Abstract
The main purpose of this thesis is to develop basic theories and results for a class of first-order partial differential equations with spatial convolution, which describe the population dynamics of species with age structure and nonlocal dispersal. First, we introduce our models; that is, age-structured population models with nonlocal diffusion and provide some basic knowledge and background. Secondly, from the view of solution flows we investigate the existence, uniqueness, and continuous dependence of solutions on the parameters and initial data of the abstract equation and define the infinitesimal generator of semigroups generated by the solution flows. Next from the view of an abstract operator we study its spectrum properties in an appropriate Banach space and search for conditions to ensure that the operator generates a strongly continuous semigroup and investigate the stability of linear equations by spectrum analysis. Then, we develop the basic theory for age-structured population models with nonlocal diffusion and nonlocal boundary conditions via the theory of integrated semigroups and non-densely defined operators. Afterwards, we study further the principal spectral theory of age-structured models with nonlocal diffusion without nonlocal boundary conditions by the theory of resolvent operators with their perturbations. Finally, we summarize the results of the thesis and discuss some potential future studies.