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

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.

32 pages, 13 figures

References in corpus (2)

• A predator-2 prey fast-slow dynamical system for rapid predator evolution
• On Bifurcation Delay: An Alternative Approach Using Geometric Singular Perturbation Theory

Cited by in corpus (3)

• A Criterion for the Existence of Relaxation Oscillations with Applications to Predator-Prey Systems and an Epidemic Model
• Relaxation Oscillations and the Entry-Exit Function in Multi-Dimensional Slow-Fast Systems
• A geometric analysis of the Bazykin-Berezovskaya predator-prey model with Allee effect in an economic framework