Evolutionary stability of optimal foraging: Partial preferences in the diet and patch models

In this article the patch and diet choice models of the optimal foraging theory are reanalyzed with respect to evolutionary stability of the optimal foraging strategies. In their original setting these fundamental models consider a single consumer only and the resulting fitness functions are both frequency and density independent. Such fitness functions do not allow us to apply the classical game theoretical methods to study an evolutionary stability of optimal foraging strategies for competing animals. In this article frequency and density dependent fitness functions of optimal foraging are derived by separation of time scales in an underlying population dynamical model and corresponding evolutionarily stable strategies are calculated. Contrary to the classical foraging models the results of the present article predict that partial preferences occur in optimal foraging strategies as a consequence of the ecological feedback of consumer preferences on consumer fitness. In the case of the patch occupation model these partial preferences correspond to the ideal free distribution concept while in the case of the diet choice model they correspond to the partial inclusion of the less profitable prey type in predators diet.     

Density‐Dependent Foraging and Interference Competition by Common Gartersnakes are Temperature Dependent

The ideal free distribution (IFD) predicts that optimal foragers will select foraging patches to maximize food rewards and that groups of foragers should thus be distributed between food patches in proportion to the availability of food in those patches. Because many of the underlying mechanisms of foraging are temperature dependent in ectotherms, the distribution of ectothermic foragers between food patches may similarly depend on temperature because the difference in fitness rewards between these patches may change with temperature. We tested the hypothesis that the distribution of Common Gartersnakes (Thamnophis sirtalis) between food patches can be explained by an IFD, but that conformance to an IFD weakens as temperature departs from the optimal temperature because fitness rewards, interference competition and the number of individuals foraging are highest at the optimal temperature. First, we determined the optimal temperature for foraging. Second, we examined group foraging at three temperatures and three density treatments. Search time was optimized at 27°C, handling time at 29°C and digestion time at 32°C. Gartersnakes did not match an IFD at any temperature, but their distribution did change with temperature: snakes at 20°C and at 30°C selected both food patches equally, while snakes at 25°C selected the low food patch more at low density and the high food patch more at high density. Food consumption and competition increased with temperature, and handling time decreased with temperature. Temperature therefore had a strong impact on foraging, but did not affect the IFD. Future work should examine temperature‐dependent foraging in ectotherms that are known to match an IFD.     

Taking the rough with the smooth: foraging for particulate food in continuous time

This paper describes the development of the general dynamical model of foraging developed by Ollason (1980, Theoret. Popul. Biol. 18, 44-65) to predict foraging for particulate food in three different types of environment. In an environment containing particles of different types of food, the model predicts the selection of an approximately optimal diet; in an environment in which the particles occur in patches, the model predicts a time budget of patch occupancy that approximates to the optimal time budget; and in an environment containing patches of particles that regenerate by the addition of particles of food at constant rates, the model predicts that animals will dispose themselves among the patches approximately as predicted by the ideal free distribution. Where the predictions of the model depart from the predictions of optimal foraging theory, they are qualitatively similar to the observed departures of the behaviour of real animals from the predictions of optimal foraging theory. The model provides a general representation of the foraging decisions of animals whether they feed strictly continuously or discontinuously on particles of food, and does so without explicit reference to optimization processes.     

Effects of habitat complexity,dominance and personality on habitat selection: Ideal despotic cichlids

