i

'

! i

(& ! j

(xi − x¯ )2 $

vij xj − x¯

(2)

'& % ! j

$

') %

vij yj − y¯

& '2  " #1/2 " #1/2 ! ! ! !  vij  (xi − x¯ )2 (yi − y¯ )2 i

j

i

c = 1; d = 0.1 c = 0.1; d = 1 c = 0.1; d = 0.1

to assure the results converged. Since all association measures appeared to converge to a semi-steady-state around 500 timesteps or sooner, we ran all simulations for 1000 time-steps, and then computed the mean and variance of the last 100 time steps to approximate the associations in (2). We simultaneously simulated and spatially analyzed populations of predators and prey with identical initial conditions and behavioral rules in the absence of one another. In this way we created a type of null model with which we compared the results when predators and prey reacted to one another. We implemented the entire model in the software package Matlab (Favret and Dewalt, 2002). 3. Results

(xi − x¯ ) (yi − y¯ )

#1/2 " #1/2 ! 2 2 ¯ ¯ (xi − x) (yi − y) i i % !! $ C vij xj − x¯ (xi − x¯ ) !

Low DD conspecific avoidance and high affinity to resources; low DD attraction to prey High DD conspecific avoidance and low affinity to resources; low DD attraction to prey Low DD conspecific avoidance and high affinity to resources; high DD attraction to prey High DD conspecific avoidance and low affinity to resources; high DD attraction to prey

i

Variables x and y, which are densities, are the quantities of interest for association which represented spatial association between predators and prey, predators with resources, or prey with resources. The variable C is the total number of patches in the cellular automata, and vij is a weighting factor that determined the spatial proximity of patch i to patch j. To identify numerical associations irrespective of spatial pattern, we calculated Pearson’s r correlation coefficient. We found the spatial correlation between two populations (e.g. predators and prey) using Lee’s L (Lee, 2001, 2004), which, unlike Pearson’s r, quantifies bivariate aggregation (i.e. two populations) across neighboring cells. In order to explore links between prey and predator aggregation and the spatial correlation between the two populations, we computed the spatial autocorrelation of prey and predators using Moran’s I (Cliff and Ord, 1981). For Moran’s I and Lee’s L, we weighted the spatial proximity parameter (v ) up to first-order neighbors (distance of one movement), in order to track small scale bivariate aggregation (v = 1 for the same cell and first-order neighbors, zero otherwise). For all three indices, values approaching −1, 0, and +1 indicate negative association, no association, and positive association, respectively. We calculated all indices at each time step for all simulations. We ran each simulation for all combinations of predator and prey movement behavior (see Table 1), resource distributions, and densities (total simulations = 288) and iterated each treatment

In all runs, predators and prey were either positively associated or randomly associated, never negatively associated (panels 1A, 2F1 ). Generally, prey were positively associated with their resources (panels 1D, 2F), and predators were positively associated with prey. These patterns seem intuitively reasonable, though in some ways both can be viewed as surprising insights. If prey avoided predators strongly and successfully, then we would expect prey to be negatively associated with predators and not necessarily associated with their resources. Despite behavioral rules where prey tend to avoid predators, prey were ultimately anchored by the spatial distribution of their resources; predators and prey were both generally positively associated with resources (panels 1D, 1E, 2I, 2J). Several patterns of spatial association, generated by various combinations of predator and prey movement rules (see Table 1), fit a priori expectations. These patterns were clear within three types of resource distributions and abundances: abundant patchy resources, scarce patchy resources, and uniformly distributed resources. We next describe the results of each of these resource distributions in more detail. 3.1. Abundant patchy resources Prey clumped more with each other if resources were aggregated, there was less density-dependent avoidance of other prey, and less density-dependent avoidance of predators (behaviors LH and LL in the model; see panels 1B,). This result was most clearly manifest at higher predator to prey ratios (higher from left to right in panels 1B). Predators were overdispersed, especially when they showed strong avoidance of each other (behaviors hl and hh; see panels 1C). However, they tended to clump if there was less densitydependent avoidance of other predators (behaviors ll and lh), fewer predators, and if resources were more clumped (i.e. when prey aggregated in patches with abundant prey resources; panels 1C, compare 1CI and 1CIII). Thus, predators aggregated in patches with more resources, presumably because these patches typically have more prey. The degree of predator association with resources

