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.     

The interplay between behavior and morphology in the evolutionary dynamics of resource specialization

We analyze the consequences of diet choice behavior for the evolutionary dynamics of foraging traits by means of a mathematical model. The model is characterized by the following features. Consumers feed on two different substitutable resources that are distributed in a fine-grained manner. On encounter with a resource item, consumers decide whether to attack it so as to maximize their energy intake. Simultaneously, evolutionary change occurs in morphological traits involved in the foraging process. The assumption here is that evolution is constrained by a trade-off in the consumer's ability to forage on the alternative resources. The model predicts that flexible diet choice behavior can guide the direction of evolutionary change and mediate coexistence of different consumer types. Such polymorphisms can evolve from a monomorphic population at evolutionary branching points and also at points where a small genetic change in a trait can provoke a sharp instantaneous and nongenetic change in choice behavior. In the case of weak trade-offs, the evolutionary dynamics of a dimorphic consumer population can lead to alternative evolutionarily stable communities. The robustness of these predictions is checked with individual-based simulations and by relaxing the assumption of optimally foraging consumers.     

Evolutionary stability of ideal free dispersal strategies in patchy environments

A central question in the study of the evolution of dispersal is what kind of dispersal strategies are evolutionarily stable. Hastings (Theor Pop Biol 24:244-251, 1983) showed that among unconditional dispersal strategies in a spatially heterogeneous but temporally constant environment, the dispersal strategy with no movement is convergent stable. McPeek and Holt's (Am Nat 140:1010-1027, 1992) work suggested that among conditional dispersal strategies in a spatially heterogeneous but temporally constant environment, an ideal free dispersal strategy, which results in the ideal free distribution for a single species at equilibrium, is evolutionarily stable. We use continuous-time and discrete-space models to determine when the dispersal strategy with no movement is evolutionarily stable and when an ideal free dispersal strategy is evolutionarily stable, both in a spatially heterogeneous but temporally constant environment.     

Adaptive feeding across environmental gradients and its effect on population dynamics

This paper analyzes a consumer's adaptive feeding response to environmental gradients. We consider a consumer-resource system where resources are distributed among many discrete resource patches. Each consumer exhibits a feeding morphology allowing it to remove resources from a patch down to some threshold density (or level) before having to seek resources elsewhere. Assuming consumers trade off resource extraction with patch access and predation, we show that for a given environment there often exists a single evolutionarily stable feeding threshold and it is an evolutionary attractor. We then investigate how the population dynamics of the resource and the consumer change as the environment changes. Two cases are considered: (i) all consumers exhibit a fixed feeding threshold that is adaptive for an intermediate environment; and (ii) the consumer population adapts and adopts the evolutionarily stable feeding threshold associated with the current environment. In less harsh environments (i.e., environments where consumers experience a lower risk of predation, or environments where resource patches are more abundant) the adaptive consumer population is predicted to evolve so that resources within a patch are depleted to lower densities. We show that the change in consumer density due to environmental change can be rather different depending on whether or not the population can adapt. In some situations we observe that when the consumer's environment becomes harsher, the consumer population may increase in density before a rapid crash to extinction. This result has implications for monitoring and managing a population.     

Smart,smarter, smartest: foraging information states and coexistence

Animals possess different abilities to gain and use information about the foraging patches they exploit. When ignorant of the qualities of encountered patches, a smart forager should leave all patches after the same amount of fixed search time. A smarter forager can be Bayesian by using information on cumulative harvest and time spent searching a patch to better inform its patch‐departure decision. The smartest forager has immediate and continuous knowledge about patch quality, and can make a perfect decision about when to leave each patch. Here we let each of these three strategies harvest resources from a slowly regenerating environment. Eventually a steady‐state distribution of prey among patches arises where the environment‐wide resource renewal just balances the environment‐wide harvest of the foragers. The fixed time forager creates a distribution with the highest mean and highest variance of patch qualities, followed by the Bayesian and the prescient in that order. The less informed strategies promote distributions with both more resources and more exploitable information than the more informed strategies. While it is true that a better‐informed strategy will always out‐perform a less well‐informed, its increase in performance may not compensate it for any costs associated with being better informed. We imagine that the fixed time strategy may be least expensive and the prescient strategy most expensive in terms of sensory organs and associated assess and respond capabilities. To consider competition between such strategies with varying costs, we introduced a single individual of each of the strategies into the environments created by populations of the other strategies. There are threshold costs associated with the better‐informed strategy such that it can or cannot outcompete a less‐informed strategy. However, over a relatively narrow range of foraging costs, less‐informed and better‐informed strategies will coexist. Furthermore, for the prescient and the Bayesian strategies, some combinations of foraging costs produce alternate stable states – whichever strategy establishes first remains safe from invasion by the other.     

