Abstract

PIP: This paper investigated some deterministic models of multidimensional population growth using the Markovian assumption. These models are formulated as initial and boundary problems of simultaneous Von Foerster equations. In particular, if a boundary condition is given by a functional relation between boundary value and age-density functions, the model can be formulated as an initial value problem. The multiregional population model which was first proposed by Herve Le Bras and Andrei Rogers is an important example of models which can be constructed as initial value problems. The authors in this case expand the model into a case where the multiregional net maternity function depends on time. And they manage to prove the existence theorem of the continuous solution in this more general case. Then they show sufficient conditions which make the balanced growth solution possible for the case that the multiregional net maternity function is time-independent. The balanced growth solution (or balanced-growth population) is an exponentially growing population with a time-invariant age-by-region distribution. However, it is remarkable that a balanced-growth population can be defined regardless of whether its time-invariant distribution is stable or not. If a balanced-growth population has a stable time-invariant distribution, it follows that this population process has what we call strong ergodicity. In such a case, the concept of balanced-growth population coincides with the classical concept of stable population. Therefore, the strong ergodic property of population growth is grasped as the stability of the time-invariant distribution under the balanced growth. (author's modified)