Migration dynamics for the ideal free distribution

This article verifies that the ideal free distribution (IFD) is evolutionarily stable, provided the payoff in each patch decreases with an increasing number of individuals. General frequency-dependent models of migratory dynamics that differ in the degree of animal omniscience are then developed. These models do not exclude migration at the IFD where balanced dispersal emerges. It is shown that the population distribution converges to the IFD even when animals are nonideal (i.e., they do not know the quality of all patches). In particular, the IFD emerges when animals never migrate from patches with a higher payoff to patches with a lower payoff and when some animals always migrate to the best patch. It is shown that some random migration does not necessarily lead to undermatching, provided migration occurs at the IFD. The effect of population dynamics on the IFD (and vice versa) is analyzed. Without any migration, it is shown that population dynamics alone drive the population distribution to the IFD. If animal migration tends (for each fixed population size) to the IFD, then the combined migration-population dynamics evolve to the population IFD independent of the two timescales (i.e., behavioral vs. population).     

The migration game in habitat network: the case of tuna

Long-distance migration is a widespread process evolved independently in several animal groups in terrestrial and marine ecosystems. Many factors contribute to the migration process and of primary importance are intra-specific competition and seasonality in the resource distribution. Adaptive migration in direction of increasing fitness should lead to the ideal free distribution (IFD) which is the evolutionary stable strategy of the habitat selection game. We introduce a migration game which focuses on migrating dynamics leading to the IFD for age-structured populations and in time varying habitats, where dispersal is costly. The model predicts migration dynamics between these habitats and the corresponding population distribution. When applied to Atlantic bluefin tunas, it predicts their migration routes and their seasonal distribution. The largest biomass is located in the spawning areas which have also the largest diversity in the age-structure. Distant feeding areas are occupied on a seasonal base and often by larger individuals, in agreement with empirical observations. Moreover, we show that only a selected number of migratory routes emerge as those effectively used by tunas.     

Linking Resource Matching and Dispersal

We study theoretically the effect of inter-habitat migration on the distribution of population sizes between two habitats, and compare this distribution with the expected ideal free distribution (IFD). Whenever emigration from the two habitats is asymmetric, or when there is a survival cost during migration, the resulting equilibrium distribution of population sizes deviates from the IFD. This result holds irrespective of emigration rule, even though a density-dependent fraction of emigrants generally produces a distribution closer to the IFD than a constant fraction of emigrants. Environmental stochasticity causes a linear relation between population sizes in the two habitats, with slope and intercept only identical to the IFD when net inter-habitat exchange is zero. The type and asymmetry of inter-habitat migration will influence how we should interpret data on population distribution in different habitats. The resulting resource matching is also critically contingent on the relative time-scales of population renewal and dispersal, and when population size is measured in relation to reproduction and dispersal. Therefore, data on population sizes cannot be used uncritically to assess habitat quality.     

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.     

Evolutionary games and two species population dynamics

Competition between species has long been modeled by population dynamics based on total numbers of each species. Recently, the evolution of strategy frequencies has been used successfully for competition models between individuals. In this paper, we illustrate that these two views of competition are compatible. It is shown that the rate of intra and interspecific competitions between individuals largely determines the population dynamics. Competition models over a single common resource and predator-prey models are developed from this individual competition approach. In particular, the equilibrium strategies in a co-evolving predator-prey system are shown to be more stable than the predicted strategy cycling of standard evolutionary game theory.     

Effect of predator density dependent dispersal of prey on stability of a predator-prey system

This work presents a predator-prey Lotka-Volterra model in a two patch environment. The model is a set of four ordinary differential equations that govern the prey and predator population densities on each patch. Predators disperse with constant migration rates, while prey dispersal is predator density-dependent. When the predator density is large, the dispersal of prey is more likely to occur. We assume that prey and predator dispersal is faster than the local predator-prey interaction on each patch. Thus, we take advantage of two time scales in order to reduce the complete model to a system of two equations governing the total prey and predator densities. The stability analysis of the aggregated model shows that a unique strictly positive equilibrium exists. This equilibrium may be stable or unstable. A Hopf bifurcation may occur, leading the equilibrium to be a centre. If the two patches are similar, the predator density dependent dispersal of prey has a stabilizing effect on the predator-prey system.     

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.     

