## Chris Cosner

# Modeling the evolution of dispersal with reaction-advection-diffusion equations and their discrete and nonlocal analogues

**Abstract:** This talk will describe the formulation and analysis of some models related to the evolution of dispersal in the context of reaction-diffusion-advection systems and the analogous discrete or nonlocal models, where the differential operator describing dispersal is replaced with a matrix or an integral operator. It will present some basic ideas and results in an approach to the evolution of dispersal motivated by adaptive dynamics. Those ideas include the notions of evolutionarily stable, convergent stable, and neighborhood invader strategies. These types of strategies are characterized by their ability to give a population the ability to invade populations using other strategies or to resist invasion by them. For example, an evolutionarily stable strategy has the property that a population using it cannot be invaded by another small population that uses a different strategy but which is otherwise ecologically similar to the first population. Modeling the invasion of one population by another using a different dispersal strategy leads to competition models with different diffusion, advection, or nonlocal dispersal terms for the two competitors. The analysis of those models is similar in general terms to that of reaction-diffusion models but the technical details and predictions of the analysis can be significantly different. The analysis typically involves various ideas in the theory of differential and integral equations, including monotone methods, eigenvalue estimates, and Lyapunov functions. In situations where the environment varies in space but not in time, it turns out in many cases that the evolutionarily stable strategies are those that lead to an ideal free distribution of the population. An ideal free distribution of a population is one where all individuals at all locations have equal fitness and there is no net movement between locations. The talk will describe how the ideal free distribution can be characterized and seen to be evolutionarily stable in different modeling frameworks.