## 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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### References:

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