Center-focus problem and limit cycles bifurcations for a class of cubic Kolmogorov model

Abstract

The problem of limit cycles for the Kolmogorov model is interesting and significant both in theory and applications. In this paper, we investigate the center-focus problems and limit cycles bifurcations for a class of cubic Kolmogorov model with three positive equilibrium points. The sufficient and necessary condition that each positive equilibrium point becomes a center is given. At the same time, we show that each one of point (1,2) and point (2,1) can bifurcate 1 small limit cycles under a certain condition, and 3 limit cycle can occur near (1,1) at the same step. Among the above limit cycles, 4 limit cycles can be stable. The limit cycles bifurcations problem for Kolmogorov model with several positive equilibrium points are hardly seen in published references. Our result is new and interesting.