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.

Exploring the dynamics of a Holling–Tanner model with cannibalism in both predator and prey population

Cannibalism is an intriguing life history trait, that has been considered primarily in the predator, in predator–prey population models. Recent experimental evidence shows that prey cannibalism can have a significant impact on predator–prey population dynamics in natural communities. Motivated by these experimental results, we investigate a ratio-dependent Holling–Tanner model, where cannibalism occurs simultaneously in both the predator and prey species. We show that depending on parameters, whilst prey or predator cannibalism acting alone leads to instability, their joint effect can actually stabilize the unstable interior equilibrium. Furthermore, in the spatially explicit model, we find that depending on parameters, prey and predator cannibalism acting jointly can cause spatial patterns to form, while not so acting individually. We discuss ecological consequences of these findings in light of food chain dynamics, invasive species control and climate change.