Coexistence of Periodic Oscillations Induced by Predator Cannibalism in a Delayed Diffusive Predator-Prey Model

This paper is concerned with the effect of predator cannibalism in a delayed diffusive predator-prey system. We aim for the case where the corresponding linear system has two pairs of purely imaginary eigenvalues at a critical point, leading to Hopf-Hopf bifurcation. An approach of center manifold reduction is applied to derive the normal form for such nonresonant Hopf-Hopf bifurcations. We find that the system exhibits very rich dynamics, including the coexistence of periodic and quasi-periodic oscillations. Numerically, we show that Hopf-Hopf bifurcation is induced if the strength of the predator cannibalism term belongs to an appropriate interval.