Age-dependent models of population growth

P. H. Leslie (1945, Biometrika 33, 183-212) (and others) introduced vector-matrix growth difference equations to model populations in which birth and death rates are age-dependent. We develop differential versions of these equations in both the unrestricted growth and logistic cases. We find that the vector logistic equations (difference and differential) are explicitly solvable in terms of solutions of the unrestricted equations, even when vital rates vary with time. These explicit solution formulas make it easy to determine the behavior of solutions as time goes on.