Propagation threshold of a cooperative–competitive reaction–diffusion system

Abstract

This paper focuses on the propagation threshold of a cooperative–competitive reaction–diffusion system based on a sexually transmitted disease model, which involves two stains and four unknown functions. Considering the competition‐exclusion process between two different strains, it is possible to obtain a cooperative system in proper domain after taking a linear transformation. However, the corresponding cooperative system is not always subhomogeneous and the invariant region is not always a rectangle. Using the propagation theory for monotone semiflows, we obtain a unique threshold no matter whether it is linearly selected, which allows us to numerically estimate the speed. The threshold is proved to be linearly selected under certain conditions by constructing proper upper solutions. Moreover, we numerically show that the threshold may be nonlinearly selected.