Bifurcation of Limit Cycles for 3D Lotka-Volterra Competitive Systems
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Bifurcation of Limit Cycles for 3D Lotka-Volterra Competitive Systems Qinlong Wang · Wentao Huang · Haotao Wu
Received: 18 May 2010 / Accepted: 13 March 2011 / Published online: 24 March 2011 © Springer Science+Business Media B.V. 2011
Abstract Bifurcation of limit cycles is discussed for three-dimensional Lotka-Volterra competitive systems. A recursion formula for computation of the singular point quantities is given for the corresponding Hopf bifurcation equation. Some new results are obtained for 6 classes 26–31 in Zeeman’s classification, especially, an example with four limit cycles in class 29 is given for the first time. The algorithm applied here is effective for solving the above general cyclicity. Keywords 3D Lotka-Volterra competitive system · Hopf bifurcation · Singular point quantities · Center manifold Mathematics Subject Classification (2000) 34C12 · 34C23 · 92D25
1 Introduction We consider the existence of limit cycles for the three-dimensional Lotka-Volterra system: 3 dxi = xi b i − aij xi , dt j =1
i = 1, 2, 3.
(LV)
Q. Wang () · H. Wu School of Information and Mathematics, Yangtze University, Jingzhou 434023, P.R. China e-mail: [email protected] Q. Wang Ecological Complexity and Modeling Laboratory, University of California, Riverside, CA 92521-0124, USA W. Huang School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin, Guangxi 541004, P.R. China
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Q. Wang et al.
At times, the problem is restricted to the assumption of competitiveness meaning that bi > 0, aij > 0 (i, j = 1, 2, 3), namely it is a 3D Lotka-Volterra competitive system which describes the competition between three species that share and compete for the same resources, habitat or territory (interference competition), and has been paid close attention of both biologists and mathematicians. As one knows, many and many valuable results have been obtained (see [1, 2]). Based on the remarkable result of Hirsch, Zeeman [3] defined a combinatorial equivalence relation on the set of all 3-dimensional LV competitive systems and identified 33 stable equivalence classes. Of these, only classes 26–31 may have limit cycles. Xiao and Li [4] have proved that the number of limit cycles of the 3-dimensional LV competitive system is finite if the system does not have a heteroclinic polycycle. However, the question of how many limit cycles can appear in Zeeman’s six classes 26–31 remains open. Especially, in [5] authors conjectured the number of limit cycles is at most two for the competitive equation (LV) and gave the first example in class 27 with a heteroclinic polycycle and two limit cycles surrounding the interior equilibrium. So far as we know, recently there are some good results as follow: two limit cycles in three classes 26, 28 and 29 in [6]; three limit cycles in three classes 30 and 31 in [7]; three limit cycles for class 29 in [8] and four limit cycles for class 27 in [1]. In this paper, we also consider Hofbauer and So’s problem for classes 26–31 in Zeeman’s classification. Usuall
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