The logistic population model with slowly varying carrying capacity

Many single-species differential equation population models feature a carrying capacity---the limiting population supportable by the environment. For constant carrying capacities an exact solution may often be found, representing the evolving population in time. However, for time varying carrying capacities, exact solution is rarely possible, and numerical techniques must be used. We demonstrate that when the carrying capacity varies slowly with time, a multiple time scale analysis leads to approximate closed form solutions that, apart from being explicit, are comparable to numerically generated ones and which are valid for a range of parameter values.