Nonlocal Dispersal Equations and Convergence to Random Dispersal Equations

Abstract

This dissertation is devoted to the study of the dynamics of nonlocal and random dispersal evolution equations. Dispersal evolution equations are widely used to model the diffusions of organisms or individuals in many biological and ecological systems. More precisely, random and nonlocal dispersal equations arise in modeling the dynamics of diffusive systems which exhibit random or local, and nonlocal internal interactions, respectively. In this dissertation, we study the dynamics of such equations complemented with Dirichlet, Neumann, and periodic types of boundary condition in a unified way. It is mainly concerned with principal spectral theory of nonlocal dispersal operators and the approximations of random dispersal operators/equations by nonlocal dispersal operators/equations.

Regarding the principal spectral theory of nonlocal dispersal operators, we investigate the dependence of the principal spectrum points of nonlocal dispersal operators on the underlying parameters and its applications. In particular, we study the effects of the spatial inhomogeneity, the dispersal rate, and the dispersal distance on the existence of the principal eigenvalues, the magnitude of the principal spectrum points, and the asymptotic behaviors of the principal spectrum points of time homogeneous nonlocal dispersal operators with Dirichlet type, Neumann type, and periodic boundary conditions. We also discuss the applications of the principal spectral theory of nonlocal dispersal operators to the asymptotic dynamics of two species competition systems.

About the approximations of random dispersal operators/equations by nonlocal dispersal operators/equations, we first prove that the solutions of properly rescaled nonlocal dispersal initial-boundary value problems converge to the solutions of the corresponding random dispersal initial-boundary value problems. Next, we prove that the principal spectrum points of time periodic nonlocal dispersal operators with properly rescaled kernels converge to the principal eigenvalues of the corresponding random dispersal operators. Thirdly, we prove that the unique positive time periodic solutions of nonlocal dispersal KPP type evolution equations with properly rescaled kernels converge to the unique positive time periodic solutions of the corresponding random dispersal KPP type evolution equations. We also discuss the applications of the approximation results to the effects of the rearrangements with equimeasurability on principal spectrum point of nonlocal dispersal operators.