Time delays in single species growth models

A general model is considered for the growth of a single species population which describes the per unit growth rate as a general functional of past population sizes. Solutions near equilibrium are studied as functions of ε = 1/b, the reciprocal of the inherent per unit growth rate b of the population in the absense of any density constraints. Roughly speaking, it is shown that for large ε the equilibrium is asymptotically stable and that for ε small the solutions show divergent oscillations around the equilibrium. In the latter case a first order approximation is obtained by means of singular perturbation methods. The results are illustrated by means of a numerically integrated delay-logistic model.