摘要
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.
本文针对具有群体防御的捕食者-食饵系统,研究其极限环的个数问题.该系统最早由Wolkowicz提出,并在20世纪80年代由Rothe-Shafer进一步展开研究.当猎物增长率较大时,系统变为奇异摄动系统,其极限环可通过几何奇异摄动法进行分析,本质上是对慢散度积分的研究.本文完善了Li-Zhu和Hsu两篇论文的部分结果,给出所有可能产生极限环的奇异环的完整列表,并针对“canard”爆炸中出现的九类不同环族的慢散度积分进行分析.基于这些结果,我们得出该系统在所有情况下从奇异环摄动产生的极限环个数至多为2,从而证实Rothe-Shafer和Xiao-Ruan论文中提出的猜想.
出处
《数学理论与应用》
2025年第3期1-52,共52页
Mathematical Theory and Applications
