Bifurcations in Holling-Tanner model with generalist predator and prey refuge

Abstract

Refuge provides an important mechanism for preserving many ecosystems. Prey refuges directly benefit prey but also indirectly benefit predators in the long term. In this paper, we consider the complex dynamics and bifurcations in Holling-Tanner model with generalist predator and prey refuge. It is shown that the model admits a nilpotent cusp or focus of codimension 3, a nilpotent elliptic singularity of codimension at least 4, and a weak focus with order at least 3 for different parameter values. As the parameters vary, the model can undergo three types degenerate Bogdanov-Takens bifurcations of codimension 3 (cusp, focus and elliptic cases), and degenerate Hopf bifurcation of codimension 3. The system can exhibit complex dynamics, such as multiple coexistent periodic orbits and homoclinic loops. Moreover, our results indicate that the constant prey refuge prevents prey extinction and causes global coexistence. A preeminent finding is that refuge can induce a stable, large-amplitude limit cycle enclosing one or three positive steady states. Numerical simulations are provided to illustrate and complement our theoretical results.