Equilibrium Equations for Human Populations with Immigration

The objective of this article is to create a framework to study asymptotic equilibria in human populations with a special focus on immigration. We present a new model, based on Resource Dependent Branching Processes, which is now broad enough to cope with the goal of finding equilibrium criteria under reasonable hypotheses. Our equations are expressed in terms of natality rates, mean productivity and mean consumption of the home-population and the immigrant population as well as policies of the Society to distribute resources among individuals. We also study the impact of integration of one sub-population into the other one, and in a third model, the additional influence of a continuous stream of new immigrants. Proofs of the results are based on classical limit theorems, on Borel-Cantelli type arguments, on the Theorem of envelopment of Bruss and Duerinckx (2015), on a maximum inequality of Bruss and Robertson (1991) and, in particular, on an extension of J.M. Steele (2016) of the latter. Conditions for the existence of an equilibrium often prove to be severe, and sometimes surprisingly sensitive. This underlines how demanding the real world of immigration can be for politicians trying to make sound decisions. Our main objective is to provide help through insights from an adequate theory. Another objective of the present study is to learn which of the possible control measures are best for combining feasibility and efficiency to reach an equilibrium, and to recognise the corresponding steps one has to take towards controls. We also make preliminary suggestions to envisage ways to optimal control. As far as the author is aware, all results are new.