Non-autonomous logistic equations as models of populations in a deteriorating environment

Abstract

Abstract The non-autonomous logistic equation d x(t) d t = r(t)x(t)[1 − x(t) K(t) ] is studied under conditions that include an environment which is completely deteriorating. In this setting, when the population's growth rate, r, is large on the average, solutions track the environment with a consequent extinction of the population. However, when both r and rK−1 are small in the sense that they are in L1[0,∞) then an asymptotic equivalence, where all solutions tend to positive limits as t approaches infinity, results and the population is persistent, independent of initial density. The asymptotic equivalence produces an unreasonable overshoot of carrying capacity which leads to concern about employing the logistic equation in the above form as a population model when growth rates are close to zero. A re-interpretation of the parameters of the logistic equation leads to the alternative logistic formulation d x(t) d t = x(t)[r(t) − c B(t) x(t)], (c > 0) . A biological interpretation of the parameters is presented and this equation is compared with the classical logistic model in the case where the parameters are constant. If the alternative logistic model is applied in a situation with time-varying parameters, then a deteriorating environment always leads to extinction of the population regardless of the behavior of r. Similarly, a growth rate which is small on the average results in extinction regardless of the behavior of B. Furthermore, r and B have limiting values as t approaches infinity then so does x and the terminal value of x is equal to the terminal value of the carrying capacity of the population. In general, the alternative formulation seems to be the more reasonable model in situations where perturbations lead to severe decreases in environmental quality and growth rates.