The effect of travel costs on the ideal free distribution in stickleback

The ideal free distribution (IFD) theory, which predicts that a population of individuals will match the distribution of a patchily distributed resource, is widely used in ecology to describe the spatial distribution of animals. While many studies have shown general support of its habitat matching prediction, others have described a systematic pattern of undermatching, where too many animals feed at patches with fewer resources, and too few animals feed in richer patches. These results have been attributed to deviations from several of the assumptions of the IFD. One possible variable, the cost of travelling between patches, has received little attention. Here, we investigated the impact on resource matching when travel costs were manipulated in a simple laboratory experiment involving two continuous input patches. This experiment allowed us to control for extraneous variables and decouple time costs from energetic costs of travel. Two experiments examined the impact of varying travel costs on movement rates between foraging patches and how these travel costs impact conformity to the IFD. Our data demonstrated that there was less movement between patches and greater discrepancies from the IFD predictions as the cost of travel increased.     

Integrating individual behavior and population ecology: the potential for habitat-dependent population regulation in a reef fish

We used the predictions of the ideal free and ideal despoticdistributions (IFD and IDD, respectively) as a basis to evaluatethe link between spatial heterogeneity, behavior, and populationdynamics in a Caribbean coral reef fish. Juvenile three-spotdamselfish (Stegastes planifrons) were more closely aggregatedin patch reef habitat than on continuous back reef. Agonisticinteractions were more frequent but feeding rates were lowerin the patch versus the continuous reef habitat. Growth rateswere lower in patch reef habitat than on the continuous reef,but mortality rates did not differ. A separate experiment usingstandard habitat units demonstrated that the patterns observedin natural habitat were the result of the spatial distributionof the habitat patches rather than resource differences between habitats. Our results do not follow the predictions of simpleIFD or IDD models. This deviation from IFD and IDD predictionsmay be the result of a number of factors, including lack ofperfect information about habitat patches, high movement costs,and higher encounter rates of dispersed patches. Our resultsdemonstrate that behavioral interactions are an integral partof population dynamics and that it is necessary to considerthe spatial organization of the habitat in both behavioraland ecological investigations.     

Foraging on spatially distributed resources with sub‐optimal movement,imperfect information,and travelling costs: departures from the ideal free distribution

Ideal free distribution (IFD) theory offers an important baseline for predicting the distribution of foragers across resource patches. Yet it is well known that IFD theory relies on several over‐simplifying assumptions that are unlikely to be met in reality. Here we relax three of the most critical assumptions: (1) optimal foraging moves among patches, (2) omniscience about the utility of resource patches, and (3) cost‐free travelling between patches. Based on these generalizations, we investigate the distributions of a constant number of foragers in models with explicit resource dynamics of logistic type. We find that, first, when foragers do not always move to the patch offering maximum intake rate (optimal foraging), but instead move probabilistically according to differences in resource intake rates between patches (sub‐optimal foraging), the distribution of foragers becomes less skewed than the IFD, so that high‐quality patches attract fewer foragers. Second, this homogenization is strengthened when foragers have less than perfect knowledge about the utility of resource patches. Third, and perhaps most surprisingly, the introduction of travelling costs causes departures in the opposite direction: the distribution of sub‐optimal foragers approaches the IFD as travelling costs increase. We demonstrate that these three findings are robust when considering patches that differ in the resource's carrying capacity or intrinsic growth rate, and when considering simple two‐patch and more complex multiple‐patch models. By overcoming three major over‐simplifications of IFD theory, our analyses contribute to the systematic investigation of ecological factors influencing the spatial distribution of foragers, and thus help in deriving new hypotheses that are testable in empirical systems. A confluence of theoretical and empirical studies that go beyond classical IFD theory is essential for improving insights into how animal distributions across resource patches are determined in nature.     

Ideal free distributions, evolutionary games, and population dynamics in multiple-species environments

In this article, we develop population game theory, a theory that combines the dynamics of animal behavior with population dynamics. In particular, we study interaction and distribution of two species in a two-patch environment assuming that individuals behave adaptively (i.e., they maximize Darwinian fitness). Either the two species are competing for resources or they are in a predator-prey relationship. Using some recent advances in evolutionary game theory, we extend the classical ideal free distribution (IFD) concept for single species to two interacting species. We study population dynamical consequences of two-species IFD by comparing two systems: one where individuals cannot migrate between habitats and one where migration is possible. For single species, predator-prey interactions, and competing species, we show that these two types of behavior lead to the same population equilibria and corresponding species spatial distributions, provided interspecific competition is patch independent. However, if differences between patches are such that competition is patch dependent, then our predictions strongly depend on whether animals can migrate or not. In particular, we show that when species are settled at their equilibrium population densities in both habitats in the environment where migration between habitats is blocked, then the corresponding species spatial distribution need not be an IFD. Thus, when species are given the opportunity to migrate, they will redistribute to reach an IFD (e.g., under which the two species can completely segregate), and this redistribution will also influence species population equilibrial densities. Alternatively, we also show that when two species are distributed according to the IFD, the corresponding population equilibrium can be unstable.     

