| Abstract: |
| Predator-prey models with small predator death rate can be understood as slow-fast systems. Using geometric singular perturbation theory and the theory of Floquet multipliers, I will derive characteristic functions that determine the location and the stability of relaxation oscillations. For systems with prey isocline possessing a single interior local extremum, I will show 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, I will show that either the positive equilibrium is globally stable, or the system has exact two periodic orbits. In particular, for the Holling type IV functional response there is 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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