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.     

Evolutionary consequences of asymmetric dispersal rates

We study the consequences of asymmetric dispersal rates (e.g., due to wind or current) for adaptive evolution in a system of two habitat patches. Asymmetric dispersal rates can lead to overcrowding of the "downstream" habitat, resulting in a source-sink population structure in the absence of intrinsic quality differences between habitats or can even cause an intrinsically better habitat to function as a sink. Source-sink population structure due to asymmetric dispersal rates has similar consequences for adaptive evolution as a source-sink structure due to habitat quality differences: natural selection tends to be biased toward the source habitat. We demonstrate this for two models of adaptive evolution: invasion of a rare allele that improves fitness in one habitat but reduces it in the other and antagonistic selection on a quantitative trait determined by five additive loci. If a habitat can sustain a population without immigration, the conditions for adaptation to that habitat are most favorable if there is little or no immigration from the other habitat; the influence of emigration depends on the magnitude of the allelic effects involved and other parameters. If, however, the population is initially unable to persist in a given habitat without immigration, our model predicts that the population will be most likely to adapt to that habitat if the dispersal rates in both directions are high. Our results highlight the general message that the effect of gene flow upon local adaptation should depend profoundly on the demographic context of selection.     

Habitat selection,interspecific interactions and landscape composition

Summary I argue here that, from the perspective of any individual, most landscapes are composed of only three basic types of habitats. These are: (1) source habitat in which reproduction exceeds mortality and the expected per capita growth rate is greater than one; (2) sink habitat, in which limited, reproduction is possible but will not on average, compensate for mortality and the per capita rate of growth is between zero and one; and (3) unusable habitat, which comprises the matrix of all habitats that are never exploited by the species in question, and in which patches of source and sink habitats are embedded. Unlike earlier source-sink models, this model explicitly considers the effects that substituting one type of habitat for another has on the equilibrium size of a population and the interactions between species which can use both source and sink habitats. The model demonstrates that the equilibrium size of a species' population can sometimes be increased by substituting unusable habitat for sink habitat. Thus, even though the average patch quality in the landscape may be decreased, the overall quality of the landscape can increase. For two species with distinct habitat preferences, interactions between species can vary qualitatively as well as quantitatively as a function of the relative abundances of each of the habitat types. The model also shows that the interactions between species are particularly sensitive to the relative costs of moving between patches and sampling patches to determine their quality. Recent fragmentation of natural landscapes may increase the cost of searching for usable (source or sink) patches. Under some conditions, the interspecific interactions may be substantially more negative (competitive) than the interactions that evolved in the original natural landscape, further reducing population sizes and increasing the likelihood of competitive exclusion in fragmented modern landscapes.     

Dispersal among habitats varying in fitness: reciprocating migration through ideal habitat selection

Current evolutionary models of dispersal set the ends of a continuum where the number of individuals emigrating from a habitat either equals the number of individuals immigrating (balanced dispersal) or where emigrants flow from a source habitat to a corresponding sink. Theories of habitat selection suggest a more sophisticated conditional strategy where individuals disperse from habitats where they have the greatest impact on fitness to habitats where their per capita impact is lower. Asymmetries between periods of population growth and decline result in a reciprocating dispersal strategy where the direction of migration is reversed as populations wax and wane. Thus, for example, if net migration of individuals flows from high- to low-density habitats during periods of population growth, net migration will flow in the opposite direction during population decline. Stochastic simulations and analytical models of reciprocating dispersal demonstrate that fitness, carrying capacity, stochastic dynamics, and interference from dominants interact to determine whether dispersal is balanced between habitats, or whether one habitat or the other acts as a net donor of dispersing individuals. While the pattern of dispersal may vary, each is consistent with an underlying strategy of density-dependent habitat selection.     

