Spreading speeds in an asymptotic autonomous system with application to an epidemic model

This article studies the initial value problem in an asymptotic autonomous reaction-diffusion system. Namely, when the time goes to infinity, these parameters converge to constants. The system may be nonmonotonic. With different decaying initial conditions, the leftward and rightward spreading speeds are estimated by constructing auxiliary functions and systems. As an application, we present the propagation properties in a diffusive epidemic model with different decaying initial values, which does not satisfy the classical comparison principle of mixed quasimonotone systems.