Multiple limit cycles for three dimensional Lotka-Volterra equations. (English) Zbl 0816.34021
This paper considers the Lotka-Volterra equations which can be given in the form \(dx_ i/dt = - (1 + x_ i) (\sum^ n_{j=1} a_{ij} x_ j)\), \(i = 1, \dots, n\). It is well-known that the two dimensional Lotka- Volterra equations cannot have limit cycles. However, for equations of higher dimensions the dynamics are more complicated. In this paper, a three dimensional competitive Lotka-Volterra equation with two limit cycles is constructed. Also, given are arguments which lead the authors to believe that such systems cannot have more than two limit cycles.
Reviewer: R.Lee (Fredericton)
MSC:
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
References:
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