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

Supercritical Hopf Bifurcation of Cooperative Predation