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.

DOI:
https://doi.org/10.1090/S0002-9939-96-03355-2

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 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 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 -dimensional system with strictly positive initial conditions are asymptotic to .

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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:
https://doi.org/10.1090/S0002-9939-96-03355-2

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