Temporal partitioning and aggression among foragers: modeling the effects of stochasticity and individual state

In many natural systems, individuals compete with conspecificsand heterospecifics for food and in some cases, individualshave been observed to partition their foraging times or fightover food. In this study, I investigated when it is optimalfor a consumer to partition time and be aggressive. I formulatedan individual-based model of foraging and used game theoryto find evolutionarily stable strategies (ESSs) that maximizethe probability that consumers survive each day and acquiretheir daily food requirements. Consumers choose when to forageand when to behave aggressively during confrontations overfood. Consumers are each associated with a state variable,representing the amount of food eaten, and a dominance ranking, which describes how likely they are to forage and fight forfood. The ESS is sensitive to food abundance, consumer state,and the dominance ranking. When food is abundant, temporalpartitioning is often an ESS where the dominant consumer foragesfirst; however, partitioning is unlikely to be an ESS when food abundance is low. Fights over food are typically avoidedbut may be part of an ESS when food abundance is low, bothconsumers are hungry, or the time available for foraging eachday is drawing to a close. Because the ESS is sensitive toconsumer state, the stochastic nature of finding food often results in considerable variation in observed foraging dynamicsfrom one day to the next, even when consumers adopt the samestate-dependent strategy each day. Results are compared withempirical observations, and I discuss implications for consumercoexistence.     

Evolution of handling time can destroy the coexistence of cycling predators

Several consumers (predators) with Holling type II functional response may robustly coexist even if they utilize the same resource (prey), provided that the population exhibits nonequilibrium dynamics and the handling time of predators is sufficiently different. We investigate the evolution of handling time and, in particular, its effect on coexistence. Longer handling time is costly in terms of lost foraging time, but allows more nutrients to be extracted from a captured prey individual. Assuming a hyperbolically saturating relationship between handling time and the number of new predators produced per prey consumed, we obtain three results: (i) There is a globally evolutionarily stable handling time; (ii) At most two predator strategies can coexist in this model; (iii) When two predators coexist, a mutant with intermediate handling time can always invade. This implies that there is no evolutionarily stable coexistence, and the evolution of handling time eventually leads to a single evolutionarily stable predator. These results are proven analytically and are valid for arbitrary (not only small) mutations; they however depend on the relationship between handling time and offspring production and on the assumption that predators differ only in their prey handling strategy.     

Why do some nectar foragers perch and others hover while probing flowers?

Diurnal hawkmoths, Hemaris fuciformis, and bumblebees, Bombus pasquorum, were observed foraging for nectar in flowers of Viscaria vulgaris. The hawkmoths hovered in front of the flowers, while the bees perched on them. The hawkmoths had a faster probing rate than the bees, and consequently also had higher gross and net rates of energy gain. A model is presented that shows that hovering only yields a higher net rate of energy gain (NREG) than perching when nectar volumes are high due to low competition for the resource. The difference in NREG of perchers and hoverers decreases with an increase of competition, and eventually perching yields the highest NREG. This is an effect of the higher cost of hovering. The results suggest that hovering can only evolve as a pure evolutionarily stable strategy (ESS) if competition is reduced, for example by co-evolutionary specializations with plants. The possibility that it has evolved as a mixed ESS (i.e. individuals can both hover and perch depending on the resource level) is discussed. The evolution of optimal foraging strategies is discussed, and it is pointed out that the rate of gain of an animal is independent of the strategy used when all competing foragers use the same strategy, but competitively superior strategies will nevertheless evolve because they are ESSs. Competition between strategies with different energy costs are special, because resource availability determines which strategy is competitively superior. A high-cost strategy can only evolve as a pure ESS at high resource levels, or as a mixed ESS at intermediate levels.     

The ideal free distribution as an evolutionarily stable state in density‐dependent population games

In classical games that have been applied to ecology, individual fitness is either density independent or population density is fixed. This article focuses on the habitat selection game where fitness depends on the population density that evolves over time. This model assumes that changes in animal distribution operate on a fast time scale when compared to demographic processes. Of particular interest is whether it is true, as one might expect, that resident phenotypes who use density‐dependent optimal foraging strategies are evolutionarily stable with respect to invasions by mutant strategies. In fact, we show that evolutionary stability does not require that residents use the evolutionarily stable strategy (ESS) at every population density; rather it is the combined resident–mutant system that must be at an evolutionary stable state. That is, the separation of time scales assumption between behavioral and ecological processes does not imply that these processes are independent. When only consumer population dynamics in several habitats are considered (i. e. when resources do not undergo population dynamics), we show that the existence of optimal foragers forces the resident‐mutant system to approach carrying capacity in each habitat even though the mutants do not die out. Thus, the ideal free distribution (IFD) for the single‐species habitat selection game becomes an evolutionarily stable state that describes a mixture of resident and mutant phenotypes rather than a strategy adopted by all individuals in the system. Also discussed is how these results are affected when animal distribution and demographic processes act on the same time scale.     

