Spreading speed for an epidemic system modelling plant disease with adaptation

This work is devoted to the study of the spreading speed for a multi-dimensional reaction-diffusion equation coupled with a system of ODE, modelling the spatial propagation a plant disease epidemic. Here the pathogen is able to evolve in a phenotype trait space according to mutation-selection processes in order to adapt to the environment. Here we devise an epidemic threshold value, with respect to one, characterizes either the extinction or the propagation of the disease. We show that the phenotype adaptation decouples from the spatio-temporal dynamics in the large times. Moreover, in the endemic case, we also describe the spreading speed of the disease when an initial compactly supported amount of infection is introduced in the environment.