Nonlocal Dispersal Equations with Almost Periodic Dependence
Type of DegreePhD Dissertation
Mathematics and Statistics
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Nonlocal dispersal equations are used to model the population dynamics of species that exhibit long-range dispersal mechanisms. This model of spatial spread is obtained by replacing the Laplacian in the usual reaction-diffusion equation with an integral operator. Most studies on this model were done for temporally homogeneous or periodic environments. However, nature is typically heterogeneous, and even when seasonal, the variations could exhibit disproportionate periods. Therefore, it seems more appropriate to incorporate the time-dependent variability of these factors using almost periodicity. The asymptotic behavior of solutions with strictly positive initials is among the fundamental issues for such population models and the stability of the zero solution is crucial in investigating these asymptotic dynamics. Thus the principal spectral theory of the linearization of the model at the zero solution is important in its own right but vital for investigating the asymptotic dynamics. This dissertation is devoted to the study of the spectral theory and asymptotic dynamics of nonlocal dispersal equations with almost periodic dependence. First, the principal spectral theory of linear nonlocal dispersal equations is investigated from three aspects: top Lyapunov exponents, principal dynamical spectrum point, and generalized principal eigenvalues. Among others, we established the equality of the top Lyapunov exponents and the principal dynamical spectrum point, provided various characterizations of the top Lyapunov exponents and generalized principal eigenvalues, established the relations between them, and studied the effect of time and space variations on them. Secondly, employing the principal spectral theory developed in the first part, we studied the asymptotic dynamics of nonlinear nonlocal dispersal equations with almost periodic dependence. In particular, we established the existence, uniqueness, and stability of a strictly positive, bounded, entire, almost periodic solution of the Fisher-KPP equation with nonlocal dispersal and almost periodic reaction term. Finally, when the domain is the whole space, we investigated the spatial spreading speeds of positive solutions with front-like initials.