Habitat structure can impede visibility and movement, resulting in lower resource monopolization and aggression. Consequently, dominant individuals may prefer open habitats to maximize resource gain, or complex habitats to minimize predation risk. We explored the role of dominance on foraging, aggression and habitat choice using convict cichlids (Amatitlania nigrofasciata) in a two‐patch ideal free distribution experiment. Groups of six fish of four distinct sizes first competed for shrimp in one‐patch trials in both an open and complex habitat; half the groups experienced each habitat type first. Following these one‐patch trials, each group then chose between habitat types in a two‐patch trial while competing for food. Finally, each fish underwent an individual behavioural assessment using a battery of “personality” tests to determine if behaviour when alone accurately reflected behaviour within a social context. In the one‐patch trials, dominant fish showed similar food consumption between habitats, but chased more in the complex habitat. In the two‐patch choice trials, dominants preferred and defended the complex habitat, forming an ideal despotic distribution with more than half the fish and competitive weight in the open habitat. Within the groups, individual fish differed in foraging and chasing, with repeatabilities of 0.45 and 0.23 across all treatments. Although a higher foraging rate during the individual assessment predicted foraging rate and use of the complex habitat during the group trials, aggression and boldness tests were not reflective of group behaviour. Across groups, heavier dominants and those with higher foraging rate in the open habitat used the open habitat more, suggesting that both risk and energetic state affect habitat preference in dominant convict cichlids.     

Learning to forage in a regenerating patchy environment: Can it fail to be optimal?

The behaviour of animals foraging along closed traplines of regenerating patches of food has been simulated using a learning rule that determines when an animal should leave the patch at which it is currently feeding to search for another one. The rule causes the animal to stay at the patch as long as it is feeding faster than it remembers doing. The foraging behaviour of one animal, and of two or more animals together, feeding in traplines containing patches of the same and of differing types has been simulated, and in all cases the foraging behaviour generated by the rule allowed the animals to exploit the food very efficiently. The learning model is also responsible for indirect social interactions among animals sharing the same trapline because the feeding of each animal reduces the availability of food for the others. This causes a population of animals to disperse themselves, on average, among patches of food according to the ideal free distribution. The relationship between the learning model and conventional optimal foraging models is examined and it is shown that it is pointless to try to account for learned behaviour in the context of optimal foraging theory.     

Foraging Guppies and the Ideal Free Distribution: The Influence of Information on Patch Choice

10 male and 10 female guppies were each examined foraging for food at 2 separate feeders. The feeders provided food over a wide area so that individuals had equal access to the food. As the proportion of food at a feeder was altered, the proportion of fish using that feeder changed to match the availability of food. Thus, both groups of fish conformed to the predictions of the ideal free distribution (IFD) theory. However, the behaviour of the two groups differed. During these experiments, males continued to shuttle between these feeders whereas females tended to remain at a feeder. Also, the female group conformed less well to an IFD on the first trial of each day but conformed more closely to an IFD in successive exposures to the same food distribution. The male group exhibited no improvement with increasing experience. For both groups, the relative influence of the previous day's experience on foraging decisions declined with time. However, unlike the male group, the female group relied on experience for their initial foraging decisions. It is possible that sampling by males provides the additional benefit of increasing their probability of encountering receptive females. This may account for the difference in information use by the male and female group.     

Foraging on patchily distributed prey by a cichlid fish (Teleostei,Cichlidae): A test of the ideal free distribution theory

Cichlid fish (Aequidens curviceps) distributed themselves and allocated their foraging time between two drift food patches in close approximation to the patch profitability ratio, as predicted by the ideal free distribution theory. The fish thereby achieved similar average feeding rates in the two patches, in two of three patch profitability ratio experiments. However, one major assumption of the ideal free model was violated, since individual fish differed in their competitive abilities for limited food resources, which resulted in unequal payoffs among individuals within each patch. Individual variation in feeding rates, and thus in competitive ability, was not related to despotism, but perhaps rather to individual differences in perceptual ability and in the ability to learn which patch was currently the more profitable. The strategy used by the fish to assess patch profitability included sampling available patches. However, individual fish switched (sampled) patches with varying frequency. Sampling had an associated cost, since high-frequency switchers had lower feeding rates on average than low-frequency switchers. Differences in foraging strategy among the fish therefore contributed to the observed in-equality in individual payoffs within patches.     

