Stability Study On Two Kinds Of The Plant Epidemic Model With Time Delay

Research on plant epidemic model is one of the hot issues in biomathematics. Recently, many experts and scholars at home and abroad have paid more attention on it. In this paper, we consider a kind of soil-borne delayed differential equations(abbr. DDEs) model and a kind of DDEs model with cultural control strategy, and then discuss the dynamical behaviors of them. The main contents are summarized as follows:Firstly, based on a kind of ordinary differential equations(abbr. ODEs) model with the characteristics of soil-borne plant diseases, we formulate a class of soil-borne DDEs model by employing time delay to describe the latent period of the disease. We discuss the stability of the disease-free equilibrium and the positive equilibrium by the eigenvalue theory, the center manifold theorem, the Lyapunov stability theory and other means. Then, we obtain the conditions under which a Hopf bifurcation occurs by the Hopf bifurcation theory, and estimate the domain of attraction of the positive equilibrium by the optimization method and obtain its numerical results by Maple and other software. And then, we carry out the sensitive analysis of the basic reproduction number 0with respect to the parameters of the model.Secondly, based on a kind of DDEs plant epidemic model with cultural control strategy, we found a plant epidemic model with the survival rate of the plant during the latent period of the disease. We analyze the existence of equilibrium points of the model. Then we respectively discuss the local stability of the disease-free equilibrium and the positive equilibrium by the eigenvalue theory, the Lyapunov stability theory and other methods. In addition, by MATLAB, et al, numerical simulations are carried out to support the obtained theoretical results.