Lloyd, N. G.; Pearson, J. M.; Sáez, E.; Szántó, I. Limit cycles of a cubic Kolmogorov system. (English) Zbl 0858.34023 Appl. Math. Lett. 9, No. 1, 15-18 (1996). 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. Reviewer: Chen Lan Sun (Beijing) Cited in 27 Documents 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 Keywords:limit cycles; cubic Kolmogorov systems; computer algebra procedure FINDETA; focal values; fine foci Citations:Zbl 0702.68072 PDF BibTeX XML Cite \textit{N. G. Lloyd} et al., Appl. Math. Lett. 9, No. 1, 15--18 (1996; Zbl 0858.34023) Full Text: DOI OpenURL 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) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.