Exact solution of a non-autonomous logistic population model

Growth models such as the logistic equation are widely studied and applied in population and ecological modelling. The carrying capacity in the logistic equation is usually regarded as a constant which is not often realistic. Functional forms of the carrying capacities are used to describe changes in the environment. The purpose of this study is to derive an exact solution of the non-autonomous logistic equation with a saturating carrying capacity. The solution is found via a power series resulting from a straightforward algebraic method. For practical applications the power series may be truncated, a simple criterion is established that leads to a good approximate solution. The approximate solution is in good agreement with the numerical simulations, even though only a small number of terms are used.