1 Note abbreviations, panels 1A = Fig. 1 row A; later in the document, Fig. 2 row B column III = panel 2BIII

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Fig. 1. Spatial patterns for high resource systems. Predator (lowercase) and prey (uppercase) behavioral combinations are along the x-axis (see Table 1). The measures that track spatial patterns, Lee’s L, Moran’s I, and Pearson’s r (see text for definitions), are along the rows. The first row tracks predator–prey association, second row prey autocorrelation, third row predator autocorrelation, forth row prey–resource association, and fifth row predator–resource association. Columns of panels represent a set density treatment represented by relative densities C :K :N :P, where C = total patch number, K = carrying capacity, N = prey, P = predator. For resources, highly aggregated (dotted curves), aggregated (dashed curves), and uniform distribution types (solid curves) are represented. Alongside un-bolded curves where predators were present, bolded curves in row D represent results from simulations where predators and prey act independently (the null treatment). Points along a column of panels provide a full spatial picture of predators, prey, and association with resources. See text for further discussion. auto = autocorrelation.

increased when there was weak density-dependent avoidance of other predators and when predators were at lower densities (1:6:3:1, in Fig. 1). When predators are rare relative to prey (1:6:3:1, in Fig. 1), predators can aggregate with prey even when the predator behavioral rule prescribes aggregation avoidance (panel 1AI). These logically intuitive results may be found in many systems. Aggregated resources can anchor prey populations to a region and intraspecific behavior (especially competition) may affect how predators and prey ‘‘track’’ high resource patches. However, our results also show that prey may become more associated with resources in counter-intuitive ways. 3.2. Counter-intuitive results We found that prey clumped more with resources when prey density-dependent avoidance was strong (panels 1D). That is, prey avoidance behavior can increase the probability that prey aggregate in patches with the highest resources. After some thought, it becomes clear why. In a high-resource environment where prey do not move to avoid competitors or predators, prey are ‘‘satisfied’’ and do not necessarily aggregate in patches with the highest resource availability. In an intriguing way, our spatially explicit model showed that the presence of competition can cause prey to continually move, which allowed the prey to sample more habitat patches and thus, become more likely to aggregate in patches with the highest resources. In optimal foraging theory, this never occurs since prey can disperse perfectly throughout the environment. Our results also showed that the presence of predators can increase or decrease prey aggregation to resources depending on the prey’s avoidance of predators and other prey. When prey

had a tendency to avoid other prey (behaviors HL and HH) and when predators were absent, prey frequently moved from patches with few resources (small Ki ) and aggregated in patches with abundant resources. Adding predators in a system where prey avoided predators and other prey reduced prey aggregation to resources (compare bold and non-bold curves in panels 1D). In this case, only high prey conspecific density-dependent avoidance was needed for prey to associate very highly with resources; therefore, predators and predator avoidance could only reduce prey association with resources. On the other hand, if prey did not avoid each other (LH, LL in the model) and predators were absent, they moved less and they did not show a strong aggregation to resources (panels 1D, especially 1DIII). Adding predators under these conditions increased the prey’s aggregation to their resources because predators increased rates of prey movement and their ability to detect and respond to patches with abundant resources. However, prey were highly clumped, but not strongly with resources when prey weakly avoid each other and predators (LL, panel 1DIII). As suggested above, this is because movement rates declined, reducing prey sampling rates and their ability to respond to patches with abundant resources. The attraction of predators to prey did not appear to have a great effect on predator aggregation with resources (panels 1E). This became clearer as predator densities increased and resource levels decreased (from density 1:6:3:1 to 1:6:1:3). Instead of aggregating in patches with abundant resources, predator conspecific density-dependent avoidance tended to prevent the aggregation of predators with each other and resources (behaviors hl, hh; note the zigzag pattern in panels 1E). When the densitydependent avoidance between predators was reduced (ll, lh in the model), predators did not aggregate with prey as much as one might expect (panels 1E). This occurred because predators satisfice

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Fig. 2. Spatial patterns for low resource systems. Predator (lowercase) and prey (uppercase) behavioral combinations are along the x-axis (see Table 1). The association measures, Lee’s L, Moran’s I, and Pearson’s r (see text for definitions), are along the rows. The first row tracks predator–prey association, second row prey autocorrelation, third row predator autocorrelation, forth row prey–resource association, and fifth row predator–resource association. Columns of panels represent a set density treatment represented by relative densities C :K :N :P, where C = total patch number, K = carrying capacity, N = prey, P = predator. For resources, highly aggregated (dotted curves), aggregated (dashed curves), and uniform distribution types (solid curves) are represented. Points along a column of panels provide a full spatial picture of predators, prey, and resources. auto = autocorrelation.

