Turing Patterns In A Host-Parasite Model With Density-and Frequency-Dependent Transmission

In this paper,pattern formations in a host-parasite model with density-and frequency-dependent transmissions are investigated.The main work of this disser-tation is as follows:First,by linearization method:the effect of diffusion on the endemic equilibri-um E*is analyzed.It is shown that diffusion driven Turing instability may occur for the equilibrium E*which is stable for the ordinary differential system but unstable for the reaction-diffusion system.Second,a priori estimates of positive steady states are given by the maximum principle and the standard elliptic regularity.The existence and nonexistence of nonconstant positive steady states are proved by the implicit function theorem and the Laray-Schauder degree theory,respectively.According to local and global bifur-cation theory,treating d2 as bifurcation parameter,the local and global structure of non-constant steady states are studied.Furthermore,by means of asymptotic analysis,we derive the stability results and an specific expression of non-constants steady.Finally,selecting suitable parameters and using Matlab Software,some numeri-cal simulations are given to certify different types of Turing patterns,such as spots,stripe-spot and stripe-hole patterns.The obtained results indicate that diffusion has great influence on the spatial pattern formation,which provide the numerical exam-ple to better understand the dynamic processes of epidemic in real environment.