A Nonlocal Spatial Model on Continuous Time and Space

In this work we studied a nonlocal spatial model on continuous time and space. Based on Levins’ metapopulation framework, we developed a population model with nonlocal dispersal. The dispersal is modeled by an integro-differential equation. In the first chapter, we studied the well-posedness of a single species model. We established the existence and uniqueness of solution, and proved a version of maxi- mum principal as well as comparison theorem. To study the stability of equilibria, we considered an eigenvalue problem and provided an estimation of the eigenvalue. Then we gave the condition of having a stable positive equilibrium, which biologi- cally implies the persistence of species; and we also gave the condition of a stable zero equilibrium, which means the species goes extinct. In the second chapter, we investigated the two species competition model. We did the stability analysis for the zero equilibrium and two semi-trivial equilibria. Also we have a sufficient condition for the existence of a coexistence equilibrium. Then we studied the evolutionarily stable strategy for this model. Ideal free dispersal is a kind of conditional strategy which feature dependence on environments and leads to an equilibrium distribution where there is no net movement of individuals and any location has the same environmental fitness. Suppose two competing species are identical except their dispersal strategy. We showed that a species with ideal free dispersal can invade when rare while the other species’ dispersal is not ideal free. In chapter three we are interested in the spreading speed on a infinite domain. The case of single species has been treated in an SIS epidemiology model. For two species competition, we proved the existence of spreading speed and showed that for each wave speed greater than the spreading speed, there exists a traveling wave solution connecting the two semi-trivial equilibria for the system