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International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 43, Pages 2747-2758
http://dx.doi.org/10.1155/S0161171203209133

An equivalence theorem concerning population growth in a variable environment

1Department of Mathematics, University of California, Los Angeles 90095-1555, CA, USA
2Department of Organismic Biology, Ecology and Evolution, University of California, Los Angeles 90095-1606, CA, USA

Received 16 September 2002

Copyright © 2003 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We give conditions under which two solutions x and y of the Kolmogorov equation x˙=xf(t,x) satisfy limy(t)/x(t)=1 as t. This conclusion is important for two reasons: it shows that the long-time behavior of the population is independent of the initial condition and it applies to ecological systems in which the coefficients are time dependent. Our first application is to an equation of Weissing and Huisman for growth and competition in a light gradient. Our second application is to a nonautonomous generalization of the Turner-Bradley-Kirk-Pruitt equation, which even before generalization, includes several problems of ecological interest.