Stochastic models and their deterministic analogues

Phil Pollett

Abstract: The predominant method in applied mathematical modelling is to suppose the system in question is governed by a set of differential equations. This approach is often justified by the "Law of Mass Action", that rapid microscopic changes result in little fluctuation in the overall (macroscopic) behaviour. In cases where random effects outweigh mean effects, this variation is at worst ignored or more often, these days, accounted for by adding noise to an existing deterministic model. I will indicate some of the limitations of this approach. I will then describe a class of stochastic models (density-dependent Markov chains) for which there are identifiable deterministic analogues. I will delimit conditions under which a deterministic approximation is justified and then identify an approximating diffusion process that can be used to model fluctuations about the deterministic mean path.

Acknowledgement: This work is supported by the ARC Centre of Excellence for Mathematics and Statistics of Complex Systems

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Last modified: 19th January 2006