On the evolutionary stability of sink populations

The evolution of adaptive behaviours can influence population dynamics. Conversely, population dynamics can affect both the rate and direction of adaptive evolution. This paper examines reasons why sink populations – populations maintained by immigration, preventing local extinction – might persist in the habitat repertoire of a species over evolutionary time-scales. Two such reasons correspond to standard explanations for deviations from an ideal free habitat distribution: organisms may not be free to settle in whichever habitat has the highest potential fitness, and may be constrained by costs, perceptual limitations, or mode of dispersal in the acuity of their habitat selectivity. Here, I argue that a third general reason for persistent sink populations is provided by unstable population dynamics in source habitats. I present a simple model illustrating how use of a sink habitat may be selectively advantageous, when a source population has unstable dynamics (which necessarily reflects temporal variation in local fitnesses). Species with unstable local dynamics in high-quality habitats should be selected to utilize a broader range of habitats than species with stable local dynamics, and in particular in some circumstances should utilize sink habitats. This observation has implications for the direction of niche evolution, and the likelihood of niche conservatism.     

Source–sink dynamics: how sinks affect demography of sources

Models of source–sink population dynamics have to make assumptions about whether, and eventually how, demographic parameters in source habitats are dependent on the demography in sink habitats. However, the empirical basis for making such assumptions has been weak. Here we report a study on experimental root vole populations, where estimates of demographic parameters were contrasted between source patches in source–sink (treatment) and source–source systems (control). In the presence of a sink patch (simulated by a pulsed removal of immigrants), source‐patch populations failed to increase over the breeding season, mainly due to a high spatially density‐dependent dispersal rate from source to sink patches. The per capita recruitment rate was almost two times higher in source–sink than in the source–source systems, but this did not compensate for the loss rate due to dispersal from source to sink patches. Sex ratio in the source–sink systems became less female biased, probably as a result of an enhanced frequency of dispersal movements in females. Good knowledge of the degree of density‐and habitat‐dependent dispersal is critical for predicting the dynamics of source–sink populations.     

The role of sink to source re-colonisation in the population dynamics of insects living in unstable habitats: an example of terrestrial chironomids

Certain species of terrestrial chironomids (Diptera) are specialised on open patches in initial stages of primary or secondary succession (early fallow, lichens and mosses on rocks, etc.). These "source" habitats provide good quality food for their larvae and most offspring are produced here, but they are sensitive to summer desiccation. This often results in extinction of the summer larval population, followed in winter by re-colonisation from less suitable, but more stable "sink" habitats in the surrounding landscape. Soil dwelling and long-lived larvae are poor migrants; short-lived, winged adult females select patches for their development. Proper choice of oviposition sites and consequent distribution of eggs among individual habitats is thus critical for the success of these species. A mathematical model was developed in order to find out whether this re-colonisation strategy could ensure population persistence at the landscape level. The model was verified using long-term data on Smittia atterima abundance in old fields. The results indicate that even a small proportion of eggs laid in a sink habitat can ensure a successful re-colonisation of the source habitat. Thus, re-colonisation of source habitats from sink habitats is concluded to be one of the reasons for persistence of the latter. The model indicates that this re-colonisation may ensure population persistence even in conditions when exclusive use of only one habitat leads to population extinction either due to environmental stress or to a negative growth rate.     

Sex-biased dispersal and adaptation to marginal habitats

If gene flow occurs through both sexes but only females contribute to population growth, adaptation to marginal (sink) habitats should be differentially affected by male versus female dispersal. Here I address this problem with two models. First, I consider the fate of a rare allele that improves fitness in the marginal habitat but reduces fitness in the core (source) habitat. Then I study the evolution of a polygenic character mediating a trade-off in fitness between the habitats. Both approaches led to qualitatively similar predictions. The effect of a difference in the dispersal rate between the sexes depends on the degree to which immigration from the core habitat boosts the reproductive output from the marginal habitat. This boost is slight if the marginal habitat is able to sustain well a population without immigration. In that case, both female- and male-biased dispersal is more favorable for adaptation to marginal habitats than equal dispersal of both sexes (assuming that the dispersal rate averaged over the sexes is kept constant). In contrast, if the marginal habitat is an absolute sink unable to sustain a population without immigration, the conditions for adaptation to that habitat are least favorable under highly male-biased dispersal and most favorable under highly female-biased dispersal. Under some circumstances, high average (male+female) dispersal is more favorable than low dispersal. Thus, gene flow should not be seen solely as thwarting adaptation to marginal habitats. The results are interpreted in terms of how male and female dispersal affects the relative rate of gene flow from the source to the sink habitat and in the opposite direction. This study predicts that ecological niches of taxa with female-biased dispersal should tend to be broader and more evolutionarily flexible.     

