Dynamic Analysis Of A Predator-Prey Model With A Prey Refuge

Predator-prey model is a type of ecological model to study the relationship be-tween different populations,and the epidemiological model is a model to reveal the dynamics mechanism of epidemic diseases based on the process of disease transmis-sion and change rule within the population.The spread of diseases can have a great impact on ecosystems and even make the species extinction.The establishment of prey refuge can balance the relationship between species effectively and prevent the prey extinction.In this paper,the predator-prey model is combined with the epidemiological model to establish an ecological epidemiological predator-prey model where only the prey population is infected and the model with infected prey refuge.A modified predator-prey model of Leslie-Gower and Holling II are considered hereFirst,the upper and lower solution method and comparison principle are used to give the boundedness,and the extinction of infected prey is obtained by using the Gronwall inequality.Secondly,the local stability of the positive constant equilibria is analyzed by using eigenvalue theory,and the global stability of constant equilib-ria is proved by constructing Lyapunov function.We also have studied the priori estimate of positive steady-state solution.By making use of the Poincare inequality and topological degree theory,we prove the nonexistence and the existence of non-constant steady-state solutions are discussed under a certain of conditions.Finally,the local structure of the bifurcation solution from the semi-trivial constant solu-tion is proved by the local bifurcation theorem.Further,the existence of periodic solution is analyzed by the Hopf bifurcation theorem.