Effects of density-dependent migrations on stability of a two-patch predator-prey model

We consider a predator-prey model in a two-patch environment and assume that migration between patches is faster than prey growth, predator mortality and predator-prey interactions. Prey (resp. predator) migration rates are considered to be predator (resp. prey) density-dependent. Prey leave a patch at a migration rate proportional to the local predator density. Predators leave a patch at a migration rate inversely proportional to local prey population density. Taking advantage of the two different time scales, we use aggregation methods to obtain a reduced (aggregated) model governing the total prey and predator densities. First, we show that for a large class of density-dependent migration rules for predators and prey there exists a unique and stable equilibrium for migration. Second, a numerical bifurcation analysis is presented. We show that bifurcation diagrams obtained from the complete and aggregated models are consistent with each other for reasonable values of the ratio between the two time scales, fast for migration and slow for local demography. Our results show that, under some particular conditions, the density dependence of migrations can generate a limit cycle. Also a co-dim two Bautin bifurcation point is observed in some range of migration parameters and this implies that bistability of an equilibrium and limit cycle is possible.     

Spatial patterns of coexistence of competing species in patchy habitat

The single-species spatially realistic patch occupancy metapopulation model is, in this study, extended to a metacommunity of many competing species. Competition is assumed to reduce the local carrying capacity (effective patch area), which in turn increases local extinction rates and reduces colonization rates because of smaller population sizes. Each species is described by three parameters: pre-competitive abundance (equilibrium incidence of patch occupancy, which reflects the rate of colonization in relation to extinction rate), the spatial range of migration, and competitive ability. The model ignores spatio–temporal correlations caused by interspecific interactions, because in metacommunities of unequal competitors inhabiting heterogeneous landscapes, correlations in the occurrence of species are driven more by patch heterogeneity than by competition. The model allows the calculation of multispecies equilibria in patchy habitats without simulations. In general, the number of coexisting species in the metacommunity increases with decreasing strength of competition, increasing rate of colonization, and decreasing range of migration. Habitat heterogeneity in the form of spatial variation in patch areas tends to facilitate coexistence. Poor competitors may coexist with superior competitors in the patch network if the former have higher colonization rates (competition–colonization trade-off). When migration distances are short, competition leads to spatial pattern formation: Species tend to have restricted spatial distributions in the network, but contrary to intuitive expectations, often the distributions of many species are nested. Having more dispersive species enhances both local and global diversity, whereas more local migration decreases local but increases global diversity.     

食饵庇护所对斑块环境下Leslie-Gower捕食系统的影响

建立了一个显式含有空间庇护所的两斑块Leslie-Gower捕食者-食饵系统。假设只有食饵种群在斑块间以常数迁移率迁移,且在每个斑块上食饵间的迁移比局部捕食者-食饵相互作用发生的时间尺度要快。利用两个时间尺度,可以构建用来描述所有斑块总的食饵和捕食者密度的综合系统。数学分析表明,在一定条件下,存在唯一的正平衡点,并且此平衡点全局稳定。进一步,捕食者的数量随着食饵庇护所数量增加而降低;在一定条件下,食饵的数量随着食饵庇护所数量增加先增加后降低,在足够强的庇护所强度下,两物种出现灭绝。对比以往研究,利用显式含有和隐含空间庇护所的数学模型所得结论不一致,这意味着在研究庇护所对捕食系统种群动态影响时,空间结构可能起着重要作用。     

Behavioral choices based on patch selection: a model using aggregation methods

The aim of this work is to study the influence of patch selection on the dynamics of a system describing the interactions between two populations, generically called 'population N' and 'population P'. Our model may be applied to prey-predator systems as well as to certain host-parasite or parasitoid systems. A situation in which population P affects the spatial distribution of population N is considered. We deal with a heterogeneous environment composed of two spatial patches: population P lives only in patch 1, while individuals belonging to population N migrate between patch 1 and patch 2, which may be a refuge. Therefore they are divided into two patch sub-populations and can migrate according to different migration laws. We make the assumption that the patch change is fast, whereas the growth and interaction processes are slower. We take advantage of the two time scales to perform aggregation methods in order to obtain a global model describing the time evolution of the total populations, at a slow time scale. At first, a migration law which is independent on population P density is considered. In this case the global model is equivalent to the local one, and under certain conditions, population P always gets extinct. Then, the same model, but in which individuals belonging to population N leave patch 1 proportionally to population P density, is studied. This particular behavioral choice leads to a dynamically richer global system, which favors stability and population coexistence. Finally, we study a third example corresponding to the addition of an aggregative behavior of population N on patch 1. This leads to a more complicated situation in which, according to initial conditions, the global system is described by two different aggregated models. Under certain conditions on parameters a stable limit cycle occurs, leading to periodic variations of the total population densities, as well as of the local densities on the spatial patches.     

