Abstract. In this paper, we study a simplified version of a West Nile virus model discussed by Lewis et al. [28], which was considered as a first approximation for the spatial spread of WNv. The basic reproduction number R 0 for the non-spatial epidemic model is defined and a threshold parameter R D 0 for the corresponding problem with null Dirichlet boundary condition is introduced. We consider a free boundary problem with coupled system, which describes the diffusion of birds by a PDE and the movement of mosquitoes by a ODE. The risk index R F 0 (t) associated with the disease in spatial setting is represented. Sufficient conditions for the WNv to eradicate or to spread are given. The asymptotic behavior of the solution to system when the spreading occurs are considered. It is shown that the initial number of infected populations, the diffusion rate of birds and the length of initial habitat exhibit important impacts on the vanishing or spreading of the virus. Numerical simulations are presented to illustrate the analytical results.MSC: primary: 35R35; secondary: 35K60

In order to investigate the impact of periodically evolving domain on the mutualism interaction of two species, we study a mutualistic model on a periodically evolving domain. To overcome the difficulty caused by the advection and dilution terms, we transform the model to a reaction–diffusion problem in a fixed domain. By means of eigenvalue problems, the threshold parameters are introduced. The asymptotic profiles of the solutions on an evolving domain are studied by using the threshold parameters and the upper and lower solutions method. The impact of the domain evolution rate on the persistence or extinction of species is analyzed. Numerical simulations are performed to illustrate our analytical results.

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.

This paper is concerned with a diffusive West Nile virus model (WNv) in a heterogeneous environment. The basic reproduction number [Formula: see text] for spatially homogeneous model is first introduced. We then define a threshold parameter [Formula: see text] for the corresponding diffusive WNv model in a heterogeneous environment. It is shown that if [Formula: see text], the model admits at least one nontrivial T-periodic solution, whereas if [Formula: see text], the model has no nontrivial T-periodic solution. By means of monotone iterative schemes, the true solution can be obtained and the asymptotic behavior of periodic solutions is presented. The paper is closed with some numerical simulations to illustrate our theoretical results.