Migration dynamics for the ideal free distribution

This article verifies that the ideal free distribution (IFD) is evolutionarily stable, provided the payoff in each patch decreases with an increasing number of individuals. General frequency-dependent models of migratory dynamics that differ in the degree of animal omniscience are then developed. These models do not exclude migration at the IFD where balanced dispersal emerges. It is shown that the population distribution converges to the IFD even when animals are nonideal (i.e., they do not know the quality of all patches). In particular, the IFD emerges when animals never migrate from patches with a higher payoff to patches with a lower payoff and when some animals always migrate to the best patch. It is shown that some random migration does not necessarily lead to undermatching, provided migration occurs at the IFD. The effect of population dynamics on the IFD (and vice versa) is analyzed. Without any migration, it is shown that population dynamics alone drive the population distribution to the IFD. If animal migration tends (for each fixed population size) to the IFD, then the combined migration-population dynamics evolve to the population IFD independent of the two timescales (i.e., behavioral vs. population).     

The ideal free distribution: a review and synthesis of the game-theoretic perspective

The Ideal Free Distribution (IFD), introduced by Fretwell and Lucas in [Fretwell, D.S., Lucas, H.L., 1970. On territorial behavior and other factors influencing habitat distribution in birds. Acta Biotheoretica 19, 16-32] to predict how a single species will distribute itself among several patches, is often cited as an example of an evolutionarily stable strategy (ESS). By defining the strategies and payoffs for habitat selection, this article puts the IFD concept in a more general game-theoretic setting of the “habitat selection game”. Within this game-theoretic framework, the article focuses on recent progress in the following directions: (1) studying evolutionarily stable dispersal rates and corresponding dispersal dynamics; (2) extending the concept when population numbers are not fixed but undergo population dynamics; (3) generalizing the IFD to multiple species.For a single species, the article briefly reviews existing results. It also develops a new perspective for Parker’s matching principle, showing that this can be viewed as the IFD of the habitat selection game that models consumer behavior in several resource patches and analyzing complications involved when the model includes resource dynamics as well. For two species, the article first demonstrates that the connection between IFD and ESS is now more delicate by pointing out pitfalls that arise when applying several existing game-theoretic approaches to these habitat selection games. However, by providing a new detailed analysis of dispersal dynamics for predator-prey or competitive interactions in two habitats, it also pinpoints one approach that shows much promise in this general setting, the so-called “two-species ESS”. The consequences of this concept are shown to be related to recent studies of population dynamics combined with individual dispersal and are explored for more species or more patches.     

Dynamical facilitation of the ideal free distribution in nonideal populations

The ideal free distribution (IFD) requires that individuals can accurately perceive density‐dependent habitat quality, while failure to discern quality differences below a given perception threshold results in distributions approaching spatial uniformity. Here, we investigate the role of population growth in restoring a nonideal population to the IFD. We place a simple model of discrete patch choice under limits to the resolution by which patch quality is perceived and include population growth driven by that underlying quality. Our model follows the population's distribution through both breeding and dispersal seasons when perception limits differ in their likely influence. We demonstrate that populations of perception limited movers can approximate an IFD provided sufficient population growth; however, the emergent IFD would be temporally inconstant and correspond to reproductive events. The time to emergence of the IFD during breeding is shorter under exponential growth than under logistic growth. The IFD during early colonization of a community persists longer when more patches are available to individuals. As the population matures and dispersal becomes increasingly random, there is an oscillation in the observance of IFD, with peaks most closely approximating the IFD occurring immediately after reproductive events, and higher reproductive rates producing distributions closer to the IFD.     