Consuming and grouping: recource-mediated animal aggregation

We demonstrate that a simplistic foraging rule for a consumer in a spatially explicit resource environment leads to consumer grouping. Although consumer groups sweeping through the renewing resource environment represents the model's dynamical attractor, for short time scales (represented by a constant total consumer population) three different distributions emerge. At low consumer density, population distributions are variable and spatially fixed, but not grouped. Moving groups erupt at intermediate consumer densities. At high consumer density, there is no spatial variability in the resource and consumer densities. Similar results have been observed in a variety of empirical systems. The results suggest interesting insights will arise by examining social interactions within a resource-consumer modeling framework.     

A comparison of foraging strategies in a patchy environment.

In this paper we compare foraging strategies that might be used by predators seeking prey in a patchy environment. The strategies differ in the extent to which predators aggregate in response to prey density. The approach to the comparison is suggested by the idea of evolutionarily stable strategies. A strategy is said to be evolutionarily stable if it cannot be invaded by another strategy. Thus we examine scenarios where a small number of individuals using one strategy are introduced into a situation where a large number of individuals using the other strategy are already present. However, our foraging models do not explicitly incorporate predator population dynamics, so we use net energy uptake as a surrogate for reproductive fitness. In cases where all of the patches visited by predators sustain prey populations, we find that for any pair of strategies one of them will have a higher net energy uptake than the other whether it is the resident or the introduced strain. However, which one is higher will typically depend on the total predator population, which is determined by the resident strain. If the predators leave prey densities high, the more aggregative strain will have the advantage. If the predators reduce prey densities to low levels the less aggregative strain will have the advantage. In cases where one strain of predators aggregates in response to prey density and the other does not, then there might be patches which do not contain prey but do contain (non-aggregating) predators. In those cases, there is the possibility that whichever strategy is used by the introduced strain will yield a higher energy uptake than that used by the resident strain. This suggests that if some patches are empty of prey then aggregative and non-aggregative strategies may be able to coexist.     

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.     

Foraging for intermittently refuged prey: theory and field observations of a parasitoid

Many insect herbivores feed in concealed locations but become accessible intermittently, creating windows of greater vulnerability to attack, and generating a proportion of the prey population that is readily accessible to foraging natural enemies. We incorporated accessible prey into an extant optimal foraging model, and found that this addition allowed opportunistic exploitation of prey that have already emerged from refugia (the leaving strategy) as a viable strategy, in addition to waiting at refugia for prey to emerge (the waiting strategy). We parameterized the model empirically for the parasitoid Macrocentrus grandii and its host, Ostrinia nubilalis, under field conditions. The model predicted that M. grandii should adopt a leaving strategy when host patch density is high (travel time between patches is short), but a waiting strategy when host patch density is low (travel time between patches is long). Field observations of M. grandii patch tenure were consistent with model predictions, indicating that M. grandii exhibited flexible behaviour based on experience within a foraging bout, and that these behavioural shifts improved foraging efficiency. Behaviour of M. grandii was responsive to heterogeneity in host emergence rates, and appeared to be driven by the relatively small proportion of the host population that became accessible at a fast rate. Therefore understanding forager responses to intermittently refuged prey may require characterization of the behaviour of a subset of the prey population, rather than the average prey individual. The model can potentially be used as a framework for comparative studies across forager taxa, to understand when foragers on intermittently accessible prey should adopt fixed waiting or leaving strategies vs. a flexible strategy that is responsive to the current environment.     

Evolutionary Convergence to Ideal Free Dispersal Strategies and Coexistence

We study a two species competition model in which the species have the same population dynamics but different dispersal strategies and show how these dispersal strategies evolve. We introduce a general dispersal strategy which can result in the ideal free distributions of both competing species at equilibrium and generalize the result of Averill et al. (2011). We further investigate the convergent stability of this ideal free dispersal strategy by varying random dispersal rates, advection rates, or both of these two parameters simultaneously. For monotone resource functions, our analysis reveals that among two similar dispersal strategies, selection generally prefers the strategy which is closer to the ideal free dispersal strategy. For nonmonotone resource functions, our findings suggest that there may exist some dispersal strategies which are not ideal free, but could be locally evolutionarily stable and/or convergent stable, and allow for the coexistence of more than one species.     

