Approximate solutions and multi-scale dynamical analysis: A study of predator-prey models under relaxation oscillations

This article aims to explore the asymptotic solutions of the spruce budworm system and conducting an in-depth analysis of its dynamical characteristics, particularly the existence of relaxation oscillations within the system. According to the geometric singular perturbation theory, the system is decomposed into fast and slow time scales, and quantitative and qualitative analyses are conducted for each. In terms of qualitative analysis, this paper delves into the stability of equilibrium points and the conditions for Hopf bifurcations and the formation of limit cycles. Furthermore, the study confirms the existence of relaxation oscillations in the system by applying Fenichel's theorem. In the quantitative analysis, this paper employs Vasil'eva's method to obtain the first-order approximate solution of the model for the first time, and based on this, derives a more precise expression for the system's period. Finally, the correctness and applicability of the theoretical analysis were verified through numerical simulations. To the best of our knowledge, this article provides, for the first time, a systematic first-order approximate solution for relaxation oscillations from a quantitative analysis perspective. While significantly improving the accuracy of the error approximation and the smoothness of the approximate solution, it also enhances the accuracy of the relaxation oscillation period. These works are unprecedented in the literature, at least for models like the one discussed in this study.