Population consequences of movement decisions in a patchy landscape

Complex, human‐dominated landscapes provide unique challenges to animals. In landscapes fragmented by human activity, species whose home ranges ordinarily consist of continuous habitat in pristine environments may be forced to forage among multiple smaller habitat patches embedded in an inhospitable environment. Furthermore, foragers often must decide whether to traverse a heterogeneous suite of landscape elements that differ in risk of predation or energetic costs. We modeled population consequences of foraging decisions for animals occupying patches embedded in a heterogeneous landscape. In our simulations, animals were allowed to use three different rules for moving between patches: a) optimal selection resulting from always choosing the least‐cost path; b) random selection of a movement path; and c) probabilistic selection in which path choice was proportional to an animal's probability of survival while traversing the path. The resulting distribution of the population throughout the landscape was dependent on the movement rule used. Least‐cost movement rules (a) produced landscapes that contained the highest average density of consumers per patch. However, optimal movement resulted in an all‐or‐none pattern of occupancy and a coupling of occupied patches into pairs that effectively reduced the population to a set of sub‐populations. Random and probabilistic rules, (b and c), in relatively safe landscapes produced similar average densities and 100% occupancy of patches. However, as the level of risk associated with travel between patches increased, random movement resulted in an all‐or‐none occupancy pattern while occupied patches in probabilistic populations went extinct independently of the other patches. Our results demonstrate strong effects of inter‐patch heterogeneity and movement decisions on population dynamics, and suggest that models investigating the persistence of species in complex landscapes should take into account the effects of the intervening landscape on behavioral decisions affecting animal movements between patches.     

State-dependent ideal free distributions

Summary The standard ideal free distribution (IFD) states how animals should distribute themselves at a stable competitive equilibrium. The equilibrium is stable because no animal can increase its fitness by changing its location. In applying the IFD to choice between patches of food, fitness has been identified with the net rate of energetic gain. In this paper we assess fitness in terms of survival during a non-reproductive period, where the animal may die as a result of starvation or predation. We find the IFD when there is a large population that can distribute itself between two patches of food. The IFD in this case is state-dependent, so that an animal's choice of patch depends on its energy reserves. Animals switch between patches as their reserves change and so the resulting IFD is a dynamic equilibrium. We look at two cases. In one there is no predation and the patches differ in their variability. In the other, patches differ in their predation risk. In contrast to previous IFDs, it is not necessarily true that anything is equalized over the two patches.     

Impact of spatial heterogeneity on a predator-prey system dynamics

This paper deals with the study of a predator-prey model in a patchy environment. Prey individuals moves on two patches, one is a refuge and the second one contains predator individuals. The movements are assumed to be faster than growth and predator-prey interaction processes. Each patch is assumed to be homogeneous. The spatial heterogeneity is obtained by assuming that the demographic parameters (growth rates, predation rates and mortality rates) depend on the patches. On the predation patch, we use a Lotka-Volterra model. Since the movements are faster that the other processes, we may assume that the frequency of prey and predators become constant and we would get a global predator-prey model, which is shown to be a Lotka-Volterra one. However, this simplified model at the population level does not match the dynamics obtained with the complete initial model. We explain this phenomenom and we continue the analysis in order to give a two-dimensional predator-prey model that gives the same dynamics as that provided by the complete initial one. We use this simplified model to study the impact of spatial heterogeneity and movements on the system stability. This analysis shows that there is a globally asymptotically stable equilibrium in the positive quadrant, i.e. the spatial heterogeneity stabilizes the equilibrium.     

The evolution of patch selection in stochastic environments

A null model for habitat patch selection in spatially heterogeneous environments is the ideal free distribution (IFD), which assumes individuals have complete knowledge about the environment and can freely disperse. Under equilibrium conditions, the IFD predicts that local population growth rates are zero in all occupied patches, sink patches are unoccupied, and the fraction of the population selecting a patch is proportional to the patch's carrying capacity. Individuals, however, often experience stochastic fluctuations in environmental conditions and cannot respond to these fluctuations instantaneously. An evolutionary stability analysis for fixed patch-selection strategies reveals that environmental uncertainty disrupts the classical IFD predictions: individuals playing the evolutionarily stable strategy may occupy sink patches, local growth rates are negative and typically unequal in all patches, and individuals prefer higher-quality patches less than predicted by their carrying capacities. Spatial correlations in environmental fluctuations can enhance or marginalize these trends. The analysis predicts that continually increasing environmental variation first selects for range expansion, then selects for persisting coupled sink populations, and ultimately leads to regional extinction. In contrast, continually increasing habitat degradation first selects for range contraction and may select for persisting coupled sink populations before regional extinction. These results highlight the combined roles of spatial and temporal heterogeneity on the evolution of habitat selection.     

