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.     

A Lyapunov function for piecewise-independent differential equations: stability of the ideal free distribution in two patch environments

In this article we construct Lyapunov functions for models described by piecewise-continuous and independent differential equations. Because these models are described by discontinuous differential equations, the theory of Lyapunov functions for smooth dynamical systems is not applicable. Instead, we use a geometrical approach to construct a Lyapunov function. Then we apply the general approach to analyze population dynamics describing exploitative competition of two species in a two-patch environment. We prove that for any biologically meaningful parameter combination the model has a globally stable equilibrium and we analyze this equilibrium with respect to parameters.      

Transition from a random to an ideal free to an ideal despotic distribution: the effect of individual difference in growth

Individual differences in growth can lead to a monopolistic form of food competition. We studied the long-term transition in the mode of competition and the distribution of individuals between food patches of the cloned salmonid fish, Oncorhynchus masou ishikawae, in the laboratory. This transition was accompanied by growth depensation, i.e., the increase over time in the variance of size between individuals resulting from the differences in individual growth rates. The 120-cm experimental tanks were divided into two compartments (patches) between which an opaque partition was placed. Fish were able to move freely between the patches and therefore were able to assess the patch quality using long-term memory, but they were not able to see the food input in the other patch directly. The distribution between the two food patches, the amount of food gained, and the growth and the agonistic behavior of four groups of six individuals were observed over 4 weeks. We found that (1) within-group variation in body weight increased with time; (2) on average, the better patch was used by more individuals than predicted by a random distribution but fewer individuals than predicted by an ideal free distribution, and (3) the distribution and pattern of resource use by the fish changed over the 4-week experimental period from a random distribution to an ideal free distribution and finally to an ideal despotic distribution. We suggest that growth depensation causes the long-term change in the spatial distribution and pattern of resource use by competitors. Received: December 19, 2000 / Accepted: March 19, 2001     

The effect of the Holling type II functional response on apparent competition

This article analyzes the classical 2-resource-1-consumer apparent competition community module with the Holling type II functional response. Two types of resource regulation (top-down vs. combined top-down and bottom-up) and two types of consumer behaviors (inflexible consumers with fixed preferences for resources vs. adaptive consumers) are considered. When resources grow exponentially and consumers are inflexible foragers, one resource is always outcompeted due to strong apparent competition. Density dependent resource growth relaxes apparent competition so that resources can coexist. As multiple attractors (either equilibria or limit cycles) coexist, population dynamics and community composition depend on initial population densities. Population dynamics change dramatically when consumers forage adaptively. In this case, the results both for top-down, and combined top-down and bottom-up regulation are similar and they show that species persistence occurs for a much larger set of parameter values when compared with inflexible consumers. Moreover, population dynamics will be chaotic when resource carrying capacities are high enough. This shows that adaptive consumer switching can destabilize population dynamics.     

The ideal free distribution as an evolutionarily stable strategy

We examine the evolutionary stability of strategies for dispersal in heterogeneous patchy environments or for switching between discrete states (e.g. defended and undefended) in the context of models for population dynamics or species interactions in either continuous or discrete time. There have been a number of theoretical studies that support the view that in spatially heterogeneous but temporally constant environments there will be selection against unconditional, i.e. random, dispersal, but there may be selection for certain types of dispersal that are conditional in the sense that dispersal rates depend on environmental factors. A particular type of dispersal strategy that has been shown to be evolutionarily stable in some settings is balanced dispersal, in which the equilibrium densities of organisms on each patch are the same whether there is dispersal or not. Balanced dispersal leads to a population distribution that is ideal free in the sense that at equilibrium all individuals have the same fitness and there is no net movement of individuals between patches or states. We find that under rather general assumptions about the underlying population dynamics or species interactions, only such ideal free strategies can be evolutionarily stable. Under somewhat more restrictive assumptions (but still in considerable generality), we show that ideal free strategies are indeed evolutionarily stable. Our main mathematical approach is invasibility analysis using methods from the theory of ordinary differential equations and nonnegative matrices. Our analysis unifies and extends previous results on the evolutionary stability of dispersal or state-switching strategies.     

