Concentration estimates in a multi-host epidemiological model structured by phenotypic traits

In this work we consider a nonlocal system modelling the evolutionary adaptation of a pathogen within a multi-host population of plants. Here we focus our analysis on the study of the stationary states. We first discuss the existence of nontrivial equilibria using dynamical system arguments. Then we introduce a small parameter 0 0 we show that the distribution of the pathogen approaches a singular measure concentrated on the maxima of fitness in each plant population. This asymptotic description allows us to show the local stability of each of the positive steady states for 1, from which we deduce a uniqueness result for the nontrivial stationary states by means of a topological degree argument. These analyses rely on a careful investigation of the spectral properties of some nonlocal operators