Non-constant positive steady states of a host-parasite model with frequency- and density-dependent transmissions

Abstract

The present work is concerned with some fundamental analytic properties of a host-parasite model with frequency- and density-dependent transmissions under homogeneous Neumann boundary condition. The global existence and the boundedness of solution are investigated. The stability of the constant positive steady state is discussed, and the Turing instability is determined. Furthermore, the framework of the occurrence of the Turing pattern of the model is established: when the diffusion coefficients of the susceptible and infected hosts are large enough, there is nonexistence of non-constant positive steady states, which impies that the diffusion is helpful for creating non-constant positive steady states to the model; when the diffusion coefficient of the infected hosts is properly chosen and other parameters are fixed, the model exhibits non-constant positive steady-states, and the stationary Turing pattern can arise as a result of diffusion. Via numerical simulations, the evolutionary processes of the pattern formation of the uninfected/infected hosts with local diffusion are presented, which shows that the model dynamics exhibits a diffusion-controlled formation growth of spots, stripes and holes pattern replication. In the viewpoint of epidemiology, the parameters must be adjusted in the specific feasible range to control the disease.