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.     

Children's competition in a natural setting: evidence for the ideal free distribution

Little is known of the foraging abilities of children in modern cultures, especially when children forage in groups. Here we present a test of optimal foraging theory in groups of street children working for money. The children we observed were selling bottles of water to drivers distributed in two lanes at a crossroad of Istanbul, Turkey. As predicted by the ideal free distribution (a model of optimal group foraging), the ratio of children working in the two lanes was sensitive to the ratio of cars (and therefore the ratio of potential buyers) present in each lane. Deviations from the ideal free model arose largely from numerical restrictions on the set of possible ratios compatible with a small group size. When these constraints were taken into account, optimal behavior emerged as a robust aspect of the children's group distribution. Our results extend to human children aspects of group foraging that were previously tested in human adults or other animal species.     

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.     

The ideal free distribution of clonal plant's ramets among patches in a heterogeneous environment

The plastic response of clonal plant to different patch quality is not always the same and the degree is different too. So the result of this kind of foraging behaviour is different. In order to make clear whether the ramtes stay in favourable patches and get the quantitative relationship between the ramets distribution among patches and the available resource amount in heterogeneous environment, we develop a theoretical work under ideal free distribution (IFD) theory framework by neglecting some morphological plasticity of the spacer in this article. The results of our general model show that the ramet distribution should obey input matching rule at equilibrium. That means the ratio of ramet number in different patches should be equal to the ratio of available resource amount in these patches. We also use the simulation to predict the distribution pattern under history mattering. The results show that the initial ramets number has significant influence on the final distribution: over matching and under matching both can occur. More initial ramets in favourable patch result in over matching and more initial ramets in unfavourable patch result in under matching. The degree of the deviation from input matching rule is great when the difference of patches is small. These results prove that ideal free distribution theory works the same with animals. The ramets can stay in favourable patches sometimes in spite of the plasticity of the spacer, and the distribution depends on both patch quality and the history factors. But these results are true only when the functional response is type II.     

Interference and the ideal free distribution: oviposition in a parasitoid wasp

The interference ideal free distribution (IFD) model of Sutherlandmakes a number of predictions that have yet to be tested andthat have implications for the validity of subsequent extensionsto the theory. We tested these predictions in a study usingdifferent densities of the parasitoid wasp, Venturia canescens,foraging on patches containing different densities of its host,Plodia interpunctella. Our results support a number of the interferenceIFD model's general predictions. Gain rate decreased becauseof increased interference at higher density. Although gain rateson the two patches differed slighdy, this would be expectedallowing for some sampling behavior and perceptual constraints.Early in each experiment when patch assessment is likely tooccur, wasp movement was higher and gain rates lower. However,the more specific prediction of Sutherland's model, that proportionalpatch use should be constant and independent of density, wasnot upheld. Contemporary IFD models use only one of severalequally valid potential relationships between gain rate, interference,and competitor density. The results of this study provide supportfor the additive model developed by Tregenza et al. (companionarticle).     

Putting competition strategies into ideal free distribution models: habitat selection as a tug of war

When resources are patchily distributed in an environment, behavioral ecologists frequently turn to ideal free distribution (IFD) models to predict the spatial distribution of organisms. In these models, predictions about distributions depend upon two key factors: the quality of habitat patches and the nature of competition between consumers. Surprisingly, however, no IFD models have explored the possibility that consumers modulate their competitive efforts in an evolutionarily stable manner. Instead, previous models assume that resource acquisition ability and competition are fixed within species or within phenotypes. We explored the consequences of adaptive modulation of competitive effort by incorporating tug-of-war theory into payoff equations from the two main classes of IFD models (continuous input (CI) and interference). In the models we develop, individuals can increase their share of the resources available in a patch, but do so at the costs of increased resource expenditures and increased negative interactions with conspecifics. We show how such models can provide new hypotheses to explain what are thought to be deviations from IFDs (e.g., the frequent observation of fewer animals than predicted in "good" patches of habitat). We also detail straightforward predictions made uniquely by the models we develop, and we outline experimental tests that will distinguish among alternatives.     

