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Limit cycles for the competitive three dimensional Lotka-Volterra system. (English) Zbl 0960.34022
Here, the following competitive three-dimensional Lotka-Volterra system is investigated \[ \dot x= X(b- Ax), \] where \(x= \text{col}(x_1, x_2, x_3)\) is a three-dimensional state vector \(X= \text{diag}(x_1, x_2, x_3)\) is a \(3\times 3\) diagonal matrix, \(b= \text{col}(b_1, b_2, b_3)\) is a positive real vector and \(A= (a_{ij})_{3\times 3}\) is a positive matrix. First, it is proved that the number of limit cycles of the system in \(\mathbb{R}^3_+\) is finite if the system has not any heteroclinic polycycles in \(\mathbb{R}^3_+\). Second, a particular 3-dimensional competitive Lotka-Volterra system with two small parameters is discussed. It is proved that there exists one parameter range in which the system is persistent and has at least two limit cycles, and there exists other parameter ranges in which the system is not persistent and has at least one limit cycle. Hence, some open questions are answered partly in this paper.

MSC:
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
92D25 Population dynamics (general)
34C60 Qualitative investigation and simulation of ordinary differential equation models
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[1] Andronov, A.A.; Leontovich, E.A.; Gordon, I.I.; Maier, A.G., Theory of bifurcations of dynamical systems on a plane, (1973), Wiley New York · Zbl 0282.34022
[2] Arneodo, A.; Coullet, P.; Tresser, C., Occurrence of strange attractors in three dimensional Volterra equations, Phys. lett. A, 79, 423-439, (1980)
[3] Aulbach, B., A classical approach to the analyticity problem of center manifolds, J. appl. math. phys., 36, 1-23, (1985) · Zbl 0564.34046
[4] Chow, Shui-Nee; Hale, J.K., Methods of bifurcation theory, (1982), Springer-Verlag New York/Berlin · Zbl 0487.47039
[5] Grauert, H.; Fritzsche, K., Several complex variables, (1976), Springer-Verlag New York/Berlin · Zbl 0381.32001
[6] Griffith, P., Algebraic curves, (1985), Peking Univ. Press
[7] Hofbauer, J.; Sigmund, K., The theory of evolution and dynamical systems, (1988), Cambridge Univ. Press Cambridge
[8] Hofbauer, J.; So, J.W.-H., Multiple limit cycles for three dimensional lotka – volterra equations, Appl. math. lett., 7, 65-70, (1994) · Zbl 0816.34021
[9] Hofbauer, J., On the occurrence of limit cycles in the lotka – volterra equation, Nonlinear anal., 5, 1003-1007, (1981) · Zbl 0477.92011
[10] Gardini, L.; Lupini, R.; Messia, M.G., Hopf bifurcation and transition to chaos in lotka – volterra equation, J. math. biol., 27, 259-272, (1989) · Zbl 0715.92020
[11] Hirsch, M.W., Systems of differential equations which are competitive or cooperative, III. competing species, Nonlinearity, 1, 51-71, (1988) · Zbl 0658.34024
[12] Li, B.; Xiao, D.; Zhang, Z., Bifurcations of a class of nongeneric quadratic Hamiltonian systems under quadratic perturbations, Lecture notes in pure and applied mathematics, (1996), Dekker New York, p. 149-162 · Zbl 0855.34048
[13] Zeeman, M.L., Hopf bifurcations in competitive three dimensional lotka – volterra systems, Dynam. stability systems, 8, 189-216, (1993) · Zbl 0797.92025
[14] Zeeman, E.C., Population dynamics from game theory, Global theory of dynamical systems, Proc. conf. northwestern univ., lecture notes in mathematics, 819, (1980), Springer-Verlag New York/Berlin, p. 472-497 · Zbl 0448.92015
[15] Zhang, Z.; Li, C.; Zheng, Z.; Li, W., Elements of bifurcation theorey of vector fields, (1997), Higher Education Press Beijing
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