When to go with the crowd: Modelling synchronization of all-or-nothing activity transitions in grouped animals

For groups of animals to keep together, the group members have to perform switches between staying in one place and moving to another place in synchrony. However, synchronization imposes a cost on individual animals, because they have to switch from one to the other behaviour at a communal time rather than at their ideal times. Here we model this situation analytically for groups in which the ideal times vary quasinormally and grouping benefit increases linearly with group size. Across the parameter space consisting of variation in the grouping benefit/cost ratio and variation in how costly it is to act too early and too late, the most common optimal solutions are full synchronization with the group staying together and zero synchronization with immediate dissolution of the group, if the group is too small for the given benefit/cost ratio. Partial synchronization, with animals at the tails of the distribution switching individually and the central core of the group in synchrony, occurs only at a narrow stripe of the space. Synchronization cost never causes splitting of the group into two as either zero, partial or full synchronization is always more advantageous. Stable solutions dictate lower degree of synchrony and lower net benefits than optimal solutions for a large range of the parameter values. If groups undergo repeated synchronization challenges, they stay together or quickly dissolve, unless the animals assort themselves into a smaller group with less variation in the ideal times. We conclude with arguing that synchronization cost is different from other types of grouping costs since it does not increase much with increasing group size. As a result, larger groups may be more stable than smaller groups. This results in the paradoxical prediction that when the grouping benefit/grouping cost ratio increases, the average group sizes might decrease, since smaller groups will be able to withstand synchronization challenges.     

Density-dependent habitat selection in migratory passerines during stopover: what causes the deviation from IFD?

We studied the distribution of migratory warblers (genus: Sylvia) in poor and high quality habitat patches at a stopover site in the northern Negev, Israel. The purpose of our study was to test predictions based on the ideal free distribution (IFD) model by using a natural ecosystem which has a high turnover of individuals moving between unfamiliar foraging patches. We trapped birds in two groves of Pistacia atlantica embedded within a coniferous forest. The fruit-density ratio between these groves was 45:1. We compared bird density, body condition and habitat matching (the ratio between bird density and resource density) at the two sites. To analyse the data we integrated two approaches to density-dependent habitat selection: the isodar method and the habitat matching rule. As predicted by the IFD model, we found that habitat suitability decreased with bird density with a high correlation between warbler densities in the two habitat patches. Contrary to IFD predictions, warbler density in the poor patch was higher than expected by the habitat-matching rule. This habitat under-matching, had a cost: in the rich habitat the average energy gain per individual bird was higher than in the poor habitat. Further analysis suggests that the apparent habitat under-matching is not due to interference or differences in warbler competitive abilities. Therefore, we suggest that this migratory bird community is not at equilibrium because the birds possess imperfect knowledge of resource distribution. We propose that this lack of knowledge leads to free, but not ideal distributions of migrant birds in unfamiliar stop over sites.     

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.     

Extension of ideal free resource use to breeding populations and metapopulations

The concept of an ideal and free use of limiting resources is commonly invoked in behavioural ecology as a null model for predicting the distribution of foraging consumers across heterogeneous habitat. In its original conception, however, its predictions were applied to the longer timescales of habitat selection by breeding birds. Here I present a general model of ideal free resource use, which encompasses classical deterministic models for the dynamics in continuous time of feeding aggregations, breeding populations and metapopulations. I illustrate its key predictions using the consumer functional response given by Holling's disc equation. The predictions are all consistent with classical population dynamics, but at least two of them are not usually recognised as pertaining across all scales. At the fine scale of feeding aggregations, the steady state of an equal intake for all ideal free consumers may be intrinsically unstable, if patches are efficiently exploited by individuals with a non-negligible handling time of resources. At coarser scales, classical models of population and metapopulation dynamics assume exploitation of a homogeneous environment, yet they can yield testable predictions for heterogeneous environments too under the assumption of ideal free resource use.     

