Broer, Henk W.; Gaiko, Valery A. Global qualitative analysis of a quartic ecological model. (English) Zbl 1364.34070 Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 2, 628-634 (2010). Summary: We complete the global qualitative analysis of a quartic ecological model. In particular, studying global bifurcations of singular points and limit cycles, we prove that the corresponding dynamical system has at most two limit cycles. Cited in 5 Documents MSC: 34C60 Qualitative investigation and simulation of ordinary differential equation models 34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations 34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations 34C23 Bifurcation theory for ordinary differential equations 92D40 Ecology Keywords:quartic ecological model; field rotation parameter; bifurcation; singular point; limit cycle; separatrix cycle; Wintner-Perko termination principle PDFBibTeX XMLCite \textit{H. W. Broer} and \textit{V. A. Gaiko}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 2, 628--634 (2010; Zbl 1364.34070) Full Text: DOI arXiv Link References: [1] Bazykin, A. D., Nonlinear Dynamics of Interacting Populations (1998), World Scientific: World Scientific Singapore · Zbl 0605.92015 [2] Broer, H. W.; Naudot, V.; Roussarie, R.; Saleh, K., Bifurcations of a predator-prey model with non-monotonic response function, C. R. Acad. Sci. Paris Ser. I, 341, 601-604 (2005) · Zbl 1096.34532 [3] Broer, H. W.; Naudot, V.; Roussarie, R.; Saleh, K., A predator-prey model with non-monotonic response function, Regul. Chaotic. Dyn., 11, 155-165 (2006) · Zbl 1164.37318 [4] Broer, H. W.; Naudot, V.; Roussarie, R.; Saleh, K., Dynamics of a predator-prey model with non-monotonic response function, Discrete Contin. Dyn. Syst. Ser. A, 18, 221-251 (2007) · Zbl 1129.92061 [5] Broer, H. W.; Naudot, V.; Roussarie, R.; Saleh, K.; Wagener, F. O.O., Organizing centers in the semi-global analysis of dynamical systems, Int. J. Appl. Math. Stat., 12, 7-36 (2007) · Zbl 1211.37020 [6] Holling, C. S., Some characteristics of simple types of predation and parasitm, Can. Entomolog., 91, 385-398 (1959) [7] Kuznetsov, Yu. A., Elements of Applied Bifurcations Theory (2004), Springer: Springer New York [8] Lamontagne, Y.; Coutu, C.; Rousseau, C., Bifurcation analysis of a predator-prey system with generalized Holling type III functional response, J. Dyn. Differential Equations, 20, 535-571 (2008) · Zbl 1160.34047 [9] Li, Y.; Xiao, D., Bifurcations of a predator-prey system of Holling and Leslie types, Chaos Solitons Fractals, 34, 606-620 (2007) · Zbl 1156.34029 [10] Zhu, H.; Campbell, S. A.; Wolkowicz, G. S.K., Bifurcation analysis of a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 63, 636-682 (2002) · Zbl 1036.34049 [11] N.N. Bautin, E.A. Leontovich, Methods and ways of the qualitative analysis of dynamical systems in a plane, Nauka, Moscow, 1990 (in Russian); N.N. Bautin, E.A. Leontovich, Methods and ways of the qualitative analysis of dynamical systems in a plane, Nauka, Moscow, 1990 (in Russian) · Zbl 0785.34004 [12] Perko, L., Differential Equations and Dynamical Systems (2002), Springer: Springer New York [13] Gaiko, V. A., Global Bifurcation Theory and Hilbert’s Sixteenth Problem (2003), Kluwer: Kluwer Boston · Zbl 1156.34316 [14] Gaiko, V. A., Limit cycles of quadratic systems, Nonlinear Anal., 69, 2150-2157 (2008) · Zbl 1171.34312 [15] Gaiko, V. A., Limit cycles of Liénard-type dynamical systems, Cubo, 10, 115-132 (2008) · Zbl 1163.34336 [16] Gaiko, V. A.; van Horssen, W. T., Global bifurcations of limit and separatrix cycles in a generalized Liénard system, Nonlinear Anal., 59, 189-198 (2004) · Zbl 1082.34035 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.