Travelling wave solutions for a non-local evolutionary-epidemic system

In this work we study the travelling wave solutions for a spatially distributed system of equations modelling the evolutionary epidemiology of plant-pathogen interaction. Here the mutation process is described using a non-local convolution operator in the phenotype space. Using dynamical system ideas coupled with refined estimates on the asymptotic behaviour of the profiles, we prove that the wave solutions have a rather simple structure. This analysis allows us to reduce the infinite dimensional travelling wave profile system of equations to a four dimensional ode system. The latter is used to prove the existence of travelling wave solutions for any wave speed larger than a minimal wave speed c ⋆ , provided some parameters condition expressed using the principle eigenvalue of some integral operator. It is also used to prove that any travelling wave solution connects two determined stationary states.