Turing Pattern In A Diffusive Parasite-Host Epidemic Model

The existence of spatial pattern of a diffusive host-parasite epidemic model with the standard incidencesis investigated.The content is divided into three parts:(?)Firstly,by the linear stability analysis,we discuss the effect of diffusion on the stability of positive constant steady state.Funthermore,under the conditions of diffusion driven Turing instability,wo investigate the existence of spatial pattern.It is divided into the following three steps to discuss:Step 1.Using the maximum principle and the standard elliptic regularity theory,we obtain a priori estimates of positive steady state solutions.Step 2.Using the implicit function theorem and the Laray-Schauder degree theory,the existence and nonexistence of nonconstant positive steady-states are proved respectively.Step 3.By regarding d2 as bifurcation parameter,using the Crandall-Rabinowitz branch theory,we obtain the specific expression of the local bifurcating solutions.Fi-nally,using the global bifurcation theory,we prove that the local bifurcation curves can be extended to the global branches.Finally,using numerical simulations,long time behavior of the solutions,the effect of the space and diffusion on the spatiotemporal patterns are obtained.We find that the space is more smaller and the diffusion is more bigger,it is more easier to form the pattern.These results are beneficial to control the major factor of epidemic disease.