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Extinction in nonautonomous competitive Lotka-Volterra systems. (English) Zbl 0866.34029
Summary: It is well known that for the two species autonomous competitive Lotka-Volterra model with no fixed point in the open positive quadrant, one of the species is driven to extinction, whilst the other population stabilizes at its own carrying capacity. In this paper we prove a generalization of this result to nonautonomous systems of arbitrary finite dimension. That is, for the \(n\) species nonautonomous competitive Lotka-Volterra model, we exhibit simple algebraic criteria on the parameters which guaratee that all but one of the species is driven to extinction. The restriction of the system to the remaining axis is a nonautonomous logistic equation, which has a unique solution \(u(t)\) that is strictly positive and bounded for all time, see B. D. Coleman [Math. Biosci. 45, 159-173 (1979; Zbl 0425.92013)] and S. Ahmad [Proc. Am. Math. Soc. 117, 199-204 (1993; Zbl 0848.34033)]. We prove in addition that all solutions of the \(n\)-dimensional system with strictly positive initial conditions are asymptotic to \(u(t)\).

MSC:
34C11 Growth and boundedness of solutions to ordinary differential equations
92D25 Population dynamics (general)
34A26 Geometric methods in ordinary differential equations
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