Models for the ecological effects and evolution of dispersal

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The goal of this project is to use mathematical models to gain insight into the effects that dispersal can have on ecological interactions between species and their environment or other species, and into the factors that influence the evolution of dispersal strategies. The research effort will be focused on understanding the effects and evolution of conditional dispersal strategies, that is, dispersal strategies that depend on population densities and environmental conditions. Dispersal strategies will be studied from the viewpoint of evolutionary stability. The mathematical models will primarily consist of reaction-diffusion-advection equations or systems of such equations. There has been a great deal of research on ecological models with simple diffusion, but much less on models involving more complex conditional dispersal strategies. Models based on simple diffusion describe random dispersal that does not depend on environmental conditions. By incorporating factors such as advection along environmental gradients, toward prey, or away from predators, and variable diffusion rates that depend on population densities or environmental conditions, the research in this project will extend such models to describe various types of conditional dispersal and to study their effects. The general mathematical methods that will be used include the classical theory of partial differential equations, dynamical systems theory, and nonlinear analysis. Key mathematical ideas include using bifurcation theory, persistence theory, and the theory of monotone semidynamical systems to translate estimates on eigenvalues of differential operators into conclusions about the dynamics of systems involving those operators. Since the models that will be studied include strongly coupled quasilinear parabolic systems (as opposed to standard reaction-diffusion systems which are usually weakly coupled and semilinear) it is anticipated that the project will involve the development of new mathematical results. Some of the research in the project will be aimed at understanding specific ecological questions. As an example, one aspect of the project will be the study of situations such as intraguild predation or apparent competition where a top predator preys on two other interacting species and influences their dispersal patterns. Some of the research will be aimed at understanding the selective pressures that influence the evolution of dispersal. That research will involve models for competing populations that are ecologically identical except for their dispersal strategies, which will be studied from the viewpoint of evolutionary stability. Previous research suggests that in this context an important aspect of dispersal strategies is how well they allow a population to match the resources available in the environment, so that conditional dispersal leading to something like an ideal free distribution of organisms may be favored. That idea will be explored further. The models that will be used to study evolutionary questions are in some sense special cases of the types of models used to study ecological questions.The dispersal of organisms is clearly an important aspect of many ecological processes. It drives biological invasions, allows populations to colonize empty habitats, and allows individuals to track resources and avoid predators or competitors. In this project mathematical models for interacting and dispersing species will be used to gain theoretical insights into how dispersal patterns can influence the persistence, distribution, and/or extinction of species, and what dispersal patterns are likely to appear as organisms evolve to adapt to new or changing environments. Dispersal may reflect purely random movement or may be conditioned on properties of the environment or the presence of other organisms. This project will be focused largely on conditional dispersal. Conditional dispersal has not been studied as much as random dispersal, even though there is empirical evidence that it occurs and there is some theoretical evidence that it may be favored by natural selection in some situations. Developing models that incorporate conditional dispersal will expand the scope of the current ecological and evolutionary theory of dispersal. A broader motivation for the project is that developing models for the effects and evolution of dispersal should provide insights that may be useful in formulating policy to address ecological issues such as the conservation of biodiversity under environmental change or the assessment of the possible results of introducing exotic species or altering environments.