Existence, uniqueness and asymptotic behaviour for a multi-stage evolution problem of an age-structured population

The theory of semigroups of bounded linear operators in L(1)-like Banach spaces is employed to show the existence and uniqueness of a triple of non-negative functions representing the solution of a model of evolution of a population distributed in three stages of individuals, each of them being dependent on his own age and one stage also on mother age. The differential equations involved are linear and are coupled through linear boundary conditions. The detailed study of the spectrum of the linear, closed operator related to the system allows to obtain estimates for the asymptotic time behaviour of the solution; these results may be considered as a generalization of the Sharpe-Lotka theorem. Finally, the analytical structure of the solution is given.