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Center-focus problem and limit cycles bifurcations for a class of cubic Kolmogorov model. (English) Zbl 1269.92065

Summary: The problem of limit cycles for the Kolmogorov model is interesting and significant both in theory and applications. We investigate the center-focus problems and limit cycles bifurcations for a class of cubic Kolmogorov models with three positive equilibrium points. A sufficient and necessary condition that each positive equilibrium point becomes a center is given. We show that each one of point \((1,2)\) and point \((2,1)\) can bifurcate 1 small limit cycle under a certain condition, and 3 limit cycles 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 models with several positive equilibrium points are hardly seen in published references. Our result is new and interesting.

MSC:

92D40 Ecology
34C23 Bifurcation theory for ordinary differential equations
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