Complex bifurcations and noise-induced transitions: A predation model with fear effect in prey and crowding effect in predators

Abstract

It has been well established in the literature that the predator-induced fear has indirect impact on prey but can have comparable effects on prey population as direct killing, and that the crowding effect or self-limitation of population plays pivotal roles in determining the dynamics and interactions among populations. In this paper, we first propose and investigate a deterministic prey-predator model incorporating simultaneously fear effect in prey and crowding effect in predators. The model has rich dynamics, including one up to three positive equilibria, complex bifurcations (saddle-node, Hopf and Bogdanov-Takens bifurcations), and two types of bistability (between two interior equilibria or between an interior equilibrium and an interior limit cycle). Thus the model is easily affected by external environmental fluctuations. When environmental noises are involved, some new dynamics can be observed for the developed stochastic model. Especially, for the scenarios when the deterministic model exhibits bistability, we can observe noise-induced frequent transitions between two different interior attractors (two interior equilibria or an interior equilibrium and an interior limit cycle). The tipping points of noise intensities for the occurrence of such transitions are estimated by constructing the confidence ellipse/band for the equilibrium/limit cycle. These indicate that the predators and prey can coexist in two different modes and switch randomly between them.