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

• Published in: Journal of Differential Equations Vol. 269; no. 12; pp. 11492 - 11539
• Main Authors: Burie, Jean-Baptiste, Ducrot, Arnaud, Griette, Quentin, Richard, Quentin
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 05.12.2020
• Subjects: Concentration phenomenon; Epidemiology; Nonlocal equation; Population genetics; Steady state solutions; 92D10; 35R09; 92D30; 47H11; Steady state solutions; Nonlocal equation; 35B40; Concentration phenomenon; Epidemiology; Population genetics
• ISSN: 0022-0396; 1090-2732
• DOI: 10.1016/j.jde.2020.08.029

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<ε≪1 that characterises the width of the mutation kernel, and we describe the asymptotic shape of steady states with respect to ε. In particular, for ε→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 ε≪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.