Limit cycles of a cubic Kolmogorov system. (English) Zbl 0858.34023

The authors consider limit cycles which bifurcate from a critical point in case of cubic Kolmogorov systems of the form \[ \dot x=x(x-2y+2)(Ax+y+B),\quad \dot y=y(2x-y-2)(Dx+y+C).\tag{1} \] The authors use the computer algebra procedure FINDETA which was described by the first two authors [J. Symb. Comput. 9, No. 2, 215-224 (1990; Zbl 0702.68072)] to compute the focal values at the critical point \((2,2)\) and show that for certain cubic Kolmogorov systems (1), four and not more than four limit cycles can bifurcate from a critical point in the first quadrant; moreover, three fine foci of order one can coexist and a limit cycle can bifurcate from each.


34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
92D25 Population dynamics (general)
34C23 Bifurcation theory for ordinary differential equations


Zbl 0702.68072
Full Text: DOI


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