The goal of the project is to use mathematical models to gain insight into the evolutionary causes and ecological effects of the dispersal of organisms. The individual behavior of dispersing organisms influences the spatial distribution of their population, which influences their ecological interactions with resources, predators, and members of their own species. Those interactions in turn influence the survival of individuals and populations, and hence create selective advantages for individuals with superior dispersal strategies. Selection results in evolutionary pressure on the dispersal behavior of individuals. Mathematical models are used to understand and describe the complex feedbacks at multiple scales involved in the ecology and evolution of dispersal behavior. The primary modeling approach is based on systems of reaction-advection-diffusion equations with variable and/or density dependent coefficients. Such equations can be derived from the movement mechanisms used by individuals, but they also make predictions about species interactions and evolution. Thus, they provide a framework for addressing complex interactions across different spatial and temporal scales and different levels of organization. They present significant mathematical challenges but recent advances in bifurcation theory and partial differential equations will facilitate their analysis. The analysis also employs ideas from the theory of dynamical systems. Patch models and integrodifferential equations with nonlocal dispersal are used in addition to partial differential equations. Many of the analytic approaches that work for reaction-diffusion-advection models can be applied to them, leading to conceptual unification. Traditional dispersal models typically view dispersal as an essentially random process that does not depend on environmental conditions or the internal states of individuals, but there have been numerous studies suggesting that conditional dispersal is common and may be advantageous. Thus, the research focuses on models where the diffusion and advection rates of individuals depend on environmental conditions and/or the population densities of their species and species with which they interact. That leads to new mathematical research on questions involving the dynamics and equilibria of nonlinear systems of parabolic partial differential equations with coefficients that vary in space and/or time.
The dispersal of organisms influences the persistence and interactions of populations and drives range expansions, biological invasions, and the colonization of empty habitats. Thus, understanding dispersal is relevant to addressing questions about biological conservation, invasive species, pest control, and other environmental issues, including the response of populations to global change. The ways that the dispersal patterns of organisms affect their interactions with the environment and with other species are complex and subtle, so it is difficult to assess the effects of variations in dispersal patterns on those interactions. Also, since dispersal affects survival and thus leads to natural selection, evolution may change the dispersal strategies of organisms inhabiting changing environments. That process involves complex feedback loops which are difficult to understand. The research addresses those problems by using mathematical models to describe the ecological effects and evolution of dispersal. The models are based on partial differential equations, which provide a framework for describing how quantities change and influence each other over space and time. Specifically, the models include terms describing random movements, directed movements, and ecological interactions that may vary in time and space and/or depend on population densities. The research provides new information about the ways that these processes interact. Traditional models for dispersal usually describe it as a random process that does not depend on location or environmental conditions, so this research involves the development and analysis of new types of models that may be relevant in studying various phenomena in ecology and related areas.