Dynamics and responses of a predator-prey system with competitive interference and time delay

In this paper, we have pointed out a time-delayed predator-prey system with modified Leslie-Gower and Beddington-DeAngelis type functional responses. We discuss the structure of nonnegative equilibria and their stability analysis. Also, the invariance and boundedness of the model system are investigated. It is observed that delay destabilizes the system when the value of delay crosses its critical value. When the delay parameter is taken equals to its critical value, Hopf bifurcation occurs. Conditions for the system to become globally asymptotically stable at nonzero equilibria are also obtained. By applying the normal form theory and center manifold theorem, stability, direction, and period of bifurcating solutions of the model system are calculated. It is observed that supercritical Hopf bifurcation occurs for a set of parameter values, and the bifurcating periodic solutions are stable with decreasing period. Following the global Hopf bifurcation result of Wu (Trans Am Math Soc 35:4799-4838, 1998) for functional differential equations, we have shown that the local Hopf bifurcation implies the global Hopf bifurcation. The theoretical results are justified by presenting some numerical simulations.