Integrating the roles of information and competitive ability on the spatial distribution of social foragers

Understanding and predicting the spatial distribution of social foragers among patchily distributed resources is a problem that has been addressed with numerous approaches over the 30 yr since the ideal free distribution (IFD) was first introduced. The two main approaches involve perceptual constraints and unequal competitors. Here we present a model of social foragers choosing among resource patches. Each forager makes a probabilistic choice on the basis of the information acquired through past foraging experiences. Food acquisition is determined by the forager's competitive ability. This model predicts that perceptual constraints have a greater influence on the spatial distribution of foragers than unequal competitive abilities but that competitive ability plays an important role in determining an individual's information state and behavior. Better competitors have access to more information; consequently, we find that competitive abilities and perceptual constraints are integrated through the social environment occupied by individual foragers. Relative competitive abilities influence the forager's information state, and the ability to use information determines the resulting spatial distribution.     

Foraging rate versus sociality in the starling Sturnus vulgaris

It is well established that social conditions often modify foraging behaviour, but the theoretical interpretation of the changes produced is not straightforward. Changes may be due to alterations of the foraging currency (the mathematical expression that behaviour maximizes) and/or of the available resources. An example of the latter is when both solitary and social foragers maximize rates of gain over time, but competition alters the behaviour required to achieve this, as assumed by ideal free distribution models. Here we examine this problem using captive starlings Sturnus vulgaris. Subjects had access to two depleting patches that replenished whenever the alternative patch was visited. The theoretical rate-maximizing policy was the same across all treatments, and consisted of alternating between patches following a pattern that could be predicted using the marginal value theorem (MVT). There were three treatments that differed in the contents of an aviary adjacent to one of the two patches (called the 'social' patch). In the control treatment, the aviary was empty, in the social condition it contained a group of starlings, and in a non-specific stimulus control it contained a group of zebra finches. In the control condition both patches were used equally and behaviour was well predicted by the MVT. In the social condition, starlings foraged more slowly in the social than in the solitary patch. Further, foraging in the solitary patch was faster and in the social patch slower in the social condition than in the control condition. Although these changes are incompatible with overall rate maximization (gain rate decreased by about 24% by self-imposed changes), if the self-generated gain functions were used the MVT was a good predictor of patch exploitation under all conditions. We discuss the complexities of nesting optimal foraging models in more comprehensive theoretical accounts of behaviour integrating functional and mechanistic perspectives.     

Are land bird assemblages functionally saturated? An empirical test in Mexico

According to the model of ideal free distribution, few or no assemblages will contain unused resources as long as species are free to colonize areas where resources are available. Consequently, because several food resources for birds are available throughout Mexico (nectar, seeds and fruits, vertebrates, carrion and different types of invertebrates), we predicted that all assemblages from Mexico contain birds from the corresponding foraging guilds. However, severe climates may prevent colonization by certain birds. Thus, a second prediction was that, if assemblages with unused resources were found, these would be from localities with climates that pose extreme stress (either extremely high temperature and low rainfall, or extremely low temperature). We tested these predictions on bird assemblages from 77 localities that represent the entire range of climatic conditions found in Mexico. Only two assemblages containing unused resources were found. These were from the localities with lowest temperature and lowest rainfall, respectively. We observed that land bird assemblages in Mexico do not contain unused resources, except when climate selectively restricts colonization by species from certain foraging guilds. Therefore, we conclude that even assemblages with almost no species in common have a common guild structure based on food resources. Null models of competition would be more realistic if they were to incorporate such patterns in their regional species pools.     

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.     

Assumption validity in human optimal foraging: The Barí hunters of Venezuela as a test case

Optimal foraging theory is being used increasingly as a means of understanding human foraging behavior. One of the central assumptions of optimal foraging theory is that prey items or patches are encountered sequentially and as a Poisson process. Using empirical data gathered on the Barí hunters of Venezuela, we assess the validity of this central assumption. We compare our observed distribution of encounter frequencies with an expected Poisson distribution, utilizing chisquare tests and graphic representations. The results are strikingly consonant with the expected Poisson distribution and lend support to the applicability of optimal foraging models to human hunting behavior.     

