关键词: 捕食者-食饵系统, 群体防御, 奇异摄动?, 极限环

Abstract:

This paper investigates the number of limit cycles in a predator-prey system with group defense, intially introduced by Wolkowicz and later examined by Rothe and Shafer in the 1980's. Under the assumption of large prey growth, the system reduces to a perturbed singular system, whose limit cycles can be analyzed using geometric singular perturbation methods—primarily through

the study of a slow-divergence integral. Our work completes partially the results previously obtained by Li and Zhu and by Hsu. We provide a comprehensive classification of all possible singular cycles capable of generating limit cycles and analyze the slow-divergence integral for the nine distinct types of cycle families that arise in a canard explosion. Based on these findings, we demonstrate that the maximum number of limit cycles emerging from the singular cycles is two in all cases, thereby confirming conjectures posed by Rothe-Shafer and Xiao-Ruan.

Key words: Predator-prey system, Group defense, Singular perturbation, Limit cycle