This project investigates the effects of conditional dispersal on dynamics of single and multiple interacting species. A common underlying assumption in theoretical studies of population dynamics is that dispersal rates are uniform across space. However, such simplification can yield misleading results because the surrounding environment can vary both spatially and temporally. Indeed, animals tend to sense and respond to local environmental cues by dispersing directionally, and their movements are often combinations of both random and directed ones. Reaction-diffusion-advection equations serve as one of the major approaches to understanding spatial-temporal processes such as dispersal, and are used in this research to model both random and directed movements of species and population dynamics. Three kinds of conditional dispersal strategies will be studied in this project. The first one is the direct movement of populations along environmental gradients or along density-dependent growth rate gradients, and the goal is to determine the effects of such biased movement on both single population and multiple competing species. The second dispersal strategy concerns area-restricted search of predators, and the purpose is to understand how biased foraging behaviors of predators can induce the aggregation of predators. The third is a dynamical model for ideal free distribution theory, which assumes that species choose habitats in such a way that each individual tries to maximize its reproduction fitness. The aim is to obtain a better understanding of interactions between such dispersal strategy and population dynamics. To address these biological questions, the principal investigator will use mathematical methods which include regularity theory for elliptic and parabolic operators, analysis of eigenvalue problems, maximum principles, bifurcation analysis, monotone system theory, permanence theory, and perturbation analysis.
The purpose of this project is to increase our understanding of how populations disperse in response to spatially varying environments, to determine which patterns of dispersal strategy can confer some selective or ecological advantage, and to provide insights on biodiversity issues such as habitat fragmentation and invasions of new species. Preliminary investigations show that the geometry of habitat can play important roles in the evolution of dispersal, and also that strong directed movement of a species can induce coexistence with its competitors. Materials from this project will be modified and used as team projects for a Mathematical Biosciences Institute Summer Program for college teachers and graduate students in mathematics and biology.
