An Equivalence Theorem concerning Population Growth in a Variable Environment

We give conditions under which two solutionsx andy of the Kolmogorov equation ẋ = 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.