Stability and bifurcation analysis of the Bazykin's predator-prey ecosystem with Holling type Ⅱ functional response

Abstract

In the paper, stability and bifurcation behaviors of the Bazykin's predator-prey ecosystem with Holling type Ⅱ functional response are studied theoretically and numerically. Mathematical theory works mainly give some critical threshold conditions to guarantee the existence and stability of all possible equilibrium points, and the occurrence of Hopf bifurcation and Bogdanov-Takens bifurcation. Numerical simulation works mainly display that the Bazykin's predator-prey ecosystem has complex dynamic behaviors, which also directly proves that the theoretical results are effective and feasible. Furthermore, it is easy to see from numerical simulation results that some key parameters can seriously affect the dynamic behavior evolution process of the Bazykin's predator-prey ecosystem. Moreover, limit cycle is proposed in view of the supercritical Hopf bifurcation. Finally, it is expected that these results will contribute to the dynamical behaviors of predator-prey ecosystem.

Keywords

• bazykin's predator-prey ecosystem
• stability
• hopf bifurcation
• bogdanov-takens bifurcation
• limit cycle