One‐Dimensional Birth‐Death Process and Delbrück‐Gillespie Theory of Mesoscopic Nonlinear Chemical Reactions

Abstract

As a mathematical theory for the stochastic, nonlinear dynamics of individuals within a population, Delbrück‐Gillespie process (DGP) is a birth–death system with state‐dependent rates which contain the system size V as a natural parameter. For large V, it is intimately related to an autonomous, nonlinear ODE as well as a diffusion process. For nonlinear dynamical systems with multiple attractors, the quasi‐stationary and stationary behavior of such a birth–death process can be understood in terms of a separation of time scales by a T*∼eαV (α > 0): a relatively fast, intra‐basin diffusion for t≪T* and a much slower inter‐basin Markov jump process for t≫T*. In this paper for one‐dimensional systems, we study both stationary behavior (t=∞) in terms of invariant distribution , and finite time dynamics in terms of the mean first passsage time (MFPT) . We obtain an asymptotic expression of MFPT in terms of the “stochastic potential”. We show in general no continuous diffusion process can provide asymptotically accurate representations for both the MFPT and the for a DGP. When n1 and n2 belong to two different basins of attraction, the MFPT yields the T*(V) in terms of Φ (x, V) ≈φ0(x) + (1/V)φ1(x). For systems with saddle‐node bifurcations and catastrophe, discontinuous “phase transition” emerges, which can be characterized by Φ (x, V) in the limit of . In terms of timescale separation, the relation between deterministic local nonlinear bifurcations, and stochastic global phase transition is discussed. The one‐dimensional theory is a pedagogic first step toward a general theory of DGP.