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For a slow-fast system of the form $\dot{p}=\epsilon f(p,z,\epsilon)+h(p,z,\epsilon)$, $\dot{z}=g(p,z,\epsilon)$ for $(p,z)\in \mathbb R^n\times \mathbb R^m$, we consider the scenario that the system has invariant sets $M_i=\{(p,z): z=z_i\}$, $1\le i\le N$, linked by a singular closed orbit formed by trajectories of the limiting slow and fast systems. Assuming that the stability of $M_i$ changes along the slow trajectories at certain turning points, we derive criteria for the existence and stability of relaxation oscillations for the slow-fast system. Our approach is based on a generalization of the entry-exit relation to systems with multi-dimensional fast variables. We then apply our criteria to several predator-prey systems with rapid ecological evolutionary dynamics to show the existence of relaxation oscillations in these models., Comment: 32 pages

We derive characteristic functions to determine the number and stability of relaxation oscillations for a class of planar systems. Applying our criterion, we give conditions under which the chemostat predator-prey system has a globally orbitally asymptotically stable limit cycle. Also we demonstrate that a prescribed number of relaxation oscillations can be constructed by varying the perturbation for an epidemic model studied by Li et al. [SIAM J. Appl. Math, 2016]., Comment: 21 pages, 8 figures

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., Comment: 32 pages, 13 figures