EVOLUTION OF DISPERSAL RATES IN METAPOPULATION MODELS: BRANCHING AND CYCLIC DYNAMICS IN PHENOTYPE SPACE

We study the evolution of dispersal rates in a two patch metapopulation model. The local dynamics in each patch are given by difference equations, which, together with the rate of dispersal between the patches, determine the ecological dynamics of the metapopulation. We assume that phenotypes are given by their dispersal rate. The evolutionary dynamics in phenotype space are determined by invasion exponents, which describe whether a mutant can invade a given resident population. If the resident metapopulation is at a stable equilibrium, then selection on dispersal rates is neutral if the population sizes in the two patches are the same, while selection drives dispersal rates to zero if the local abundances are different. With non-equilibrium metapopulation dynamics, non-zero dispersal rates can be maintained by selection. In this case, and if the patches are ecologically identical, dispersal rates always evolve to values which induce synchronized metapopulation dynamics. If the patches are ecologically different, evolutionary branching into two coexisting dispersal phenotypes can be observed. Such branching can happen repeatedly, leading to polymorphisms with more than two phenotypes. If there is a cost to dispersal, evolutionary cycling in phenotype space can occur due to the dependence of selection pressures on the ecological attractor of the resident population, or because phenotypic branching alternates with the extinction of one of the branches. Our results extend those of Holt and McPeek (1996), and suggest that phenotypic branching is an important evolutionary process. This process may be relevant for sympatric speciation.     

Coexistence of competitors in metacommunities due to spatial variation in resource growth rates; does R * predict the outcome of competition?

Simple mathematical models are used to investigate the coexistence of two consumers using a single limiting resource that is distributed over distinct patches, and that has unequal growth rates in the different patches. Relatively low movement rates or high demographic rates of an inefficient resource exploiter allow it to coexist at a stable equilibrium with a more efficient species whose ratio of movement to demographic rates is lower. The range of conditions allowing coexistence depends on the between‐patch heterogeneity in resource growth rates, but this range can be quite broad. The between‐patch movement of the more efficient consumer turns patches with high resource growth rates into sources, while low‐growth‐rate patches effectively become sinks. A less efficient species can coexist with or even exclude the more efficient species from the global environment if it is better able to bias its spatial distribution towards the source patches. This can be accomplished with density independent dispersal if the less efficient species has a lower ratio of per capita between‐patch movement rate to demographic rates. Conditions that maximize the range of efficiencies allowing coexistence of two species are: a relatively high level of heterogeneity in resource growth conditions; high dispersal (or low demographic rates) of the superior competitor; and low dispersal (or high demographic rates) of the inferior competitor. Global exclusion of the more efficient competitor requires that the inferior competitor have sufficient movement to also produce a source‐sink environment.     

Distribution of a naturally fluctuating ungulate population among heterogeneous plant communities: ideal and free?

1. Herbivore distribution is often assumed to follow the ideal free distribution (IFD) model. This assumes that organisms are omniscient about forage quality and availability within the area available to them and are free to move, with negligible cost, throughout this environment. If this were the case we would expect that, at lowest densities, all animals would be found in the best habitat patches, with less desirable habitats being occupied stepwise as population density increases. We test this using data from a naturally fluctuating population of feral Soay sheep. 2. We show that, although the distribution of individuals is correlated positively with food quality, in line with patterns reported for hill sheep in Scotland, their distribution does not conform to the predictions of the IFD model. We argue that it is the dynamic nature of their food resource that causes this departure from the predictions of the IFD model and make the case that the IFD model, in its unmodified form, is inappropriate for use in modelling distribution among patches containing dynamic resources.     

How does adaptive consumer movement affect population dynamics in consumer-resource metacommunities with homogeneous patches?

This article uses simple models to explore the impact of adaptive movement by consumers on the population dynamics of a consumer-resource metacommunity consisting of two identical patches. Consumer-resource interactions within a patch are described by the Rosenzweig-MacArthur predator-prey model, and these dynamics are assumed to be cyclic in the absence of movement. The per capita movement rate from one patch to the other is an increasing function of the difference between the per capita birth minus death rate in the destination patch and that in the currently occupied patch. Several variations on this model are considered. Results show that adaptive movement frequently creates anti-phase cycles in the two patches; these suppress the predator-prey cycle and lead to low temporal variation of the total population sizes of both species. Paradoxically, even when movement is very sensitive to the fitness difference between patches, perfect synchrony of patches is often much less likely than in comparable systems with random movement. Under these circumstances adaptive movement of consumers often generates differences in the average properties of the two patches. In addition, mean global densities and responses to global perturbations often differ greatly from similar systems with no movement or random movement.     

