The Allee-type ideal free distribution

The ideal free distribution (IFD) in a two-patch environment where individual fitness is positively density dependent at low population densities is studied. The IFD is defined as an evolutionarily stable strategy of the habitat selection game. It is shown that for low and high population densities only one IFD exists, but for intermediate population densities there are up to three IFDs. Population and distributional dynamics described by the replicator dynamics are studied. It is shown that distributional stability (i.e., IFD) does not imply local stability of a population equilibrium. Thus distributional stability is not sufficient for population stability. Results of this article demonstrate that the Allee effect can strongly influence not only population dynamics, but also population distribution in space.     

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.     

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.     

The approximately ideal, more or less free distribution

We present the minimum set of requirements necessary and sufficient to represent the foraging behaviour of an animal, and its utilisation of food, in order to explore the emergent properties of behaviour that allow animals to reduce their hunger. We present an individual-based model of foraging that provides a simple quantification of the requirements, which is sufficiently simple to yield some analytical results. Complex interactions beyond the scope of analysis have been explored through simulating animals foraging in regenerating patchy environments. In most cases the populations pass into equilibrium distributions which appear to be stable. The equilibria always approximate closely to the ideal free distribution, although typically with a small degree of undermatching. (Undermatching is the term applied to the departure from the ideal free distribution caused by a smaller proportion of the population than expected occupying areas with a higher than average regeneration rate). The model therefore implies that the distribution, hitherto accounted for in terms of ESSs may, in fact, be simply an effect of the animal's utilization of the food it collects to reduce its hunger. The model defines a specific feeling rate, v, the rate at which an animal can feed on a unit of food. This is a function of three parameters, v1, the specific feeding rate when alone, v(infinity), the rate, possibly zero, at which it can feed in the presence of an indefinitely large number of conspecifics, and n1/2, the number of conspecifics that cause v to take the value (v1+v(infinity)/2. Exploitation competition in the absence of interference is represented by setting v1 = v(infinity). Differences in competitive ability in exploitation have been represented by simulating animals with a range of values of v1, those with the larger values, feeding more rapidly, being the more effective competitors, and those with the lower values being the less effective. Interference competition is represented by setting v1 > v(infinity) and social facilitation by v1 < v(infinity). Individual differences in the strength of interaction are represented by different values of n(1/2). In competition, the animals with the larger values of n(1/2) are the more effective competitors: in facilitation, they are the less effective facilitators. The addition of physiological and behavioural detail makes very little alteration to the emergent equilibria, always close to the ideal free distribution, almost always showing undermatching.     

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 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.     

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.     

An experimental investigation of a new ideal free distribution model

Summary We investigate a new continuous input ideal free distribution model which removes the assumption that resources are consumed as soon as they enter a patch. The model makes predictions about the standing crop of resources and allows consideration of the effects of simultaneous exploitation and interference competition. Using a group of cichlid fish competing for food items, we show that consistent with the model, standing crops can vary in continuous input situations. As predicted, higher standing crops are associated with increased intake rates. Furthermore, with greater numbers of competitors, standing crops are higher, suggesting that there is interference as well as exploitation competition in our system. An experiment to investigate the effects of fish density on the level of movement revealed that the reported interference competition could not be attributed to increased fish movement at higher density.     

The ideal free distribution and bacterial growth on two substrates

A population dynamical model describing growth of bacteria on two substrates is analyzed. The model assumes that bacteria choose substrates in order to maximize their per capita population growth rate. For batch bacterial growth, the model predicts that as the concentration of the preferred substrate decreases there will be a time at which both substrates provide bacteria with the same fitness and both substrates will be used simultaneously thereafter. Preferences for either substrate are computed as a function of substrate concentrations. The predicted time of switching is calculated for some experimental data given in the literature and it is shown that the fit between predicted and observed values is good. For bacterial growth in the chemostat, the model predicts that at low dilution rates bacteria should feed on both substrates while at higher dilution rates bacteria should feed on the preferred substrate only. Adaptive use of substrates permits bacteria to survive in the chemostat at higher dilution rates when compared with non-adaptive bacteria.     

Vertical distribution of zooplankton: density dependence and evidence for an ideal free distribution with costs

Background  

In lakes with a deep-water algal maximum, herbivorous zooplankton are faced with a trade-off between high temperature but low food availability in the surface layers and low temperature but sufficient food in deep layers. It has been suggested that zooplankton (Daphnia) faced with this trade-off distribute vertically according to an "Ideal Free Distribution (IFD) with Costs". An experiment has been designed to test the density (competition) dependence of the vertical distribution as this is a basic assumption of IFD theory.     

Learning rules for social foragers: implications for the producer-scrounger game and ideal free distribution theory

In population games, the optimal behaviour of a forager depends partly on courses of action selected by other individuals in the population. How individuals learn to allocate effort in foraging games involving frequency-dependent payoffs has been little examined. The performance of three different learning rules was investigated in several types of habitats in each of two population games. Learning rules allow individuals to weigh information about the past and the present and to choose among alternative patterns of behaviour. In the producer-scrounger game, foragers use producer to locate food patches and scrounger to exploit the food discoveries of others. In the ideal free distribution game, foragers that experience feeding interference from companions distribute themselves among heterogeneous food patches. In simulations of each population game, the use of different learning rules induced large variation in foraging behaviour, thus providing a tool to assess the relevance of each learning rule in experimental systems. Rare mutants using alternative learning rules often successfully invaded populations of foragers using other rules indicating that some learning rules are not stable when pitted against each other. Learning rules often closely approximated optimal behaviour in each population game suggesting that stimulus-response learning of contingencies created by foraging companions could be sufficient to perform at near-optimal level in two population games.     

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.     

Global asymptotic stability and the ideal free distribution in a starvation driven diffusion

We study a logistic model with a nonlinear random diffusion in a Fokker-Planck type law, but not in Fick’s law. In the model individuals are assumed to increase their motility if they starve. Any directional information to resource is not assumed in this starvation driven diffusion and individuals disperse in a random walk style strategy. However, the non-uniformity in the motility produces an advection toward surplus resource. Several basic properties of the model are obtained including the global asymptotic stability and the acquisition of the ideal free distribution.     

The information of food distribution realizes an ideal free distribution: Support of perceptual limitation

Previous tests of ideal free distribution (IFD) under continuous input conditions have demonstrated that more profitable patches tend to be relatively underused compared to that predicted by the theory. We tested the hypothesis that competitors’ perceptual constraints of resource distribution cause this deviation from the IFD. A laboratory experiment was conducted to determine whether additional information on food distribution by a light cue that indicates the food input point improves the IFD theory’s fit to the distribution of clone red-spotted masu salmons (Salmonids),Oncorhynchus masou ishikawai, that had been conditioned to the light as a cue indicating the site with a higher input rate. In the treatments without a light cue, the distribution of fish was closer to a random pattern than an IFD. In contrast, in the treatments with light cue, the distribution of fish was closer to the expected value of an IFD rather than to a random pattern, supporting the perception-limit hypothesis. The distribution and the pattern of resource use by fish in the treatments without the light cue were best explained by the perception-limit model. Our results suggest that it is perceptual constraints that cause deviation from the IFD.     

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.