BIASED RANDOM WALKS, PARTIAL DIFFERENTIAL EQUATIONS AND UPDATE SCHEMES

• Published in: The ANZIAM journal Vol. 55; no. 2; pp. 93 - 108
• Main Authors: HYWOOD, JACK D., LANDMAN, KERRY A.
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.10.2013
• Subjects: Biology; Communities; Density; Lattices; Mathematical analysis; Mathematical models; Partial differential equations; Random walk; Stochasticity; update schemes; 60G07; partial differential equations; asymmetric random walkers; Fokker–Planck
• ISSN: 1446-1811; 1446-8735
• DOI: 10.1017/S1446181113000369

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.