Bifurcations of a predator-prey system with cooperative hunting and Holling III functional response

TL;DR

This paper analyzes a predator-prey model with cooperative hunting and Holling III response, revealing complex bifurcation behaviors and demonstrating that cooperative hunting enhances predator persistence and ecosystem diversity.

Contribution

It provides a detailed bifurcation analysis of the predator-prey system with cooperative hunting, uncovering richer dynamics than models without cooperation.

Findings

Presence of weak focus of multiplicity three

Existence of degenerate equilibrium with double zero eigenvalues

Various bifurcations including transcritical, pitchfork, saddle-node, Bogdanov-Takens, and Hopf

Abstract

In this paper, we consider the dynamics of a predator-prey system of Gause type with cooperative hunting among predators and Holling III functional response. The known work numerically shows that the system exhibits saddle-node and Hopf bifurcations except homoclinic bifurcation for some special parameter values. Our results show that there are a weak focus of multiplicity three and a degenerate equilibrium with double zero eigenvalues (i.e., a cusp of codimension two) for general parameter conditions and the system can exhibit various bifurcations as perturbing the bifurcation parameters appropriately, such as the transcritical and the pitchfork bifurcations at the degenerate boundary equilibrium, the saddle-node and the Bogdanov-Takens bifurcations at the degenerate positive equilibrium and the Hopf bifurcation around the weak focus. The comparative study demonstrates that the dynamics are far richer and more complex than that of the system without cooperative hunting among predators. The analysis results reveal that appropriate intensity of cooperative hunting among predators is beneficial for the persistence of predators and the diversity of ecosystem.