Coexistence of a cross-diffusive West Nile virus model in a heterogenous environment

This paper is concerned with a strongly-coupled elliptic system, which describes a West Nile virus (WNv) model with cross-diffusion in a heterogeneous environment. The basic reproduction number is introduced through the next generation infection operator and some related eigenvalue problems. The existence of coexistence states is presented by using a method of upper and lower solutions. The true positive solutions are obtained by monotone iterative schemes. Our results show that a cross-diffusive WNv model possesses at least one coexistence solution if the basic reproduction number is greater than one and the cross-diffusion rates are small enough, while if the basic reproduction number is less than or equal to one, the model has no positive solution. To illustrate the impact of cross-diffusion and environmental heterogeneity on the transmission of WNv, some numerical simulations are given.