Stability and Bifurcation Analysis of Bazykin’s Model with Holling I Functional Response and Allee Effect

Abstract

In this paper, we introduce Allee effect and predator competition in the Bazykin’s model with Holling I functional response. Theoretically, we analyze the existence and stability of equilibria, and derive the existence conditions of saddle-node bifurcation and Hopf bifurcation. In addition, in order to determine the stability of limit cycles, we explicitly calculate the first Lyapunov coefficient and prove that the positive equilibrium is not a center, but a weak focus with a multiplicity of at least two. Therefore, the system has Hopf bifurcation and Bautin bifurcation with two limit cycles. Our results indicate that the Allee effect and predator competition lead to a series of complex dynamic phenomena. Finally, numerical simulation verifies the effectiveness of the theoretical results.