Population dynamics based on the McKendrick-von Foerster model.

Abstract

The current state of information concerning the classical model of deterministic, age-dependent population dynamics - the McKendrick von Foerster equation - is overviewed. This model and the related Renewal equation are derived and the parameters involved in both are elaborated upon. Fundamental theorems concerning existence, uniqueness and boundedness of solutions are outlined. A necessary and sufficient condition concerning the stability of equilibrium age-distributions is rederived along different lines. Attention is then given to generalizations of the McKendrick-von Foerster model that have arisen from the inclusion of density- dependence into the parameters of the system; the inclusion of harvesting terms; and the extension of the model to describe the dynamics of a two-sex population. A technique which reduces the model, under certain conditions on the mortality and fertility functions, to a system of ordinary differential equations is discussed and applied to specific biochemical population models. Emphasis here is on the possible existence of stable limit cycles.The Kolmogorov system of ordinary differential equations and its use in describing the dynamics of predator-prey systems is examined. The Kolmogorov theorem is applied as a simple alternative to a lengthy algorithm for determining whether limit cycles are stable. Age-dependence is incorporated into this system by means of a McKendrick - von Foerster equation and the effects on the system of different patterns of age-selective predation are demonstrated. Finally, brief mention is made of recent work concerning the use of the McKendrick - von Foerster equation to describe the dynamics of both predator and prey. A synthesis of the theory and results of a large number of papers is sought and areas valuable to further research are pointed out.