How can maladaptive habitat choice generate source‐sink population dynamics?

Several theoretical models have been proposed to describe population dynamics in a spatially heterogeneous environment. The source-sink model is among the most popular. Diffendorfer recently summarized its assumptions and predictions. Given the model reviewed, he argued that source-sink population dynamics arises if dispersal is somehow constrained. I offer an additional mechanism by suggesting that source-sink population dynamics can be generated by anthropogenic changes in landscapes that occur so quickly that organisms no longer make optimal habitat selection decisions. Individuals select the same habitats as their ancestors but these decisions no longer provide high fitness because of human-induced changes in habitat quality, such as increased rates of predation and/or parasitism. Provided that some of the habitats selected are turned by human-induced changes into sink habitats, source-sink population dynamics can emerge.     

The effects of habitat fragmentation on persistence of source-sink metapopulations in systems with predators and prey or apparent competitors.

We consider systems with one predator and one prey, or a common predator and two prey species (apparent competitors) in source and sink habitats. In both models, the predator species is vulnerable to extinction, if productivity in the source is insufficient to rescue demographically deficient sink populations. Conversely, in the model with two prey species, if the source is too rich, one of the prey species may be driven extinct by apparent competition, since the predator can maintain a large population because of the alternative prey. Increasing the rate of predator movement from the source population has opposite effects on prey and predator persistence. High emigration rate exposes the predator population to danger of extinction, reducing the number of individuals that breed and produce offspring in the source habitat. This may promote coexistence of prey by relaxing predation pressure and apparent competition between the two prey species. The number of sinks and spatial arrangement of patches, or connectivity between patches, also influence persistence of the species. More sinks favor the prey and fewer sinks are advantageous to the predator. A linear pattern with the source at one end is profitable for the predator, and a centrifugal pattern in which the source is surrounded by sinks is advantageous to the prey. When the dispersal rate is low, effects of the spatial structure may exceed those of the number of sinks. In brief, productivity in patches and patterns of connectivity between patches differentially influence persistence of populations in different trophic levels.     

Landscape configuration determines gene flow and phenotype in a flightless forest-edge ground-dwelling bush-cricket, Pholidoptera griseoaptera

Spatial configuration of habitats influences genetic structure and population fitness whereas it affects mainly species with limited dispersal ability. To reveal how habitat fragmentation determines dispersal and dispersal-related morphology in a ground-dispersing insect species we used a bush-cricket (Pholidoptera griseoaptera) which is associated with forest-edge habitat. We analysed spatial genetic patterns together with variability of the phenotype in two forested landscapes with different levels of fragmentation. While spatial configuration of forest habitats did not negatively affect genetic characteristics related to the fitness of sampled populations, genetic differentiation was found higher among populations from an extensive forest. Compared to an agricultural matrix between forest patches, the matrix of extensive forest had lower permeability and posed barriers for the dispersal of this species. Landscape configuration significantly affected also morphological traits that are supposed to account for species dispersal potential; individuals from fragmented forest patches had longer hind femurs and a higher femur to pronotum ratio. This result suggests that selection pressure act differently on populations from both landscape types since dispersal-related morphology was related to the level of habitat fragmentation. Thus observed patterns may be explained as plastic according to the level of landscape configuration; while anthropogenic fragmentation of habitats for this species can lead to homogenization of spatial genetic structure.     

Spatial structure, dispersal, and management of a recovering raptor population

The recovery of the peregrine falcon (Falco peregrinus anatum) in California has taken place amid strong geographical differences in habitat quality, potentially creating a sink population in the southern coastal habitat and source populations in the northern interior and urban habitats. We analyzed long-term monitoring data to investigate the mechanisms and consequences of spatial structuring for the recovery of this set of nonstable subpopulations. Dispersal rates between habitats were asymmetric, with extremely limited dispersal out of the interior habitat and a strong tendency for birds in the southern coast to disperse to the urban habitats. We used these dispersal estimates and habitat-specific productivity rates to build a set of regional population models that describe population growth within and dispersal between each subpopulation. We tested for the existence of habitat-specific survival and territory acquisition rates by comparing model projections with the number of breeding pairs censused annually in each subpopulation. Our analyses indicate a high rate of survival for interior birds and suggest that both the interior and urban subpopulations were regulated by territory availability over the study period. The inherent spatial structure of this regional peregrine falcon population has had a considerable influence on its recovery and management.     

