Slow variation in the Gompertz model

Abstract

In many differential, single species, population models, the intrinsic model parameters are assumed to be given constants. Typical examples are the ``carrying capacity" and ``rate constant" arising in the logistic model. Such constant parameters usually allow exact solution of the model equations, to completely describe the evolving population. However, this assumption of constancy is not realistic since such parameters may exhibit a wide range of variation with time. This investigation considers a particularly useful population model---the Gompertz model---in the case where the model parameters vary slowly with time. Multi-timing methods construct an approximate expression for the population that has the advantages of being both explicit and giving results comparable to those obtained from numerical calculations.