Hebert, DJ_, Simulations of stochastic reaction-diffusion systems, Mathematics and Computers in Simulation 34 (1992) 411-432 .
Direct simulation of a discrete stochastic model of reaction-diffusion provides a means of studying the fluctuations which occur when populations are finite . This paper introduces the mathematical model along with the computational model which implements it, and shows the relationship to some standard random methods for PDE simulation . Two examples arc then given . First, the stochastic phase-field model is introduced . This attempt to study the influence of randomness in liquid-solid interface problems leads easily to meaningful extensions of the standard PDE model . Second, some experiments with a model threshold reaction problem are described . These illustrate many of the effects of the stochastic model which the PDE idealization does not address . In particular. varying the population scale in the model leads to a variety of new observations and conjectures .
1 . Introduction
The solutions of partial differential equations (PDEs) which describe fluid flow, chemical reaction-diffusion, and population interaction may be understood as approximations to -or expected values of -quantities determined by the chaotic or random motion of microscopic interacting particles . It may be argued that, as computers become more capable, we may substitute direct simulation of microscopic models for the numerical solution of partial differential equations . In certain biological and ecological models this is certainly feasible and fruitful, but in physical problems differences in scale may be enormous and direct simulation of molecular motion is wasteful if not impossible . In these cases meaningful simulation must be done at an intermediate level where laws of large numbers, central limit theorems, and self-similarity give a reflection of the molecular randomness in microscopic and macroscopic observable fluctuations .
In the derivation of partial differential equations from microscopic models the classical and modern mathematical results usually address limiting cases of infinite population with infinitesimal space and time scale, with few results concerning the effect of known population sizes and with no sharp estimates of PDE approximations .