Effects of Density Dependent Migrations on the Dynamics of a Predator Prey Model

We study the effects of density dependent migrations on the stability of a predator-prey model in a patchy environment which is composed with two sites connected by migration. The two patches are different. On the first patch, preys can find resource but can be captured by predators. The second patch is a refuge for the prey and thus predators do not have access to this patch. We assume a repulsive effect of predator on prey on the resource patch. Therefore, when the predator density is large on that patch, preys are more likely to leave it to return to the refuge. We consider two models. In the first model, preys leave the refuge to go to the resource patch at constant migration rates. In the second model, preys are assumed to be in competition for the resource and leave the refuge to the resource patch according to the prey density. We assume two different time scales, a fast time scale for migration and a slow time scale for population growth, mortality and predation. We take advantage of the two time scales to apply aggregation of variables methods and to obtain a reduced model governing the total prey and predator densities. In the case of the first model, we show that the repulsive effect of predator on prey has a stabilizing effect on the predator-prey community. In the case of the second model, we show that there exists a window for the prey proportion on the resource patch to ensure stability.     

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.     

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.     

Competition and coexistence in regional habitats

Habitat heterogeneity plays a key role in the dynamics and structures of communities. In this article, a two-species metapopulation model that includes local competitive dynamics is analyzed to study the population dynamics of two competing species in spatially structured habitats. When local stochastic extinction can be ignored, there are, as in Lotka-Volterra equations, four outcomes of interspecific competition in this model. The outcomes of competition depend on the competitive intensity between the competing pairs. An inferior competitor and a superior competitor, or two strongly competing species, can never stably coexist, whereas two weak competitors (even if they are very similar species) may coexist over the long term in such environments. Local stochastic extinction may greatly affect the outcomes of interspecific competition. Two competing species can or cannot stably coexist depending not only on the competitive intensity between the competing pairs but also on their precompetitive distributions. Two weak competitors that have similar precompetitive distributions can always regionally coexist. Two strongly competing species that competitively exclude each other in more stable habitats may be able to stably coexist in highly heterogenous environments if they have similar precompetitive distributions. There is also a chance for an inferior competitor to coexist regionally or even to exclude a superior competitor when the superior competitor has a narrow precompetitive distribution and the inferior competitor has a wide precompetitive distribution.     

Mutualism as a constraint on invasion success for legumes and rhizobia

Because hereditary symbiont transmission is normally absent in the mutualism of legume plants and root‐nodule bacteria (rhizobia), dispersing plants may often arrive at new habitats where mutualist partners are too rare to provide full benefits. Factors governing invasion success were explored by analysing a system of two coupled pairwise competition models: a legume invader competing with a resident non‐mutualistic plant, and a rhizobial population competing with a resident population of nonsymbiotic bacteria. The non‐linear dependence of benefits on partner abundance in this mutualism creates the possibility of two alternative population size equilibria, so that a threshold density can exist for invasion. If legumes and rhizobia exceed a critical population size, both species achieve rapid population growth, while if initial densities of both species are below their respective thresholds, they remain rare and are thus vulnerable to extinction in the presence of competitors. Overall, the results indicate that legumes may often fail at colonization attempts within habitats where mutualist partners are scarce. Data on legume prevalence in island floras and rates of geographical spread by legume weeds are consistent with this inference. Predictive insights about invasiveness may emerge from comparative research on key traits identified by the model, especially the shape of the function determining the number of nodules formed at low rhizobial density.     

Allee effects can both conserve and create spatial heterogeneity in population densities

In order to determine conditions which allow the Allee effect (caused by biparental reproduction) to conserve and create spatial heterogeneity in population densities, we studied a deterministic model of a symmetric two-patch metapopulation. We proved that under certain conditions there exist stable equilibria with unequal population densities in the two patches, a situation which can be interpreted as conserved heterogeneity. Furthermore, the Allee effect can lead to instability of the equilibrium with equal population densities if some degree of competition is assumed to occur between the subpopulations (non-local competition). This indicates the potential of the Allee effect to create spatial heterogeneity. Neither of these effects appear under biologically realistic parameter values in a model where uniparental reproduction is assumed. We proved that both the between-patch migration intensity and the degree of non-local competition are decisive in determining boundaries between these types of behaviour of the spatial system with Allee effect. Therefore, we propose that the Allee effect, migration intensity, and non-local competition should be considered jointly in studies focusing on problems like pattern formation in space and invasions of spreading species.     

Grazers and Diggers: Exploitation Competition and Coexistence among Foragers with Different Feeding Strategies on a Single Resource

A mathematical model is presented that describes a system where two consumer species compete exploitatively for a single renewable resource. The resource is distributed in a patchy but homogeneous environment; that is, all patches are intrinsically identical. The two consumer species are referred to as diggers and grazers, where diggers deplete the resource within a patch to lower densities than grazers. We show that the two distinct feeding strategies can produce a heterogeneous resource distribution that enables their coexistence. Coexistence requires that grazers must either move faster than diggers between patches or convert the resources to population growth much more efficiently than diggers. The model shows that the functional form of resource renewal within a patch is also important for coexistence. These results contrast with theory that considers exploitation competition for a single resource when the resource is assumed to be well mixed throughout the system.