Large-scale Input Matching by Urban Feral Pigeons (Columba livia)

Ideal Free Distribution (IFD) theory predicts the number of animals choosing habitats of differing quality. Most experimental tests of the IFD have been conducted at small spatial scales (i.e. smaller than maximum daily movement of animals) by comparing the number of animals foraging at adjacent food patches of different quality. Urban pigeons ( Columba livia ) feed in large, open aggregations, and can distribute according to predictions of the IFD at alternative food patches. In this study, we test IFD predictions over a much larger spatial scale by comparing the abundance of urban feral pigeons at four sites spread over the city centre of Montréal, Québec, Canada, to the amount of anthropogenically provided food in each site. We found that the pigeons' distribution among the four sites qualitatively matched that of resources available at these sites. After controlling for the effect of stochastic variation in food resources, two pair-wise comparisons between sites indicated undermatching, one indicated matching and three indicated overmatching of consumers to resources. These results suggest that the pigeons inhabiting the downtown area of Montréal may behave as a single population that distributes qualitatively among foraging sites in proportion to the quantity of food offered, and that deviations from expectations cannot be attributed simply to stochastic variation in the food levels at the sites.     

Spatial covariation of fish population vital rates in a stream network

Animal populations are spatially structured in heterogeneous landscapes, in which local patches with differing vital rates are connected by dispersal of individuals to varying degrees. Although there is evidence that vital rates differ among local populations, much less is understood about how vital rates covary among local patches in spatially heterogeneous landscapes. In this study, we conducted a nine-year annual mark–recapture survey to characterize spatial covariation of survival and growth for two Japanese native salmonids, white-spotted charr Salvelinus leucomaenis japonicus and red-spotted masu salmon Oncorhynchus masou ishikawae, in a headwater stream network composed of distinctly different tributary and mainstem habitats. Spatial structure of survival and growth differed by species and age class, but results provided support for negative covariation between vital rates, where survival was higher in the tributary habitat but growth was higher in the mainstem habitat. Thus, neither habitat was apparently more important than the other, and local habitats with complementary vital rates may make this spatially structured population less vulnerable to environmental change (i.e. portfolio effect). Despite the spatial structure of vital rates and possibilities that fish can exploit spatially distributed resources, movement of fish was limited due partly to a series of low-head dams that prevented upstream movement of fish in the study area. This study shows that spatial structure of vital rates can be complex and depend on species and age class, and this knowledge is likely paramount to elucidating dynamics of spatially structured populations.     

Dispersal-mediated coexistence of competing predators

Models of metapopulations have often ignored local community dynamics and spatial heterogeneity among patches. However, persistence of a community as a whole depends both on the local interactions and the rates of dispersal between patches. We study a mathematical model of a metacommunity with two consumers exploiting a resource in a habitat of two different patches. They are the exploitative competitors or the competing predators indirectly competing through depletion of the shared resource. We show that they can potentially coexist, even if one species is sufficiently inferior to be driven extinct in both patches in isolation, when these patches are connected through diffusive dispersal. Thus, dispersal can mediate coexistence of competitors, even if both patches are local sinks for one species because of the interactions with the other species. The spatial asynchrony and the competition-colonization trade-off are usual mechanisms to facilitate regional coexistence. However, in our case, two consumers can coexist either in synchronous oscillation between patches or in equilibrium. The higher dispersal rate of the superior prompts rather than suppresses the inferior. Since differences in the carrying capacity between two patches generate flows from the more productive patch to the less productive, loss of the superior by emigration relaxes competition in the former, and depletion of the resource by subsidized consumers decouples the local community in the latter.     

The influence of spatio-temporal resource fluctuations on insular rat population dynamics

Local spatio-temporal resource variations can strongly influence the population dynamics of small mammals. This is particularly true on islands which are bottom-up driven systems, lacking higher order predators and with high variability in resource subsidies. The influence of resource fluctuations on animal survival may be mediated by individual movement among habitat patches, but simultaneously analysing survival, resource availability and habitat selection requires sophisticated analytical methods. We use a Bayesian multi-state capture-recapture model to estimate survival and movement probabilities of non-native black rats (Rattus rattus) across three habitats seasonally varying in resource availability. We find that survival varies most strongly with temporal rainfall patterns, overwhelming minor spatial variation among habitats. Surprisingly for a generalist forager, movement between habitats was rare, suggesting individuals do not opportunistically respond to spatial resource subsidy variations. Climate is probably the main driver of rodent population dynamics on islands, and even substantial habitat and seasonal spatial subsidies are overwhelmed in magnitude by predictable annual patterns in resource pulses. Marked variation in survival and capture has important implications for the timing of rat control.