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arXiv · 1801.02590

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

Ting-Hao Hsu
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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.

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BibTeXRIS

Ting-Hao Hsu. 2018-01-08. Number and Stability of Relaxation Oscillations for Predator-Prey Systems with Small Death Rates. https://doi.org/10.1137/18m1166705

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