One-Dimensional Birth-Death Process and Delbr\"{u}ck-Gillespie Theory of Mesoscopic Nonlinear Chemical Reactions

As a mathematical theory for the stochasstic, nonlinear dynamics of individuals within a population, Delbr"{u}ck-Gillespie process (DGP) $n(t)\in {\mathbb{Z}}^N$, 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 $T^{\ast }\sim e^{\alpha V}$ $(\alpha >0)$: a relatively fast, intra-basin diffusion for $t\ll T^{\ast }$ and a much slower inter-basin Markov jump process for $t\gg T^*$. In the present paper for one-dimensional systems, we study both stationary behavior ($t=\mathrm{\infty }$) in terms of invariant distribution $p_n^{ss}(V)$, and finite time dynamics in terms of the mean first passsage time (MFPT) $T_{n_1\to n_2}(V)$. We obtain an asymptotic expression of MFPT in terms of the “stochastic potential” $\Phi(x,V)=-(1/V)\ln p^{ss}_{xV}(V)$. We show in general no continuous diffusion process can provide asymptotically accurate representations for both the MFPT and the $p_n^{ss}(V)$ for a DGP. When $n_1$ and $n_2$ belong to two different basins of attraction, the MFPT yields the $T^{\ast }(V)$ in terms of $\Phi(x,V)\approx \phi_0(x)+(1/V)\phi_1(x)$. For systems with a saddle-node bifurcation and catastrophe, discontinuous “phase transition” emerges, which can be characterized by $\mathrm{\Phi }(x,V)$ in the limit of $V\to \mathrm{\infty }$. 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.