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     

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.     

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.     

A stochastic model of the distribution of unequal competitors between resource patches

We present a stochastic model of individuals' movements between two patches of resources. The population is made up of two types of individual with differing competitive abilities, and two types of movements occur, with individuals moving either to increase their intake rate or at random. Several previous models have used simulations to evaluate the likely distribution of individuals. We instead derive equations for the equilibrium distribution of the population, which can be solved numerically. This avoids the need to choose an initial distribution for the population, and enables us to obtain the probability with which rare events occur. This may not be possible when simulations are used, since a rare event may not occur at all. We find that when random movements are rare, an increase in the rate of random movements out of a patch can increase the number of individuals on that patch. We consider an approximation to the model with rare random movements, which provides an explanation for this phenomenon.     

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.     

The differing effects of interference on individuals of different rank

A widely studied ecological model considers the distribution of individuals of different competitive abilities between a number of resources, such that each opti-mizes its reward rate. Recently this model has been criticised for making the predic-tion that the reward rate of an individual can increase under some circumstances with increasing competitor number. The originators of the model have challenged the suggestion that this prediction is biologically implausible and challenged empiricists to test this prediction. Here it is shown that a previous study on hares is consistent with the controversial prediction of the original model, but not with that of an alternative formulation.     

Spatial relationships between mate acquisition probability and aggregation size in a gregarious coreid bug, (Colpula lativentris): A case of the ideal free distribution under perceptual constraints

Spatial relationships of mate acquisition probability for individuals of both sexes of a gregariously-mating coreid bug, Colpula lativentris, were studied in relation to aggregation size. Operational sex ratio was always strongly male biased. Mate acquisition probability of females was rather constant and independent of aggregation size, as predicted by an ideal free distribution. Moreover laboratory experiments showed that both multiple mating and rearing density little affected female fecundity, suggesting ideal free distribution of females in terms of reproductive success. On the other hand, mate acquisition probability of males was higher in larger aggregations, where more receptive females were available. This male discrepancy from an ideal free distribution was similar to the patterns predicted by an ideal free distribution under perceptual constraints (Abrahams, 1986), but not by that under unequal competitive ability.     

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.     

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.     

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.     

Do haddock select habitats to maximize condition?

Haddock Melanogrammus aeglefinus in the North sea increased their distributional range when more abundant, but this density dependent habitat selection (DDHS) explained only a small part of the year‐on‐year variation in distribution patterns. The condition of haddock was examined at 24 sites in the North Sea in August and September 2004 and related to their abundance, to examine if the ideal free distribution theory (IFD), which assumes that organisms select habitats that maximize their rate of food intake, can be used to explain this variation in large scale distribution patterns. At a given temperature, condition (hepato‐somatic index, I _H) was better at stations where haddock were most abundant. Therefore, haddock were not distributed perfectly according to the IFD in 2004. The positive correlation between abundance and I _H, however, indicated there was some habitat selection by haddock, as in the total absence of habitat selection no correlation between I _H and abundance, and no spatial variation in abundance was expected. DDHS may only explain a small part of the yearly variation in the distribution because haddock did not equalize and maximize their fitness at the scale of the North Sea. In addition, stable isotope analysis of muscle samples showed that haddock did not avoid competition for food when at high abundance by feeding at a lower or wider range of trophic levels.     

Evolutionary stability of ideal free nonlocal dispersal

We study the evolutionary stability of nonlocal dispersal strategies that can produce ideal free population distributions, that is, distributions where all individuals have equal fitness and there is no net movement of individuals at equilibrium. We find that the property of producing ideal free distributions is necessary and often sufficient for evolutionary stability. Our results extend those already developed for discrete diffusion models on finite patch networks to the case of nonlocal dispersal models based on integrodifferential equations. The analysis is based on the use of comparison methods and the construction of sub- and supersolutions.     

Physiological Ecology Meets the Ideal-free Distribution: Predicting the Distribution of Size-structured Fish Populations Across Temperature Gradients

We describe a habitat selection model that predicts the distribution of size-structured groups of fish in a habitat where food availability and water temperature vary spatially. This model is formed by combining a physiological model of fish growth with the logic of ideal free distribution (IFD) theory. In this model we assume that individuals scramble compete for resources, that relative competitive abilities of fish vary with body size, and that individuals select patches that maximize their growth rate. This model overcomes limitations in currently existing physiological and IFD-based models of habitat selection. This is because existing physiological models do not take into account the fact that the amount of food consumed by a fish in a patch will depend on the number of competitors there (something that IFD theory addresses), while traditional IFD models do not take into account the fact that fish are likely to choose patches based on potential growth rate rather than gross food intake (something that physiological models address). Our model takes advantage of the complementary strengths of these two approaches to overcome these weaknesses. Reassuringly, our model reproduces the predictions of its two constituent models under the simple conditions where they apply. When there is no competition for resources it mimics the physiological model of habitat selection, and when there is competition but no temperature variation between patches it mimics either the simple IFD model or the IFD model for unequal competitors. However, when there are both competition and temperature differences between patches our model makes different predictions. It predicts that input-matching between the resource renewal rate and the number of fish (or competitive units) in a patch, the hallmark of IFD models, will be the exception rather than the rule. It also makes the novel prediction that temperature based size-segregation will be common, and that the strength and direction of this segregation will depend on per capita resource renewal rates and the manner in which competitive weight scales with body size. Size-segregation should become more pronounced as per capita resource abundance falls. A larger fish/cooler water pattern is predicted when competitive ability increases more slowly than maximum ration with body size, and a smaller fish/cooler water pattern is predicted when competitive ability increases more rapidly than maximum ration with body size.     

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.     

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.