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BIASED RANDOM WALKS, PARTIAL DIFFERENTIAL EQUATIONS AND UPDATE SCHEMES

Part of: Stochastic processes

Published online by Cambridge University Press:  18 March 2014

JACK D. HYWOOD  and
KERRY A. LANDMAN
JACK D. HYWOOD
Affiliation:
Department of Mathematics and Statistics, University of Melbourne, Victoria 3010, Australia email j.hywood@hotmail.com
KERRY A. LANDMAN*
Affiliation:
Department of Mathematics and Statistics, University of Melbourne, Victoria 3010, Australia email j.hywood@hotmail.com
*
✉kerryl@unimelb.edu.au
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Abstract

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There is much interest within the mathematical biology and statistical physics community in converting stochastic agent-based models for random walkers into a partial differential equation description for the average agent density. Here a collection of noninteracting biased random walkers on a one-dimensional lattice is considered. The usual master equation approach requires that two continuum limits, involving three parameters, namely step length, time step and the random walk bias, approach zero in a specific way. We are interested in the case where the two limits are not consistent. New results are obtained using a Fokker–Planck equation and the results are highly dependent on the simulation update schemes. The theoretical results are confirmed with examples. These findings provide insight into the importance of updating schemes to an accurate macroscopic description of stochastic local movement rules in agent-based models when the lattice spacing represents a physical object such as cell diameter.

Keywords

asymmetric random walkerspartial differential equationsFokker–Planckupdate schemes

MSC classification

Secondary: 60G07: General theory of processes
Type
Research Article
Information
The ANZIAM Journal , Volume 55 , Issue 2 , October 2013 , pp. 93 - 108
Copyright
Copyright © 2014 Australian Mathematical Society 

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