Relaxation oscillations in predator-prey systems with piecewise smooth functional responses

Abstract

In this paper we consider a class of predator-prey systems with a piecewise smooth functional response and a small predator’s death rate. We provide a criterion for the existence and stability of relaxation oscillations produced by slow-fast cycles, which extends the results of Hsu (2019) [9] and Ai and Yi (2024) [2] to the piecewise smooth case. Moreover, we also study upper bounds for the number of relaxation oscillations. Our methods, different and more simple, are based on geometric singular perturbation theory, entry-exit function, the construction of difference maps and the computation of divergence integrals. Finally, we apply our results to predator-prey systems with a Holling type I functional response and show the existence of exactly two relaxation oscillations.