Robert Stephen Cantrell

We study the mechanism and effects of ideal free distributions (IFDs) on an ecological community consisting of n prey and m predator species, for any positive integers n and m , by considering the corresponding diffusive Lotka-Volterra system with time-periodic coefficients. We define a notion of joint IFD in a timeperiodic environment, and give necessary and sufficient conditions for it to be achieved by a subcollection of prey and predator species with suitable dispersal strategies. Next, we show, via construction of a Lyapunov function, that such dispersal strategies are evolutionarily stable, in the sense that if a subcollection of prey and predator species adopts an ideal free dispersal strategy, then the total community must converge to an IFD for large time; if a unique combination of prey-predator species adopts an ideal free strategy, then it can drive all other species to extinction. Conversely, if a combination of prey-predator species adopts a non-ideal free dispersal strategy, then it can be invaded by some suitable mutant strategies. Our results provide insight into the evolution of spatial distribution of ecological communities with predator-prey interactions.

Roughly speaking, a population is said to have an ideal free distribution on a spatial region if all of its members can and do locate themselves in a way that optimizes their fitness, allowing for the effects of crowding. Dispersal strategies that can lead to ideal free distributions of populations that use them have been shown to exist and to be evolutionarily stable in a number of models for a single population. Those models include reaction-diffusion-advection equations and the analogous models using discrete diffusion or nonlocal dispersal described by integrodifferential equations. Furthermore, in the case of reaction-diffusion-advection models and their nonlocal analogues, for environments that are static in time there are strategies that allow populations to achieve an ideal free distribution by using only local information about environmental quality and/or gradients. In this paper, we extend some of these ideas and results to certain Lotka-Volterra type predator-prey systems. In the case of single-species models it is often possible to do the analysis via methods based on monotonicity, but in the predator-prey context those fail so we use methods based on a Lyapunov functional.

The ideal free distribution in ecology was introduced by Fretwell and Lucas to model the habitat selection of animal populations. In this paper, we revisit the concept via a mean field game system with local coupling, which models a dynamic version of the habitat selection game in ecology. We establish the existence of classical solution of the ergodic mean field game system, including the case of heterogeneous diffusion when the underlying domain is one-dimensional and further show that the population density of agents converges to the ideal free distribution of the underlying habitat selection game, as the cost of control tends to zero. Our analysis provides a derivation of ideal free distribution in a dynamical context.

It is well known that in reaction-diffusion models for a single unstructured population in a bounded, static, heterogeneous environment, slower diffusion is advantageous. That is not necessarily the case for stage structured populations. In (Cantrell et al. 2020), it was shown that in a stage structured model introduced by Brown and Lin (1980), there can be situations where faster diffusion is advantageous. In this paper we extend and refine the results of (Cantrell et al. 2020) on persistence to more general combinations of diffusion rates and to cases where either adults or juveniles do not move. We also obtain results on the asymptotic behavior of solutions as diffusion rates go to zero, and on competition between species that differ in their diffusion rates but are otherwise ecologically identical. We find that when the spatial distributions of favorable habitats for adults and juveniles are similar, slow diffusion is still generally advantageous, but if those distributions are different that may no longer be the case.

A population is said to have an ideal free distribution in a spatially heterogeneous but temporally constant environment if each of its members have chosen a fixed spatial location in a way that optimizes its individual fitness, allowing for the effects of crowding. In this paper, we extend the idea of individual fitness associated with a specific location in space to account for the full path that an individual organism takes in space and time over a periodic cycle, and extend the mathematical formulation of an ideal free distribution to general time periodic environments. We find that, as in many other cases, populations using dispersal strategies that can produce a generalized ideal free distribution have a competitive advantage relative to populations using strategies that do not produce an ideal free distribution. A sharp criterion on the environmental functions is found to be necessary and sufficient for such ideal free distribution to be feasible. In the case the criterion is met, we showed that there exist dispersal strategies that can be identified as producing a time-periodic version of an ideal free distribution, and such strategies are evolutionarily steady and are neighborhood invaders from the viewpoint of adaptive dynamics. Our results extend previous works in which the environments are either temporally constant, or temporally periodic but the total carrying capacity is temporally constant.

Trait‐mediated behavioral responses to other species, especially predators, can have important effects on the dynamics of populations. One effect is to modify dispersal patterns, which in turn can modify population dynamics and species interactions. We model a situation where a focal population responds to disturbances created by the harvesting of another population by increasing the rate of random dispersal by individuals. The model shows that in some situations this effect can result in the extinction of the focal population, even if it is not itself subject to harvesting. This observation suggests that it may be desirable to consider trait‐mediated effects when assessing the possible impacts of harvesting on conservation.Recommendations for Resource Managers: Be aware that harvesting a given crop or population can affect other populations because of their behavioral responses to the harvesting activity, even if it does not directly impose mortality on those populations or change their interactions with other species. A way that harvesting activity can affect nonharvested populations is to increase their dispersal rate. Managers should be aware that this has the potential to cause extinction in some situations even if no other effects of harvesting are present. For some populations, harvesting resources with which they interact or share an environment could in principle cause both direct and indirect effects, which could combine to have a different, perhaps greater, impact than either one would have separately.