Nonlinear age-structured population models with nonlocal diffusion and nonlocal boundary conditions

Abstract

In this paper, we develop some basic theory for age-structured population models with nonlocal diffusion and nonlocal boundary conditions. We first apply the theory of integrated semigroups and non-densely defined operators to a linear equation, study the spectrum, and analyze the asymptotic behavior via asynchronous exponential growth. Then we consider a semilinear equation with nonlocal diffusion and nonlocal boundary condition, use the method of characteristic lines to find the resolvent of the infinitesimal generator and the variation of constant formula, apply Krasnoselskii's fixed point theorem to obtain the existence of nontrivial steady states, and establish the stability of steady states. Finally we generalize these results to a nonlinear equation with nonlocal diffusion and nonlocal boundary condition.