International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 43, Pages 2747-2758
doi: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 Ray Redheffer and Richard R. Vance. 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.