Stability, bifurcation and limit cycle for a predator-prey model with some feedback control

This paper is devoted to a predator-prey model with some feedback control. We prove that there exist a unique positive equilibrium or three positive equilibria for such model if the feedback control parameters satisfy some conditions. We also show that, under some additional assumptions, the positive equilibrium is asymptotically stable. Finally, we study the existence of limit cycles as well as bifurcations in this predator-Prey model. It is further shown that there exist infinite bifurcation points as long as the parameters of the model is in some area. Numerical simulations demonstrate this asymptotic behavior depending on parameters of the species.