On the large time behaviour of the solutions of an evolutionary-epidemic system with spatial dispersal

This paper investigates some properties of the large time behaviour of the solutions of a spatially distributed system of equations modelling the evolutionary epidemiology of a plant-pathogen system. The model takes into account the phenotypic trait and the mutation of the pathogen, which is described by a non-local operator. We roughly speaking prove that the solutions separate the phenotype trait from the spatio-temporal evolution in the large time asymptotic. This feature is obtained by investigating the positive and bounded entire solutions of the problem, which are shown to exhibit such a separation of the variables property, by reformulating them as the positive solutions of suitable integral equations in some ordered Banach space. In addition, some numerical simulations are performed to support our theoretical results.