Stability and bifurcation analysis of a discrete predator-prey system with modified Holling-Tanner functional response

In this paper, we study a discrete predator-prey system with modified Holling-Tanner functional response. We derive conditions of existence for flip bifurcations and Hopf bifurcations by using the center manifold theorem and bifurcation theory. Numerical simulations including bifurcation diagrams, maximum Lyapunov exponents, and phase portraits not only illustrate the correctness of theoretical analysis, but also exhibit complex dynamical behaviors and biological phenomena. This suggests that the small integral step size can stabilize the system into the locally stable coexistence. However, the large integral step size may destabilize the system producing far richer dynamics. This also implies that when the intrinsic growth rate of prey is high, the model has bifurcation structures somewhat similar to the classic logistic one.