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

TL;DR

This paper develops a method to analyze long transient dynamics and early warning signals of predator extinction in bistable predator-prey models near a singular zero-Hopf bifurcation, combining normal form reduction and basin boundary analysis.

Contribution

It introduces a novel approach to identify early warning signals of extinction in predator-prey systems exhibiting bistability and long transients, using basin boundary analysis and normal form reduction.

Findings

Method accurately predicts basin of attraction for initial conditions.

Early warning signals can be identified well before extinction occurs.

The approach aligns with numerical simulations of the predator-prey model.

Abstract

In this paper, we develop a method of analyzing long transient dynamics in a class of predator-prey models with two species of predators competing explicitly for their common prey, where the prey evolves on a faster timescale than the predators. In a parameter regime near a {\em{singular zero-Hopf bifurcation}} of the coexistence equilibrium state, we assume that the system under study exhibits bistability between a periodic attractor that bifurcates from the singular Hopf point and another attractor, which could be a periodic attractor or a point attractor, such that the invariant manifolds of the coexistence equilibrium point play central roles in organizing the dynamics. To find whether a solution that starts in a vicinity of the coexistence equilibrium approaches the periodic attractor or the other attractor, we reduce the equations to a suitable normal form, and examine the basin boundary near the singular Hopf point. A key component of our study includes an analysis of the long transient dynamics, characterized by their rapid oscillations with a slow variation in amplitude, by applying a moving average technique. We obtain a set of necessary and sufficient conditions on the initial values of a solution near the coexistence equilibrium to determine whether it lies in the basin of attraction of the periodic attractor. As a result of our analysis, we devise a method of 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.