Extinction Times in Autocatalytic Systems

Abstract

Systems evolving under the influence of autocatalytic processes have been the subject of much study due to their appearance in a wide variety of contexts. Well known examples range from the simplest autocatalytic chemical reactions, biochemical models of ribozyme or prion replication, right up to the duplication of entire organisms in models of population dynamics. While the deterministic approach frequently taken to model such systems is adequate in the limit of large reactant numbers, the intrinsic fluctuations exert an important influence on the dynamics when the supply of reactants is limited. In particular, when combined with spontaneous degradation of the autocatalytic reactant population, such fluctuations can lead to the extinction of that population. In this paper, we study reversible autocatalytic processes of the form X + Y ⇌ 2X in the limit that there exists a surplus of Y and in the presence of a spontaneous degradation process X → Z. Through the use of the Poisson representation, we identify an exact analytical expression for the mean extinction time of the X population. We show that the exact result can be neatly approximated by an Arrhenius-like expression involving an effective activation energy separating a quasi-stationary state from the extinct state.