and therefore do not aggregate in patches with the greatest availability of prey. Like prey, this model of predators contrasts with optimal foraging theory in that predators do not distribute themselves perfectly to prey-densities. 3.3. Scarce patchy resources Associations in general became more random in environments with scarce, patchy resources compared to environments with abundant, patchy resources (compare Figs. 1 and 2). Differences between abundant and scarce resources was not as apparent when prey did not strongly avoid competitors or predators (behaviors LH and LL). The entire pattern was governed by the resource distribution, which anchored the distributions of predators and prey. When resources were abundant, the anchoring effect was strong, generating positive associations of prey and predators with resources (panels 1D, 1E). However, there was not a strong anchoring effect when resources were scarce. If prey showed a strong tendency to avoid conspecifics and predators (behaviors HL and HH; esp. avoid conspecifics) and resources were scarce, then patterns, in general, tended to be more random than when resources were abundant (Fig. 2). Strong associations and aggregation between prey and predators primarily occurred when prey did not strongly avoid competitors or predators (behaviors LH and LL; note pattern over all columns of Fig. 2). That is, prey aggregation increased when prey avoidance behavior decreased (behavior LL); thus, predators could aggregate with prey because there were patches with markedly greater prey densities. In contrast to environments with abundant resources where we saw relatively little interaction between the effects of attraction to resources, avoidance of competitors, and avoidance of predators

on prey clumping, in environments with low resources each of these had an independent effect. The anchoring effect of clumped resources on prey aggregation only arose if prey did not strongly avoid each other (behaviors LH and LL). If prey avoided other prey strongly then prey ended up random regardless of the degree of aggregation of resources (behaviors HL and HH; panels 2G). The effects of density-dependent avoidance of predators were stronger when prey only weakly avoided conspecifics. Predator behavior also matters, but in complex ways that differ depending on the exact scenario. If prey weakly repel (behaviors LH and LL) other prey, and there is not a strong spatial anchor (resources), there is little that motivates the prey to move in a nonrandom way. Predators that satisfice at a uniform distribution because of conspecific repulsion and weak density-dependent prey attraction (behavior HL) will in turn direct prey to also distribute uniformly. That is, predators determined the spatial arrangement of the prey. 3.4. Uniform resources This situation provided insight into how the predators were spatially associated with prey when there was no ‘spatial anchor’. With no anchor, did prey successfully avoid predators, or did predators successfully aggregate in patches with more prey? In contrast to the scenario with aggregated resources, predators and prey were more randomly distributed relative to others in their trophic level and relative to each other (Fig. 2). The random distribution of predators and/or prey precludes significant negative associations. On the other hand, predators and prey were also positively associated, sometimes strongly (panel 2FIII). Positive associations can be very important if they result in predators, on average, experiencing much higher prey densities

A.V. Bell et al. / Theoretical Population Biology 76 (2009) 52–58

than if predators and prey were randomly associated, and if they resulted in prey experiencing higher than random predator densities. However, when resources are uniform, the positive predator–prey associations turned out to be situations where both were overdispersed. Although they were positively associated, it would not have an important effect on the average density of prey experienced by predators and vice-versa. This distinction is important if the ratio between predators and prey does not adequately predict trophic interactions between predators and prey; e.g. predation success may depend on the absolute densities of prey, rather than the predator to prey ratio. 4. Discussion Integrating behavioral rules, mobility, and a spatial landscape can provide new insights into the trophic interactions between predators and prey. Our model of the density-dependent avoidance of competitors and predators, and different resource distributions and abundances produced expected and counter-intuitive spatial patterns of predators and prey. First, this model shows how a patchy distribution of resources acts as a spatial anchor that generally increases positive associations between prey, predators, and resources. As resources become more uniform, positive or negative associations become less manifest. Second, prey association with resources may be increased by conspecific avoidance and the absence of predator avoidance. Prey which do not avoid each other do not associate highly with resources until predators and predator avoidance are added. Prey which move often more effectively sample and respond to patches with the most abundant resources. Third, the attraction of predators to prey did not appear to have a great effect on how much predators aggregated with resources. Predator intraspecific competition prevented high association with prey and the resource base; on the other hand, without conspecific avoidance, predators sampled less than the available prey patches and consequently occupied patches with a smaller number of prey (under-matching prey densities). The results of this model add to a growing body of theory predicting departures from game-theoretic models and/or the ideal-free distribution. Prey density, vigilance, and predator sensitivity to prey vigilance are mechanisms that Jackson et al. (2006) investigated. In a two-patch one predator environment, they show that a prey density threshold exists whereby prey clump in one group or equally split into both patches. Other work shows that at the behavioral co-evolutionary stable strategy for predators and prey, when population dynamics are stable, predators will tend to aggregate disproportionately in high preydensity patches (van Baalen and Sabelis, 1993). Our model shows that IFD predictions can be further violated by introducing (i) resource heterogeneity, (ii) simple attraction–avoidance behaviors for mobile predators and prey, and (iii) small-scale movements relative to available patches. The fact that the model did not produce negative associations between predators and prey suggests that new spatial ‘‘anchors’’ should be considered to explain this empirical observation in nature (Hammond et al., 2007; Rader and McArthur, 1995; Sih, 1998, 2005). In fact, most models with predator and prey behavioral responses predict positive association unless other spatial anchors besides prey resources are taken into account (see Sih, 2005, for a review). If patches vary only in resource value, then both predators and prey should aggregate in high resource patches. Alternative resources to predators will result in negative spatial associations, because it introduces a spatial anchor independent of the focal prey (Heithaus, 2001). Using statedependent habitat selection rules may point to other important spatial anchors (e.g. Alonzo, 2002). Since in this model prey and predators have no memory, the occurrence of negative associations