Evolution of positive and negative density-dependent dispersal

Understanding the evolution of density-dependent dispersal strategies has been a major challenge for evolutionary ecologists. Some existing models suggest that selection should favour positive and others negative density-dependence in dispersal. Here, we develop a general model that shows how and why selection may shift from positive to negative density-dependence in response to key ecological factors, in particular the temporal stability of the environment. We find that in temporally stable environments, particularly with low dispersal costs and large group sizes, habitat heterogeneity selects for negative density-dependent dispersal, whereas in temporally variable environments, particularly with high dispersal costs and small group sizes, habitat heterogeneity selects for positive density-dependent dispersal. This shift reflects the changing balance between the greater competition for breeding opportunities in more productive patches, versus the greater long-term value of offspring that establish themselves there, the latter being very sensitive to the temporal stability of the environment. In general, dispersal of individuals out of low-density patches is much more sensitive to habitat heterogeneity than is dispersal out of high-density patches.     

Experimental demonstration of accelerated extinction in source‐sink metapopulations

Population extinction is a fundamental ecological process which may be aggravated by the exchange of organisms between productive (source) and unproductive (sink) habitat patches. The extent to which such source‐sink exchange affects extinction rates is unknown. We conducted an experiment in which metapopulation effects could be distinguished from source‐sink effects in laboratory populations of Daphnia magna. Time‐to‐extinction in this experiment was maximized at intermediate levels of habitat fragmentation, which is consistent with a minority of theoretical models. These results provided a baseline for comparison with experimental treatments designed to detect effects of concentrating resources in source patches. These treatments showed that source‐sink configurations increased population variability (the coefficient of variation in abundance) and extinction hazard compared with homogeneous environments. These results suggest that where environments are spatially heterogeneous, accurate assessments of extinction risk will require understanding the exchange of organisms among population sources and sinks. Such heterogeneity may be the norm rather than the exception because of both the intrinsic heterogeneity naturally exhibited by ecosystems and increasing habitat fragmentation by human activity.     

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.     

Habitat matching: Alternatives and implications to populations and communities

Summary I evaluate habitat matching rules based on ideal distribution models of density-dependent habitat use. Recent approaches and the ideal free continuous input matching rule on which they depend, are restricted to only those habitats that are jointly occupied across the full range of population sizes. These assumptions may often be inappropriate to field applications of habitat matching. I develop alternatives that can be applied to a wide array of ideal forms of habitat selection, including the ideal free, continuous input example. Input matching can be distinguished from assumptions of consumer-resource models and preemptive habitat use by regressions of density between paired habitats (isodars). Isodars for continuous input models should be linear on a logarithmic scale, while those for consumer-resource models should be linear on an arithmetic scale. Pre-emptive isodars can be distinguished from the others by dramatic non-linearities at both low and high densities. Field data on white-footed mice support the consumer-resource theory. Implications of the rules for population regulation and community organization are highlighted by new models that specify how the fitness of pre-emptive habitat selectors should decline with increasing density. Strong non-linearities produced by comparisons between variable and homogeneous habitats produce reversing source-sink population regulation and a new form of cyclical community dynamics. Variable habitats act as a source of emigrants at low density and a sink for immigrants at high density. Subordinate species may occupy only the variable habitat at both low and high density.     

