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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.

MSC:

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

Citations:

Zbl 0702.68072
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References:

[1] Hirsch, M. W., Systems of differential equations that are competitive or cooperative. V: Convergence in 3-dimensional systems, J. Differential Equations, 80, 94-106 (1989) · Zbl 0712.34045
[2] Ye, Y.; Ye, W., Cubic Kolmogorov differential systems with two limit cycles surrounding the same focus, Ann. Differential Equations, 1, 2, 201-207 (1985) · Zbl 0597.34020
[3] Coleman, C. S., Hilbert’s 16th problem: How many cycles?, (Lucas, W., Differential Equations Models, Volume 1 (1978), Springer-Verlag), 279-297
[4] Lloyd, N. G.; Pearson, J. M., REDUCE and the bifurcation of limit cycles, J. Symbolic Comput., 9, 215-224 (1990) · Zbl 0702.68072
[5] Lloyd, N. G.; Pearson, J. M., Algorithmic derivation of centre conditions (1994), University of Wales: University of Wales Aberystwyth, Preprint · Zbl 0876.34033
[6] Pearson, J. M., Hilbert’s sixteenth problem: An approach using Computer Algebra, (Ph.D. Dissertation (1992), University of Wales: University of Wales Aberystwyth)
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