Age-dependent Population Growth

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Abstract
This paper considers a non-linear deterministic model, in which the death rate rises as the population grows. The model is in the form of a semi-linear first-order partial differential equation, with respect to time and age. It is an age-dependent version of a simple equation (the logistic equation) often used to describe populations whose growth is controlled by limited resources. The problem reduces to the solution of a non-linear integral equation; it has a constant solution, which is proved to be globally asymptotically stable. This implies that there are no steady oscillations, and that in the long run the population size and age-structure become fixed, independent of the initial conditions. Further details of the solution are discussed, including numerical results.

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