Canard, homoclinic loop, and relaxation oscillations in a Lotka–Volterra system with Allee effect in predator population

Abstract

In this paper, we study the dynamics of a Lotka–Volterra model with an Allee effect, which is included in the predator population and has an abstract functional form. We classify the original system as a slow–fast system when the conversion rate and mortality of the predator population are relatively low compared to the prey population. In comparison to numerical simulation results that indicate at most three limit cycles in the system [Sen et al., J. Math. Biol. 84(1), 1–27 (2022)], we prove the uniqueness and stability of the slow–fast limit periodic set of the system in the two-scale framework. We also discuss canard explosion phenomena and homoclinic bifurcation. Furthermore, we use the enter–exit function to demonstrate the existence of relaxation oscillations. We construct a transition map to show the appearance of homoclinic loops including turning or jump points. To the best of our knowledge, the homoclinic loop of fast slow jump slow type, as classified by Dumortier, is uncommon. Our biological results demonstrate that under certain parameter conditions, population density does not change uniformly, but instead presents slow–fast periodic fluctuations. This phenomenon may explain sudden population density explosions in populations.