Bogdanov-Takens bifurcation of codimension 3 in a predator-prey model with constant-yield predator harvesting

Jicai Huang,Sanhong Liu,Shigui Ruan,Xinan Zhang

doi:10.3934/cpaa.2016.15.1053

Abstract

Recently, we (J. Huang, Y. Gong and S. Ruan, Discrete Contin. Dynam. Syst. B 18 (2013), 2101-2121) showed that a Leslie-Gower type predator-prey model with constant-yield predator harvesting has a Bogdanov-Takens singularity (cusp) of codimension 3 for some parameter values. In this paper, we prove analytically that the model undergoes Bogdanov-Takens bifurcation (cusp case) of codimension 3. To confirm the theoretical analysis and results, we also perform numerical simulations for various bifurcation scenarios, including the existence of two limit cycles, the coexistence of a stable homoclinic loop and an unstable limit cycle, supercritical and subcritical Hopf bifurcations, and homoclinic bifurcation of codimension 1.

Highlights

Harvesting is commonly practiced in fishery, forestry, and wildlife management (Clark [8])
Our analytical results confirmed the conjecture in Huang, Gong and Ruan [17] about the existence of Bogdanov-Takens bifurcation of codimension 3 in system (5), so there exist some new dynamics in system (5), such as the coexistence of a stable homoclinic loop and an unstable limit cycle, two limit cycles and a semi-stable limit cycle for various parameters values, which were only numerically simulated in [17]
Notice that these complex dynamics cannot occur in the unharvested systems (2) and the case (3) with only constant-yield prey harvesting

Summary

Introduction

Harvesting is commonly practiced in fishery, forestry, and wildlife management (Clark [8]). Predator-prey model, constant-yield harvesting, Bogdanov-Takens bifurcation of codimension 3, Hopf bifurcaton, homoclinic bifurcation.

Results

Conclusion