Saturation effects in population dynamics: Use branching processes or dynamical systems?

Abstract

This chapter deals with the behavior of a branching population undergoing saturation effects when it becomes too large. We study in particular the limits of the prediction given in the setting of the deterministic dynamical system related to the stochastic branching process modeling the evolution of the population. We also generalize the usual Markovian branching processes of order one to size-dependent branching processes that may have a longer memory and give conditions leading to an almost sure extinction of the process while the dynamical system is persistent. The notion of reproductive rate is explained and generalized. We give some examples, in particular the amplification process in the polymerase chain reaction (PCR).