Coexistence and community structure in a consumer resource model with implicit stoichiometry

We combine stoichiometry theory and optimal foraging theory into the MacArthur consumer-resource model. This generates predictions for diet choice, coexistence, and community structure of heterotroph communities. Tradeoffs in consumer resource-garnering traits influence community outcomes. With scarce resources, consumers forage opportunistically for complementary resources and may coexist via tradeoffs in resource encounter rates. In contrast to single currency models, stoichiometry permits multiple equilibria. These alternative stable states occur when tradeoffs in resource encounter rates are stronger than tradeoffs in elemental conversion efficiencies. With abundant resources consumers exhibit partially selective diets for essential resources and may coexist via tradeoffs in elemental conversion efficiencies. These results differ from single currency models, where adaptive diet selection is either opportunistic or selective. Interestingly, communities composed of efficient consumers share many of the same properties as communities based on substitutable resources. However, communities composed of relatively inefficient consumers behave similarly to plant communities as characterized by Tilman’s consumer resource theory. The results of our model indicate that the effects of stoichiometry theory on community ecology are dependent upon both consumer foraging behavior and the nature of resource garnering tradeoffs.     

Evolution of conditional dispersal: evolutionarily stable strategies in spatial models

We consider a two-species competition model in which the species have the same population dynamics but different dispersal strategies. Both species disperse by a combination of random diffusion and advection along environmental gradients, with the same random dispersal rates but different advection coefficients. Regarding these advection coefficients as movement strategies of the species, we investigate their course of evolution. By applying invasion analysis we find that if the spatial environmental variation is less than a critical value, there is a unique evolutionarily singular strategy, which is also evolutionarily stable. If the spatial environmental variation exceeds the critical value, there can be three or more evolutionarily singular strategies, one of which is not evolutionarily stable. Our results suggest that the evolution of conditional dispersal of organisms depends upon the spatial heterogeneity of the environment in a subtle way.     

Size‐dependent resource allocation and sex allocation in herbaceous perennial plants

Individuals within a population often differ considerably in size or resource status as a result of environmental variation. In these circumstances natural selection would favour organisms not with a single, genetically determined allocation, but with a genetically determined allocation rule specifying allocation in relation to size or environment. Based on a graphical analysis of a simple evolutionarily stable strategy (ESS) model for herbaceous perennial plants, we aim to determine how cosexual plants within a population should simultaneously adjust their reproductive allocation and sex allocation to their size. We find that if female fitness gain is a linear function of resource investment, then a fixed amount of resources should be allocated to male function, and to post‐breeding survival as well, for individuals above a certain size threshold. The ESS resource allocation to male function, female function, and post‐breeding survival positively correlate if both male and female fitness gains are a saturating function of resource investment. Plants smaller than the size threshold are expected to be either nonreproductive or functionally male only.     

Sperm competition. II-post-copulatory guarding

A two round sperm competition model is analysed to determine which male strategy is advantageous for fertilization of a given set of eggs; guarding a particular female or searching for another copulation. A guarding male is one who would guard if he mates in the first round (which may not occur) whilst a non-guarding male decides on how much sperm to allocate if given the opportunity to inseminate a female in round one. Guarding behaviour is defined in terms of a probability of preventing a further insemination if challenged by a rival male. Sperm success with a single female obeys the "raffle principle". An evolutionarily stable strategy (ESS) approach is used to ascertain the best non-guarding ejaculation strategy. We show that for each fixed proportion of guarders in the population the strategies are ordered and that only a single guarding strategy need be considered. The model predicts that there will be evolution to either the guarding strategy or a single non-guarding strategy or a polymorphic combination of guarding and some (or all) of the non-guarding strategies. The conditions for coexistence to occur were shown to be rare in comparison to those necessary for a monomorphism. Copyright 1999 Academic Press.     

Some evolutionary properties of parental investment per offspring in a heterogeneous environment

The possibility of protected polymorphisms and of monomorphic evolutionarily stable strategies for parental investment per offspring in a heterogeneous environment is theoretically analysed. A high density two-niche model of the classical soft selection kind is used, although it incorporates the possibility of rare strategies invading an empty niche the contribution of which is not constant. Protected polymorphisms can be found whether or not both strategies included produce surviving offspring in both niches. However, a monomorphic evolutionarily stable strategy exists unless offspring of the optimal size in one of the niches cannot survive in the other. The robustness of the model is graphically illustrated under a variety of circumstances, and some evolutionary consequences are briefly discussed.