The uniqueness of limit cycles in a predator-prey system with Ivlev-type group defense

Abstract

This paper discusses the uniqueness of limit cycles in a two-dimensional autonomous Gause predator-prey model with an Ivlev-type group defense introduced by D. M. Xiao, S. G. Ruan, Codimension two bifurcations in a predator-prey system with group defense, Int. J. Bifurcat. Chaos, 11 (2001). We proved their conjecture that the system can exhibit at most one limit cycle. Furthermore, we compared the qualitative differences between this system and two similar systems with group defense: One system with the same Ivlev-type functional response function but with Leslie-Gower predator dynamics and another system with a comparable functional response function. For both systems, we show that two limit cycles can occur.

Keywords

• limit cycle
• group defense
• liénard system
• predator-prey
• gause system