Study On The Stability And Pattern Formation Of Population And Epidemiological Models

In this paper, we investigate the stability of a predator-prey model with the disease in the predator and the pattern formation of a spatial epidemic model with migration as well as a ratio-dependent predator-prey model, respectively.Firstly, we present a predator-prey model with the disease in the predator. The boundedness of solutions and the existence of the equilibria are studied, and the sufficient conditions of locally asymptotically stable of the equilibria are also obtained by the Routh-Hurwitz criterion. Furthermore, we analyze the global stability of the equilibria by using Lyapunov functions, i.e., the conditions of the disease extinction or persistence. From the numerical results, we know the epidemic can induce the oscillation of the population density. It can induce the disease eliminate or become an endemic.Secondly, we present an epidemic model with both diffusion and migration. From the mathematical analysis and numerical simulation, we reveal that traveling pattern can be obtained in the model. That is to say that, space- and time- periodic solution can emerge when combined diffusion and migration. Furthermore, according to the dispersion relation formula, we discuss the changes of the wavelength, as well as the conditions of the spatial pattern formation. Our obtained results may be helpful to understand the mechanism of the spatial-temporal epidemics and have potential application of control of the epidemics. Finally, we present a ratio-dependent predator-prey model. Based on both mathematical analysis and numerical simulations, we get the critical line of Hopf and Turing bifurcation in a spatial domain. In particular, the exact Turing domain is given. And we have found that its spatial pattern includes spotted pattern, labyrinth pattern, spotted and stripe coexisted pattern, which shows that it is useful to use reaction-diffusion model to reveal the spatial dynamics.