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may be underestimated if patch densities are stable enough through time. Memory (learning) may have an effect on optimal prey and predator movement strategies (Mitchell and Lima, 2002). In short, memory may create anchors for predators. Our model shows that negative associations will not arise when only prey resources are spatially fixed, even in a complex spatial landscape with small-scale movements. Further spatial modeling will complement the valuable heuristics of optimal foraging and game theory models, and supply further intuition into predator–prey dynamics. In this study, the departure from spatially-implicit, analytical model predictions arose because this spatially explicit approach allowed for both predator and prey movements, and intended that behavioral decisions and movements operate at a much smaller-scale than the total available patches. As a result, predators and prey often occupied less abundant prey patches than were available in the total resource space. Empirical work should focus on the behavioral responses of predators and prey (e.g. Hammond et al., 2007). Spatial anchors, such as prey resources, may also be put in the context of temporally varying behavioral states (Houston and McNamara, 1999), such as foraging efforts (as determined by hunger-levels) and finding mates. We anticipate that further consensus on the mechanisms behind movement behavior and resulting spatial patterns can better inform the conditions for stable interactions between predators and prey (reviewed in Murdoch et al., 2003). Acknowledgments We thank Amy R. Bell and Bruce Schaalje for comments on earlier drafts of this paper. We also thank the Department of Integrative Biology at Brigham Young University and the J.R. and Shauna Larsen Scholarship Award for financial support. References Alonzo, S.H., 2002. State-dependent habitat selection games between predators and prey: The importance of behavioural interactions and expected lifetime reproductive success. Evolutionary Ecology Research 4, 759–778. Brown, J.S., Vincent, T.L., 1992. Organization of predator–prey communities as an evolutionary game. Evolution 46, 1269–1283. Burger, A.E., Hitchcock, C.L., Davoren, G.K., 2004. Spatial aggregations of seabirds and their prey on the continental shelf off SW Vancouver Island. Marine Ecology-Progress Series 283, 279–292. Burnham, K., Anderson, D., 2002. Model Selection and Multi-Model Inference: A Practical Information-Theoretic Approach. Springer. Caudill, C.C., 2005. Trout predators and demographic sources and sinks in a mayfly metapopulation. Ecology 86, 935–946. Cliff, A.D., Ord, J.K., 1981. Spatial Processes: Models and Applications. Pion Limited, London. Cressman, R., Krivan, V., Garay, J., 2004. Ideal free distributions, evolutionary games, and population dynamics in multiple-species environments. American Naturalist 164, 473–489. Diehl, S., Cooper, S.D., Kratz, K.W., Nisbet, R.M., Roll, S.K., Wiseman, S.W., Jenkins, T.M., 2000. Effects of multiple, predator-induced behaviors on short-term producer–grazer dynamics in open systems. American Naturalist 156, 293–313. Doebeli, M., Killingback, T., 2003. Metapopulation dynamics with quasi-local competition. Theoretical Population Biology 64, 397–416. Favret, C., Dewalt, R.E., 2002. Comparing the Ephemeroptera and Plecoptera specimen databases at the Illinois Natural History Survey and using them to document changes in the Illinois fauna. Annals of the Entomological Society of America 95, 35–40. Flaxman, S.M., Lou, Y., 2009. Tracking prey or tracking the prey’s resource? Mechanisms of movement and optimal habitat selection by predators. Journal of Theoretical Biology 256, 187–200. Hammond, J., Luttbeg, B., Sih, A., 2007. Predator and prey space use: Dragonflies and tadpoles in an interactive game. Ecology 88, 1525–1535. Heithaus, M.R., 2001. Habitat selection by predators and prey in communities with asymmetrical intraguild predation. Oikos 92, 542–554. Honer, O.P., Wachter, B., East, M.L., Runyoro, V.A., Hofer, H., 2005. The effect of prey abundance and foraging tactics on the population dynamics of a social, territorial carnivore, the spotted hyena. Oikos 108, 544–554. Houston, A.L., McNamara, J.M., 1999. Models of adaptive behaviour. Cambridge University Press, Cambridge. Hugie, D.M., 2003. The waiting game: A ‘‘battle of waits’’ between predator and prey. Behavioral Ecology 14, 807–817.

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