Analysis of bistable behavior and early warning signals of extinction in a class of predator-prey models

Abstract

In this paper, we develop a method of detecting an early warning signal of catastrophic population collapse in a class of predator-prey models with two species of predators competing for their common prey, where the prey evolves on a faster timescale than the predators. In a parameter regime near {\em{singular Hopf bifurcation}} of a coexistence equilibrium point, we assume that the class of models exhibits bistability between a periodic attractor and a boundary equilibrium point, where the invariant manifolds of the coexistence equilibrium play central roles in organizing the dynamics. To determine whether a solution that starts in a vicinity of the coexistence equilibrium approaches the periodic attractor or the point attractor, we reduce the equations to a suitable normal form, which is valid near the singular Hopf bifurcation, and study its geometric structure. A key component of our study includes an analysis of the transient dynamics, characterized by their rapid oscillations with a slow variation in amplitude, by applying a moving average technique. As a result of our analysis, we could devise a method for identifying early warning signals, significantly in advance, of a future crisis that could lead to extinction of one of the predators. The analysis is applied to the predator-prey model considered in [\emph{Discrete and Continuous Dynamical Systems - B} 2021, 26(10), pp. 5251-5279] and we find that our theory is in good agreement with the numerical simulations carried out for this model