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 function of epsilon = 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 epsilon the equilibrium is asymptotically stable and that for epsilon 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.