Distinguishing between discreteness effects in stochastic reaction processes

The effect of discreteness on stochastic dynamics of chemically reacting systems is studied analytically. We apply the scheme bridging the chemical master equation and the chemical Fokker-Planck equation by a parameter representing the degree of discreteness previously proposed by the author for two concrete systems. One is an autocatalytic reaction system, and the other is a branching-annihilation reaction system. It is revealed that the change in characteristic time scales when discreteness is decreased is yielded between the two systems for different reasons. In the former system, it originates from the boundaries where one of the chemical species is zero, whereas in the latter system, it is due to modification of the most probable extinction path caused by discreteness loss.

Figure 1

Stationary variance in $X$ in the autocatalytic reaction system with $N=10$ and $s=1$ for different values of $\epsilon$ and the CFPE. Marks represent numerically calculated stationary variance from the stationary distribution ($\epsilon =1$: square; $\epsilon =0.1$: circle; $\epsilon =0.01$: upper triangle; $\epsilon =0.001$: lower triangle). The solid lines are Eq. (17). The dashed line is Eq. (18).

Figure 2

Correlation time of the autocatalytic reaction system with $N=10$ and $s=1$ for different values of $\epsilon$ and the CFPE. Marks represent values calculated from the numerical derivative of the numerically obtained maximum eigenvalue $\varphi \left(r\right)$ ($\epsilon =1$: square; $\epsilon =0.1$: circle; $\epsilon =0.01$: upper triangle; $\epsilon =0.001$: lower triangle). Solid lines are the combination of Eqs. (17) and (26). The dashed line is the combination of Eqs. (18) and (29).