Number and Stability of Relaxation Oscillations for Predator-Prey Systems with Small Death Rates

TL;DR

This paper analyzes predator-prey systems with small predator death rates, deriving conditions for the existence and stability of relaxation oscillations using geometric singular perturbation and Floquet theories.

Contribution

It provides a rigorous analysis of the number and stability of relaxation oscillations in predator-prey models with small death rates, including a threshold for the Holling type IV response.

Findings

Unique relaxation oscillation when prey-isocline has one extremum

Either global stability or two periodic orbits with two extrema

Threshold value of carrying capacity explains paradox of enrichment

Abstract

We consider planar systems of predator-prey models with small predator death rate $\epsilon>0$. Using geometric singular perturbation theory and Floquet theory, we derive characteristic functions that determines the location and the stability of relaxation oscillations as $\epsilon\to 0$. When the prey-isocline has a single interior local extremum, we prove that the system has a unique nontrivial periodic orbit, which forms a relaxation oscillation. For some systems with prey-isocline possessing two interior local extrema, we show that either the positive equilibrium is globally stable, or the system has exact two periodic orbits. In particular, for a predator-prey model with the Holling type IV functional response we derive a threshold value of the carrying capacity that separates these two outcomes. This result supports the so-called paradox of enrichment.