J M Cushing | PubMed Health

PubMed1977

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.

Models, Theoretical↗

PubMed1976

Predator prey interactions with time delays.

A general (Volterra-Lotka type) integrodifferential system which describes a predator-prey interaction subject to delay effects is considered. A rather complete picture is drawn of certain qualitative aspects of the solutions as they are functions of the parameters in the system. Namely, it is argued that such systems have, roughly speaking, the following features. If the carrying capacity of the prey is smaller than a critical value then the predator goes extinct while the prey tends to this carrying capacity; and if the carrying capacity is greater than, but close to this critical value then there is a (globally) asymptotically stable positive equilibrium. However, unlike the classical, non-delay Volterra-Lotka model, if the carrying capacity of the prey is too large then this equilibrium becomes unstable. In this event there are critical values of the birth and death rates of the prey and predator respectively (which hitherto have been fixed) at which "stable" periodic solutions bifurcate from the equilibrium and hence at which the system is stabilized. These features are illustrated by means of a numerically solved example.

Animals↗