  In this work we consider an epidemic system modelling the evolution of a
spore-producing 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 such nontrivial states by using the theory of global attractors.
Then we introduce a small parameter epsilon that characterises the width of the
mutation kernel, and we describe the asymptotic shape of steady states with
respect to epsilon. In particular, we show that the distribution of spores
converges to the singular measure concentrated on the maxima of fitness of the
pathogen in each plant population. This asymptotic description allows us to
show the local stability of each of the positive steady states in the regime of
narrow mutations, 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 non-local
operators.
