Bogdanov\u2013Takens bifurcation of a Holling IV prey\u2013predator model with constant-effort harvesting

Lifang Cheng,Litao Zhang

doi:10.1186/s13660-021-02597-9

Abstract

A prey–predator model with constant-effort harvesting on the prey and predators is investigated in this paper. First, we discuss the number and type of the equilibria by analyzing the equations of equilibria and the distribution of eigenvalues. Second, with the rescaled harvesting efforts as bifurcation parameters, a subcritical Hopf bifurcation is exhibited near the multiple focus and a Bogdanov–Takens bifurcation is also displayed near the BT singularity by analyzing the versal unfolding of the model. With the variation of bifurcation parameters, the system shows multi-stable structure, and the attractive domains for different attractors are constituted by the stable and unstable manifolds of saddles and the limit cycles bifurcated from Hopf and Bogdanov–Takens bifurcations. Finally, a cusp point and two generalized Hopf points are found on the saddle-node bifurcation curve and the Hopf bifurcation curves, respectively. Several phase diagrams for parameters near one of the generalized Hopf points are exhibited through the generalized Hopf bifurcation.

Highlights

1 Introduction The prey–predator model based on Lotka–Volterra model is one of the most popular models in mathematical ecology and has been widely applied in understanding population dynamics of the species, which is characterized by the complicated interaction among the species and the interaction between the species and their surroundings
We find that most of the harvested prey–predator models which have been considered by many authors are confined to three aspects: (i) constant-yield harvesting, (ii) harvesting subjected to only one species or (iii) the numerical response is proportional to the functional response, where the numerical response is the change in predator density as a function of change in prey density [26]
Besides some original dynamical behaviors, some new phenomena and bifurcations appear in the harvested model, such as a globally stable equilibrium, a cusp bifurcation and a generalized Hopf bifurcation

Summary

Introduction

The prey–predator model based on Lotka–Volterra model is one of the most popular models in mathematical ecology and has been widely applied in understanding population dynamics of the species, which is characterized by the complicated interaction among the species and the interaction between the species and their surroundings. Li and Xiao [20] proved that the model could simultaneously undergo a Bogdanov–Takens bifurcation and a subcritical Hopf bifurcation in the small neighborhoods of two different equilibria, respectively. When the system had a unique positive equilibrium, there existed parameter values such that the system had two limit cycles around it.

Results

Conclusion