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GLOBAL BIFURCATION ANALYSIS OF A GENERALIZED LOTKA­VOLTERRA SYSTEM Gaiko V.A. National Academy of Sciences of Belarus, United Institute of Informatics Problems, Belarus, 220040, Minsk, Leonid Beda Str. 6-4; E-mail: valery.gaiko@yahoo.com

We study a quartic dynamical system which models the dynamics of the populations of predators and their prey that use the group defense strategy in a given ecological, epidemiological or immunological system and which is a variation on the classical Lotka­Volterra system: x = x((1 - x)(x2 + x + 1) - y) P, (1) y = -y(( + µy)(x2 + x + 1) - x) Q , where 0, > 0, > 0, µ 0 and > -2 are parameters. Such a quartic dynamical model was studied earlier, for instance, in [1, 2]. However, its qualitative analysis was incomplete, since the global bifurcations of limit cycles could not be studied properly by means of the methods and techniques which were used earlier in the qualitative theory of dynamical systems. Together with (1), we also consider an auxiliary system x = P - Q , y = Q + P, (2)

applying to these systems new bifurcation methods and geometric approaches developed in [3] and completing the qualitative analysis of system (1). In particular, using (2), we prove the following theorem (see [4]). Theorem 1. System (1) has at most two limit cycles. Besides, we discuss how to use higher-dimensional Lotka­Volterra systems as biomedical or ecological models. References. H. Zhu, S.A. Campbell and G.S.K. Wolkowicz Bifurcation analysis of a predatorprey system with nonmonotonic functional response// SIAM J. Appl. Math. 63, 2002. Pp. 636-683. H.W. Broer, V. Naudot, R. Roussarie and K. Saleh Dynamics of a predator-prey model with non-monotonic response function// Discr. Contin. Dynam. Syst. Ser. A 18, 2007. Pp. 221-251. V.A. Gaiko Global Bifurcation Theory and Hilbert's Sixteenth Problem. -- Kluwer, 2003. 204 pages. H.W. Broer and V.A. Gaiko Global qualitative analysis of a quartic ecological model// Nonlinear Anal. 72, 2010. Pp. 628-634.

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