Spatial propagation for a two component reaction–diffusion system arising in population dynamics

In this work a two component epidemic reaction–diffusion system posed on the whole space RNRN is considered. Uniform boundedness of the solutions is proved using suitable local LpLp-estimates. The spatial invasion of a localized introduced amount of infective is studied yielding to the derivation of the asymptotic speed of spread for the infection. This part is achieved using uniform persistence ideas. The state of the population after the epidemic is further investigated using different Lyapunov like arguments. The solution is shown to converge the endemic equilibrium point behind the front in the equi-diffusional case. For general diffusion coefficient unique ergodicity of the tail of invasion is obtained by constructing a suitable sub-harmonic map.