Supercritical Hopf Bifurcation of Cooperative Predation

TL;DR

This paper rigorously analyzes a predator-prey model with cooperative predation, establishing conditions for stability, bifurcation, and oscillations, and proves that the Hopf bifurcation is always supercritical, leading to stable oscillations.

Contribution

It provides a detailed mathematical analysis of cooperative predation dynamics, including explicit criteria for equilibria stability and the nature of Hopf bifurcations, which was not previously established.

Findings

Positive equilibria existence criteria derived

Hopf bifurcation is always supercritical

Stable oscillations near bifurcation points

Abstract

In this work, we conduct a rigorous analysis on the dynamics of a predator-prey model with cooperative predation. From the root classification of an algebraic equation, we derive existence criteria of the positive equilibria. By Jacobian matrix and central manifold theory, we find critical conditions under which the positive equilibria are locally asymptotically stable or unstable. We also use a careful computation to obtain a concise and explicit formula for the first Lyapunov coefficient. Especially, we prove that the Hopf bifurcation induced by the cooperative predation is always supercritical, which means that the sustained oscillations near the Hopf bifurcation points are locally asymptotically stable.