Multi-scaling analysis of a logistic model with slowly varying coefficients

All single-species differential-equation population models incorporate parameters which define the model - for example, the rate constant  , rr, and carrying capacity  , KK, for the Logistic model. For constant parameter values, an exact solution may be found, giving the population as a function of time. However, for arbitrary time-varying parameters, exact solutions are rarely possible, and numerical solution techniques must be employed. In this work, we demonstrate that for a Logistic model in which the rate constant and carrying capacity both vary slowly with time, an analysis with multiple time scales leads to approximate closed form solutions that are explicit, are valid for a range of parameter values and compare favourably with numerically generated ones.