# zbMATH — the first resource for mathematics

Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems. (English) Zbl 0797.92025
The Lotka-Volterra system $dx_ i/dt=x_ i\cdot(b_ i-\sum^ n_{j=1}a_{ij}x_ j);\;a_{ij},b_ i>0;\quad i,j=1,2,3,\tag{1}$ is considered in the closed positive cone $$x_ i\geq 0$$. System (1) models three mutually competing species, each of which, in isolation, would exhibit logistic growth. A geometric analysis of nullclines (the lines on which $$dx_ i/dt=0)$$, Gersgorin’s theorem from matrix analysis, a theorem of Hirsch, and Hopf bifurcation from the theory of dynamical systems are used.
Algebraic inequalities on the parameters for 33 stable equivalence classes are listed. It is shown, that in 25 of these classes there are no periodic orbits and only fixed points (in case of attracting fixed point that means an eventually stable coexistence of all three species). In the other 8 classes conditions for attracting periodic orbits are given (that means an eventual coexistence of oscillatory nature).

##### MSC:
 92D25 Population dynamics (general) 34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations 37N99 Applications of dynamical systems 34C25 Periodic solutions to ordinary differential equations 34C23 Bifurcation theory for ordinary differential equations
Full Text:
##### References:
 [1] DOI: 10.1007/BF01832841 · Zbl 0489.92017 [2] DOI: 10.1016/0024-3795(77)90056-8 · Zbl 0379.15004 [3] DOI: 10.1007/BF00275158 · Zbl 0503.92021 [4] DOI: 10.1137/0136039 · Zbl 0412.92015 [5] DOI: 10.1016/0040-5809(78)90022-9 · Zbl 0387.92011 [6] DOI: 10.1016/0024-3795(78)90021-6 · Zbl 0376.15007 [7] Freedman H., Deterministic Mathematical Models in Population Ecology (1980) · Zbl 0448.92023 [8] DOI: 10.1016/0025-5564(85)90078-1 · Zbl 0584.92018 [9] DOI: 10.1086/282973 [10] DOI: 10.1007/BF00275063 · Zbl 0362.92013 [11] DOI: 10.1086/283144 [12] DOI: 10.1007/978-1-4757-3849-0 [13] Hassard B. D., Theory and Applications of the Hopf Bifurcation (1981) · Zbl 0474.34002 [14] DOI: 10.1137/0513013 · Zbl 0494.34017 [15] DOI: 10.1137/0516030 · Zbl 0658.34023 [16] Hirsch M. W., Journal fur die reine und angewandte Mathematik 383 pp 1– (1988) [17] DOI: 10.1088/0951-7715/1/1/003 · Zbl 0658.34024 [18] DOI: 10.1016/0022-0396(89)90097-1 · Zbl 0712.34045 [19] DOI: 10.1137/0521067 · Zbl 0734.34042 [20] Hirsch M. W., Ergodic Theory and Dynamical Systems 11 pp 443– (1990) [21] Hirsch M. W., Differential Equations, Dynamical Systems, and Linear Algebra (1974) · Zbl 0309.34001 [22] DOI: 10.1016/0362-546X(81)90059-6 · Zbl 0477.92011 [23] Hofbauer J., The Theory of Evolution and Dynamical Systems (1988) · Zbl 0678.92010 [24] DOI: 10.1017/CBO9780511810817 · Zbl 0576.15001 [25] Irwin M. C., Smooth Dynamical Systems (1980) · Zbl 0465.58001 [26] DOI: 10.1007/BF02547774 · Zbl 0004.06104 [27] Lotka A. J., Elements of Physical Biology (1924) · JFM 51.0416.06 [28] Lotka A. J., Elements of Mathematical Biology (1956) · Zbl 0074.14404 [29] DOI: 10.1016/0040-5809(70)90039-0 [30] May R. M., Stability and Complexity in Model Ecosystems (1975) [31] DOI: 10.1137/0129022 · Zbl 0314.92008 [32] DOI: 10.1137/0145032 · Zbl 0577.92020 [33] DOI: 10.1007/BF02476696 [34] DOI: 10.1137/0137004 · Zbl 0418.92016 [35] DOI: 10.2307/1935355 [36] DOI: 10.1137/1.9781611970258 [37] DOI: 10.1007/BFb0087009
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.