The effects of spatial food distribution and group size on foraging behaviour in a benthic fish

Animals foraging in heterogeneous environments benefit from information on local resource density because it allows allocation of foraging effort to rich patches. In foraging groups, this information may be obtained by individuals through sampling or by observing the foraging behaviour of group members. We studied the foraging behaviour of goldfish (Carassius auratus) groups feeding in pools on resources distributed in patches. First, we determined if goldfish use sampling information to distinguish between patches of different qualities, and if this allowed goldfish to benefit from a heterogeneous resource distribution. Then, we tested if group size affected the time dedicated to food searching and ultimately foraging success. The decision of goldfish to leave a patch was affected by whether or not they found food, indicating that goldfish use an assessment rule. Giving-up density was higher when resources were highly heterogeneous, but overall gain was not affected by resource distribution. We did not observe any foraging benefits of larger groups, which indicate that grouping behaviour was driven by risk dilution. In larger groups the proportion searching for food was lower, which suggests interactions among group members. We conclude that competition between group members affects individual investments in food searching by introducing the possibility for alternative strategies, such as scrounging or resource monopolisation.     

Modelling locust foraging: How and why food affects group formation

Locusts are short horned grasshoppers that exhibit two behaviour types depending on their local population density. These are: solitarious, where they will actively avoid other locusts, and gregarious where they will seek them out. It is in this gregarious state that locusts can form massive and destructive flying swarms or plagues. However, these swarms are usually preceded by the aggregation of juvenile wingless locust nymphs. In this paper we attempt to understand how the distribution of food resources affect the group formation process. We do this by introducing a multi-population partial differential equation model that includes non-local locust interactions, local locust and food interactions, and gregarisation. Our results suggest that, food acts to increase the maximum density of locust groups, lowers the percentage of the population that needs to be gregarious for group formation, and decreases both the required density of locusts and time for group formation around an optimal food width. Finally, by looking at foraging efficiency within the numerical experiments we find that there exists a foraging advantage to being gregarious.     

The group-size paradox: effects of learning and patch departure rules

In many species, foraging in groups can enhance individual fitness.However, groups are often predicted to be larger than the sizethat maximizes individual fitness. This is because individualforagers are expected to continue joining a group until thefitness in the group falls to the level experienced by solitaryforagers. If such a process were pervasive, social foraging,paradoxically, would provide little evolutionary advantages.We propose a solution to the group-size paradox by allowingforagers to learn about habitat quality and leave food patcheswhen their current intake rate falls below that expected forthe whole habitat. By using a simulation model, we show thatunder a wide range of population sizes, foragers using suchrules abandon under- and overcrowded patches, ensuring thatgroup size remains close to the optimal value. The results holdin habitats with varying patch quality, but we note that thelack of food renewal in patches can disrupt the process of groupformation. We conclude that groups of optimal sizes can occurfrequently if fitness functions are peaked and resources patchilydistributed, without the need to invoke relatedness betweenjoiners and established group members, group defense againstjoiners, or other mechanisms that were proposed earlier to preventgroups from becoming too large.     

Optimal foraging in nonpatchy habitats. I. Bounded one-dimensional resource

We present a model of optimal foraging in habitats where the food has an arbitrary density distribution (continuous or not). The classical models of foraging strategies assume that the food is distributed in patches and that the animal divides its time between the two distinct behaviors of patch exploitation and interpatch travel. This assumption is hard to accept in instances where the food distribution is continuous in space, and where travel and feeding cannot be sharply distinguished. In this paper, the habitat is assumed to be one-dimensional and bounded, and the animal is assumed to have a limited foraging time available. The problem is treated mathematically in the context of the calculus of variations. The optimal solution is to divide the habitat in two subsets according to the food density. In the richer subset, the animal equalizes the density distribution; in the poorer subset, it travels as fast as possible.