Stability And Hopf Bifurcation Analysis For A Predator-prey Model With Infected Prey

In this paper,we propose and analyze a predator-prey model with infected prey.We consider the system with and without time delay.Susceptible population growth is subject to Logistic growth.The effect of infected population for susceptible is nonlinear.Predator-prey interaction occurs following Holling type II functional response function.We hope to establish a mathematical model to simulate the predation relationship between species and the spread of disease within species.To lay a foundation for protecting species and to protect species diversity in the future.For the model without time delay,we discuss the positivity,stability and existence of the solution.By analyzing the corresponding characteristic equation,the local stability of equilibrium is investigated.In view of Lyapunov functionals,we obtain the condition of an interior equilibrium's global stability.The existence of Hopf bifurcation is established through the existence theorem of Hopf bifurcation.For the model with time delay,the infection process is not instantaneous,when the disease-free prey comes into contact with the diseased prey,it will take a period of time before it can be transformed into the diseased prey.Sufficient conditions were derived for the local stability of the positive equilibrium point and existence of Hopf bifurcation.The direction of Hopf bifurcation and the stability of period solutions bifurcating from Hopf bifurcations are derived by using normal from theory and center manifold theorem.