Density-dependent habitat selection in muskrats: a test of the ideal free distribution model

Summary Two predictions of the ideal free distribution model, a null hypothesis of habitat selection, were examined using free-ranging muskrats. We rejected the prediction that the proportion of the animals found in each of five habitats was independent of population size. Data on over-winter occupancy of muskrat dwellings tend also to refute the prediction of equal fitness reward among habitats. Habitat type and water-level had a profound effect on the suitability of a site for settlement. We concluded that the observed pattern of muskrat distribution followed more closely an ideal despotic distribution where some individuals benefited from a higher fitness because of resource monopolization. Current theories of density-dependent habitat selection, which assume an ideal free distribution, would not apply to muskrats and possibly to many other mammal species.     

The ideal free distribution when the resource is variable

On the basis of the ideal free distribution (IFD) model, twostochastic models that incorporate the uncertainty of the informationused for decision making were considered to investigate theeffects of the variability in the resource supply rate on theIFD under continuous input conditions. In the uncertain-informationmodel, competitors cannot trace the variation of the supplyrate and use the expectation of the supply rate or previouspayoffs for decision making. Both submodels predict matchingof means, in which the average number of competitors for eachpatch is proportional to the average supply rate in the patch.In the perfect-information model, competitors continuously knowand trace the environment conditions. Numerical predictionsdepend on the relative size of the resource variance betweenpatches. When the resource variance in the good patch is sufficientlylarger than that in the poor patch, it predicts undermatchingof means; when the variance of the supply rate for each patchis small and proportional to the average of the supply ratein the patch, it predicts matching of means; and when the resourcevariance in the poor patch is larger than (or equal to) thatin the good patch, it predicts overmatching of means. Theseresults indicate the importance of clarifying the assumptionon the uncertainty in information for decision making and thetype of the resource variance for the test of the IFD underconditions where the resource supply rate is stochastic.     

Interference and the ideal free distribution: models and tests

We review the assumptions and predictions of five competitivedistribution models that predict how optimal foragers will bedistributed across resource patches when gains are reduced byinterference. This review revealed a number of previously ignoredpredictions and assumptions: in particular, there should beno change in relative patch use as competitor density changes.A new model is proposed in which interference results from thecosts of encounters with other foragers and where the gainson a patch are independent of the costs of interference. Thismodel predicts that as density increases, there will be increasedproportional use of lower-quality patches. Past empirical studiesof interference distributions are reanalyzed; none of the studiesprovides strong support for any of the existing ideal free-distributionmodels. We suggest that previous results are more consistentwith the predictions of our new model.     

Consumers that are not 'ideal' or 'free' can still approach the ideal free distribution using simple patch-leaving rules

1.  The ideal free distribution (IFD) has been widely used to determine whether consumers distribute themselves optimally. However, this theory is based on three assumptions that are clearly violated in many systems. The theory assumes that all individuals know the quality of each available site, are equally free to move between all sites, and have equal competitive abilities.
2.  I examine the utility of this theory to predict the distribution of the invasive European green crab Carcinus maenas , a species that likely violates all of these assumptions. I demonstrate three main findings.
3.  First, understanding how density-dependent interference and size alter individual foraging behaviour is important for understanding the density and biomass distribution of C. maenas in invaded habitats.
4.  Second, once behavioural mechanisms of crab foraging are accurately included in the model, the IFD does a good job of predicting the distribution of C. maenas , even though C. maenas violates the theory's fundamental assumptions.
5.  Third, C. maenas ' distribution can be obtained using simple decision rules and reasonable movement patterns.     

