Hopf bifurcation problems near double positive equilibrium points for a class of quartic Kolmogorov model

Abstract

The Kolmogorov model is a class of significant ecological models and is initially introduced to describe the interaction between two species occupying the same ecological habitat. Limit cycle bifurcation problem is close to Hilbertis 16th problem. In this paper, we focus on investigating bifurcation of limit cycle for a class of quartic Kolmogorov model with two positive equilibrium points. Using the singular values method, we obtain the Lyapunov constants for each positive equilibrium point and investigate their limit cycle bifurcations behavior. Furthermore, based on the analysis of their Lyapunov constants' structure and Hopf bifurcation, we give the condition that each one positive equilibrium point of studied model can bifurcate 5 limit cycles, which include 3 stable limit cycles.

Keywords

• limit cycles
• kolmogorov model
• poincar$ \acute{e} $ succession function
• stable cycles