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Consumers must track and acquire resources in complex landscapes. Much discussion has focused on the concept of a ‘resource gradient’ and the mechanisms by which consumers can take advantage of such gradients as they navigate their landscapes in search of resources. However, the concept of tracking resource gradients means different things in different contexts. Here, we take a synthetic approach and consider six different definitions of what it means to search for resources based on density or gradients in density. These include scenarios where consumers change their movement behavior based on the density of conspecifics, on the density of resources, and on spatial or temporal gradients in resources. We also consider scenarios involving non-local perception and a form of memory. Using a continuous space, continuous time model that allows consumers to switch between resource-tracking and random motion, we investigate the relative performance of these six different strategies. Consumers’ success in matching the spatiotemporal distributions of their resources differs starkly across the six scenarios. Movement strategies based on perception and response to temporal (rather than spatial) resource gradients afforded consumers with the best opportunities to match resource distributions. All scenarios would allow for optimization of resource-matching in terms of the underlying parameters, providing opportunities for evolutionary adaptation, and links back to classical studies of foraging ecology. 

In this paper, we use an integrodifference equation model and pairwise invasion analysis to find what dispersal strategies are evolutionarily stable strategies (ESS) when there is spatial heterogeneity in habitat suitability, and there may be seasonal changes in this spatial heterogeneity, so that there are both advantages and disadvantages of dispersing. We begin with the case where all spatial locations can support a viable population, and then consider the case where there are non-viable regions in the habitat that makes dispersal really necessary for sustaining a population. Our findings generally align with previous findings in the literature that were based on other modeling frameworks, namely that dispersal strategies associated with ideal free distributions are evolutionarily stable. In the case where only part of the habitat can sustain a population, a partial occupation ideal free distribution that occupies only the viable region is shown to be associated with a dispersal strategy that is evolutionarily stable. As in some previous works, the proofs of these results make use of properties of line sum symmetric functions, which are analogous to those of line sum symmetric matrices but applies to integral operators. 

Seasonal migrations are a widespread and broadly successful strategy for animals to exploit periodic and localized resources over large spatial scales. It remains an open and largely case-specific question whether long-distance migrations are resilient to environmental disruptions. High levels of mobility suggest an ability to shift ranges that can confer resilience. On the other hand, a conservative, hard-wired commitment to a risky behavior can be costly if conditions change. Mechanisms that contribute to migration include identification and responsiveness to resources, sociality, and cognitive processes such as spatial memory and learning. Our goal was to explore the extent to which these factors interact not only to maintain a migratory behavior but also to provide resilience against environmental changes. We develop a diffusion-advection model of animal movement in which an endogenous migratory behavior is modified by recent experiences via a memory process, and animals have a social swarming-like behavior over a range of spatial scales. We found that this relatively simple framework was able to adapt to a stable, seasonal resource dynamic under a broad range of parameter values. Furthermore, the model was able to acquire an adaptive migration behavior with time. However, the resilience of the process depended on all the parameters under consideration, with many complex trade-offs. For example, the spatial scale of sociality needed to be large enough to capture changes in the resource, but not so large that the acquired collective information was overly diluted. A long-term reference memory was important for hedging against a highly stochastic process, but a higher weighting of more recent memory was needed for adapting to directional changes in resource phenology. Our model provides a general and versatile framework for exploring the interaction of memory, movement, social and resource dynamics, even as environmental conditions globally are undergoing rapid change. 

We analyze a reaction-diffusion system modeling the competition of multiple phytoplankton species which are limited only by light. While the dynamics of a single species has been well studied, the dynamics of the two- species model has only begun to be understood with the recent establishment of a comparison principle. In this paper, we show that the competition of N similar phytoplankton species, for any number N, generically leads to compe- tition exclusion. The main tool is the theory of normalized principal bundle for linear parabolic equations. 

We analyze a reaction-diffusion system modeling the competition of multiple phytoplankton species which are limited only by light. While the dynamics of a single species has been well studied, the dynamics of the twospecies model has only begun to be understood with the recent establishment of a comparison principle. In this paper, we show that the competition of N similar phytoplankton species, for any number N, generically leads to competitive exclusion. The main tool is the theory of a normalized principal bundle for linear parabolic equations. 

In this article, we study how the rates of diffusion in a reaction-diffusion model for a stage structured population in a heterogeneous environment affect the model’s predictions of persistence or extinction for the population. In the case of a population without stage structure, faster diffusion is typically detrimental. In contrast to that, we find that, in a stage structured population, it can be either detrimental or helpful. If the regions where adults can reproduce are the same as those where juveniles can mature, typically slower diffusion will be favored, but if those regions are separated, then faster diffusion may be favored. Our analysis consists primarily of estimates of principal eigenvalues of the linearized system around ( 0 , 0 ) and results on their asymptotic behavior for large or small diffusion rates. The model we study is not in general a cooperative system, but if adults only compete with other adults and juveniles with other juveniles, then it is. In that case, the general theory of cooperative systems implies that, when the model predicts persistence, it has a unique positive equilibrium. We derive some results on the asymptotic behavior of the positive equilibrium for small diffusion and for large adult reproductive rates in that case.