Temporal autocorrelation can enhance the persistence and abundance of metapopulations comprised of coupled sinks

In spatially heterogeneous landscapes, some habitats may be persistent sources, providing immigrants to sustain populations in unfavorable sink habitats (where extinction is inevitable without immigration). Recent theoretical and empirical studies of source-sink systems demonstrate that temporally variable local growth rates in sinks can substantially increase average abundance of a persisting population, provided that the variation is positively autocorrelated--in effect, temporal variation inflates average abundance. Here we extend these results to a metapopulation in which all habitat patches are sinks. Using numerical studies of a population with discrete generations (buttressed by analytic results), we show that temporal variation and moderate dispersal can jointly permit indefinite persistence of the metapopulation and that positive autocorrelation both lowers the magnitude of variation required for persistence and increases the average abundance of persisting metapopulations. These effects are weakened--but not destroyed--if variation in local growth rates is spatially synchronized and dispersal is localized. We show that the inflationary effect is robust to a number of extensions of the basic model, including demographic stochasticity and density dependence. Because ecological and environmental processes contributing to temporally variable growth rates in natural populations are typically autocorrelated, these observations may have important implications for species persistence.     

Scales and costs of habitat selection in heterogeneous landscapes

Summary Two scales of habitat selection are likely to influence patterns of animal density in heterogeneous landscapes. At one scale, habitat selection is determined by the differential use of foraging locations within a home range. At a larger scale, habitat selection is determined by dispersal and the ability to relocate the home range. The limits of both scales must be known for accurate assessments of habitat selection and its role in effecting spatial patterns in abundance. Isodars, which specify the relationships between population density in two habitats such that the expected reproductive success of an individual is the same in both, allow us to distinguish the two scales of habitat selection because each scale has different costs. In a two-habitat environment, the cost of rejecting one of the habitats within a home range can be expressed as a devaluation of the other, because, for example, fine-grained foragers must travel through both. At the dispersal scale, the cost of accepting a new home range in a different habitat has the opposite effect of inflating the value of the original habitat to compensate for lost evolutionary potential associated with relocating the home range. These costs produce isodars at the foraging scale with a lower intercept and slope than those at the dispersal scale.Empirical data on deer mice occupying prairie and badland habitats in southern Alberta confirm the ability of isodar analysis to differentiate between foraging and dispersal scales. The data suggest a foraging range of approximately 60 m, and an effective dispersal distance near 140 m. The relatively short dispersal distance implies that recent theories may have over-emphasized the role of habitat selection on local population dynamics. But the exchange of individuals between habitats sharing irregular borders may be substantial. Dispersal distance may thus give a false impression of the inability of habitat selection to help regulate population density.     

Patterns of dispersal and dynamics among habitat patches varying in quality

Both source-sink theory and extensions of optimal foraging theory ("balanced dispersal" theory) address dispersal and population dynamics in landscapes where habitat patches vary in quality. However, studying dispersal mechanisms empirically has proven difficult, and dispersal is rarely tied back to long-term spatial dynamics. We used a manipulable laboratory system consisting of bacteria and protozoa to investigate the ability of source-sink and optimal foraging theories to explain both dispersal and emergent spatial dynamics. Consistent with source-sink models and contrary to balanced dispersal models, there was a consistent net flux of protist individuals from high to low resource patches. However, unlike the simplest source-sink models, intermediate rates of dispersal led to highest abundances in low resource patches. Side experiments found strong density dependence in local population dynamics and differences in average protist body size in high and low resource patches. Parameterization and analysis of a two-patch model showed that high migration from high to low resource patches could have depressed population density in low resource patches, creating pseudosinks. The movement of individuals and biomass from sources to sinks (a form of ecosystem subsidy) resulted in the convergence of body size and population densities in sources and sinks. Our results indicate a need to carefully consider movement patterns and interaction with local dynamics in potential source-sink systems.     

Sources and sinks in population biology

The various habitats inhabited by a given species are never of the same quality. When demographic models take into account this habitat heterogeneity, the source-sink concept naturally emerges: a local demographic surplus arises in good quality habitats (source), and a local demographic deficit occurs in habitats of poor quality (sink). Within a landscape, a permanent migration of propagules or individuals from source to sink habitats may lead to a stabilization of the overall demographic system. This simple situation, explored in the recent literature, has surprising properties. In particular, it requires a change in our view of classical concepts such as ecological niche and carrying capacity, it can explain the existence and persistence of local maladaptation and it can improve conservation practice.