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

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As a mathematical theory for the stochasstic, nonlinear dynamics of individuals within a population, Delbrück-Gillespie process (DGP) n(t)ⁿ, 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 ordinary differential equation 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 underestood 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 the present paper for one-dimensional systems, we study both stationary behavior (t=∞) in terms of invariant distribution pₙss(V), and finite time dynamics in terms of the mean first passsage time (MFPT) Tₙ₁→ ₙ₂(V). We obtain an asymptotic expression of MFPT in terms of the "stochastic potential" Φ(x,V)=-(1/V) pssxV(V). We show in general no continuous diffusion process can provide asymptotically accurate representations for both the MFPT and the pₙss(V) 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 a saddle-node bifurcation and catastrophe, discontinuous "phase transition" emerges, which can be characterized by Φ(x,V) in the limit of V→∞. In terms of time scale 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.