Numerical Aspects of Computing Possible Equilibria for Resource Dependent Branching Processes with Immigration

This article studies the stability of solutions of equilibrium equations arising in so-called resource dependent branching processes. We argue that these new models, building on the model already presented by Bruss (1984 a), refined and elaborated in Bruss and Duerinckx (2015) and now extended to allow immigration, are suitable to cope with specific properties of human populations. Our main interest is here to understand under which conditions immigration may lead to an equilibrium. At the same time, we would like to advertize resource dependent branching processes as possibly the best models to study such questions. The equilibrium equations for the new models we obtain are clear and informative for several important stability questions. The goal of the study of the specific examples we provide is to see where the impact of immigration is most visible, and in how far increased efforts of integration can cope with dangers of instability. Moreover we discuss the advantages and a weaker point of our model, and also include a brief look at continuous-state, continuous-time branching processes as an alternative.