Bifurcation Analysis of a Multiple-Delayed Predator–Prey System with Habitat Complexity

This paper presents a generalized delayed predator–prey system with habitat complexity. First, the dynamical behaviors of the considered system, including positivity and boundedness, stability property and the existence of Hopf bifurcation and super-critical bifurcation without time delay, are analyzed. Second, the effect of time delay on the dynamical behavior of the presented system is considered in four cases, and sufficient conditions for the existence of Hopf bifurcation are obtained. Moreover, the explicit formulas to determine the direction, stability and period of bifurcating periodic solutions are derived by applying the center manifold theory and normal form theory. Finally, numerical simulations are conducted to verify the corresponding theoretical results.