Ideal free distribution and natal dispersal in female roe deer

We investigated whether adult ( 2 years) female roe deer Capreolus capreolus conform to an ideal free or an ideal despotic distribution, in order to understand whether natal dispersal is voluntary or socially enforced. The study was undertaken in a high-density, free-ranging population close to Stockholm, Sweden, during 1989–1994. Data on population density, habitat quality, and five parameters on female reproduction and body condition, in addition to age distribution, were obtained in two nearby located areas, the field and the forest, representing contrasting habitats. Population density was estimated by faecal pellet group counting in addition to total counts of culled animals after a major deer harvest. Density in the field area was twice that in the forest area (66 vs 33 deer/km²). Habitat quality was determined by analysing the amount of faecal nitrogen from samples of faeces collected in the two areas, and by comparing size of summer home ranges for adults of both sexes with the aid of telemetry. Both estimates indicated a higher nutritional quality and resource abundance, respectively, in the field area, with faecal nitrogen content being higher (2.96 vs 2.43%), and ranges being smaller (12.9 vs 20.9 ha). No significant differences were found in any of the parameters on reproduction or body condition obtained from culled females, i.e. number of corpora lutea (1.8 vs 1.8), proportion of females with offspring (71 vs 56%), body mass (16.9 vs 17.3 kg), kidney fat index (91.9 vs 98.5), and length of the lower mandible (152.8 vs 151.4 mm). Neither did the age distribution among females, as determined from tooth wear, differ between the areas. These results are consistent with the prediction of the ideal free hypothesis. This, in turn, suggests that female dispersal in this species is voluntary, the underlying proximate cause being maximization of resource gain.     

Movement between patches, unequal competitors and the ideal free distribution

Researchers have often commented on the ability of the original ideal free distribution (IFD) model to approximate observed animal distributions even though the critical assumption that competitors are of equal ability is usually violated. We provide an explanation by recognizing that animals will occasionally move between patches for reasons other than to simply maximize their resource payoffs, given perfect (i.e. ideal) information about the current payoff in each patch, and that these movements will continue to occur even after an equilibrium is reached. When such movements are incorporated into an unequal competitors IFD model, a single, stable distribution of each competitor type is predicted. This equilibrium will usually be characterized by under-matching of total competitive units relative to the distribution of resources (i.e. too few competitive units in the good patch). More importantly, it will often resemble the original, equal competitors IFD, in that total competitor numbers will come close to matching the distribution of resources. We argue that researchers claiming to have observed an IFD of equal competitors have actually observed this equilibrium distribution of unequal competitors. Our model predicts that the deviation from input-matching will usually be an under-matching of total competitor numbers relative to resources (i.e. too few competitors in the good patch). Examination of published data reveals that post-equilibrium movement between patches occurs frequently and, although the reported distributions are similar to those predicted by input-matching, under-matching is usually observed.     

The energetic equivalence of cover to juvenile coho salmon (Oncorhynchus kisutch): ideal free distribution theroy applied

Cover is often thought to be an important habitat characteristicfor juvenile stream salmonida. In addition to providing protectionfrom predators, cover may be associated with reduced food availability.Thus, an individual's use of cover is likely to reflect a trade-offbetween the conflicting demands of growth and survival. We measuredthe influence of cover on foraging-site selection in groupsof eight juvenile coho salmon (Oncorhynchus kisutch) by examiningtheir distribution across two stream channel patches, one providingaccess to cover but little food (the "poor" patch), the otherproviding more food but no cover (the "good" patch). Becausefish distributions in the absence of cover conformed to an idealfree distribution (IFD) for unequal competitors (i.e., the distributionof competitive abilities matched the distribution of food),we used IFD theory to quantify the energetic equivalence ofcover to the fish. In the presence of cover and a model avianpredator, use of the poor patch increased relative to the predictionsof the IFD model. Using this observed deviation from an IFD,we calculated how much extra food must be added to the goodpatch to return the distribution of fish to the previously observedIFD of unequal competitors. As predicted, adding this amountof food caused the fish to return to their previous distribution,demonstrating that IFD theory can be used to relate energy intakeand risk of predation in a common currency