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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Extinction in nonautonomous competitive Lotka-Volterra systems


Authors: Francisco Montes de Oca and Mary Lou Zeeman
Journal: Proc. Amer. Math. Soc. 124 (1996), 3677-3687
MSC (1991): Primary 34C35, 92D25; Secondary 34A26.
MathSciNet review: 1327029
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Abstract: 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 stabilises at its own carrying capacity. In this paper we prove a generalisation 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 guarantee 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 Coleman (Math. Biosci. 45 (1979), 159-173) and Ahmad (Proc. Amer. Math. Soc. 117 (1993), 199-205). We prove in addition that all solutions of the $n$-dimensional system with strictly positive initial conditions are asymptotic to $u(t)$.


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Additional Information

Francisco Montes de Oca
Affiliation: Universidad Centroccidental, Lisandro Alvarado, Barquisimeto, Venezuela

Mary Lou Zeeman
Affiliation: Division of Mathematics and Statistics, University of Texas at San Antonio, San Antonio, Texas 78249-0664
Email: zeeman@ringer.cs.utsa.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-96-03355-2
PII: S 0002-9939(96)03355-2
Keywords: Lotka-Volterra, nonautonomous, Liapunov, competition, extinction.
Received by editor(s): March 21, 1995
Additional Notes: The first author was supported in part by the Division of Mathematics and Statistics at the University of Texas at San Antonio.
The second author was supported in part by the Office of Research Development at the University of Texas at San Antonio.
Communicated by: Linda Keen
Article copyright: © Copyright 1996 American Mathematical Society