Generalized stable population theory

In generalizing stable population theory we give sufficient, then necessary conditions under which a population subject to time dependent vital rates reaches an asymptotic stable exponential equilibrium (as if mortality and fertility were constant). If chi 0 (t) is the positive solution of the characteristic equation associated with the linear birth process at time t, then rapid convergence of chi 0 (t) to chi 0 and convergence of mortality rates produce a stable exponential equilibrium with asymptotic growth rate chi 0-1. Convergence of chi 0 (t) to chi 0 and convergence of mortality rates are necessary. Therefore the two sets of conditions are very close. Various implications of these results are discussed and a conjecture is made in the continuous case.