Stochastic properties of systems controlled by autocatalytic reactions II

TL;DR

This paper investigates the stochastic behavior of autocatalytic reaction systems, revealing significant deviations from classical kinetic equations and implications for theories of natural selection.

Contribution

It derives a forward Kolmogorov equation for autocatalytic particles and shows how stochastic effects alter traditional kinetic predictions and bifurcation behavior.

Findings

Stochastic model differs from classical kinetic rate equations.

Mass action law is violated in the stochastic regime.

Bifurcation points disappear in the stochastic analysis.

Abstract

We analyzed the stochastic behavior of systems controlled by autocatalytic reaction A+X -> X+X, X+X -> A+X, X -> B provided that the distribution of reacting particles in the system volume is uniform, i.e. the point model of reaction kinetics introduced in arXiv:cond-mat/0404402 can be applied. Assuming the number of substrate particles A to be kept constant by a suitable reservoir, we derived the forward Kolmogorov equation for the probability of finding n=0,1,... autocatalytic particles X in the system at a given time moment. We have shown that the stochastic model results in an equation for the mean value of autocatalytic particles X which differs strongly from the kinetic rate equation. It has been found that not only the law of the mass action is violated but also the bifurcation point is disappeared in the well-known diagram of X particle- vs. A particle-concentration. Therefore, speculations about the role of autocatalytic reactions in processes of the "natural selection" can be hardly supported.