Relaxation Oscillations in Predator-Prey Systems

We characterize a criterion for the existence of relaxation oscillations in planar systems of the form du/dt = u(k+1) g(u, v, epsilon), dv/dt = epsilon f(u, v, epsilon) + u(k+1)h(u, v, epsilon), where k >= 0 is an arbitrary constant and epsilon > 0 is a sufficiently small parameter. Taking into account of possible degeneracy of the "discriminant" function occurred when k >= 0, this criterion generalizes and strengthens those for the case k = 0 obtained by Hsu (SIAM J Appl Dyn Syst 18:33-67, 2019) and Hsu and Wolkowicz (Discrete Contin Dyn Syst Ser B 25:1257-1277, 2020). Differing from the case of k = 0, our proof of the criterion is based on the construction of an invariant, thin annular region in an arbitrarily prescribed small neighborhood of a singular closed orbit and the establishment of an asymptotic formula for solutions near the v-axis. As applications of this criterion, we will give concrete conditions ensuring the existence of relaxation oscillations in general predator-prey systems, as well as spatially homogeneous relaxation oscillations and relaxed periodic traveling waves in a class of diffusive predator-prey systems.