General population theory in the age-time continuum

Abstract

A linear first-order partial differential equation is derived for the age density k(x,t) of a biological population. It contains factors which are time-dependent functions defining attrition and immigration. Emigration may be included in the attrition function. The general solution k(x,t) of the differential equation contains an arbitrary function that is identified as the birth rate B(t) of the community. A Volterra integral equation for B(t) arises if the total population N of the community is planned to be a prescribed function of time. Also, projection of the population on the basis of a given net maternity function and a given immigration function requires the solution B(t) of a linear integral equation. This equation can be transformed to a Volterra type, but another form in which the limits of integration are constant bounds of the fertility period has some advantages. Some properties of solutions of the integral equation are discussed, and effects of stepwise discontinuities in the kernel are considered.