Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-08T01:24:09.486Z Has data issue: false hasContentIssue false
  • English
  • Français

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 [email protected]
KERRY A. LANDMAN*
Affiliation:
Department of Mathematics and Statistics, University of Melbourne, Victoria 3010, Australia email [email protected]
*
✉[email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 

References

Alber, M., Chen, N., Glimm, T. and Lushnikov, P. M., “Multiscale dynamics of biological cells with chemotactic interactions: from a discrete stochastic model to a continuous description”, Phys. Rev. E 73 (2006) ; doi:10.1103/PhysRevE.73.051901.Google ScholarPubMed
Bhattachary, R. N. and Waymire, E. C., Stochastic processes with applications, Volume 61 of Classics in Applied Mathematics (SIAM, Philadelphia, 2009).CrossRefGoogle Scholar
Binder, B. J. and Landman, K. A., “Exclusion processes on a growing domain”, J. Theor. Biol. 259 (2009) 541–551 ; doi:10.1016/j.jtbi.2009.04.025.CrossRefGoogle ScholarPubMed
Binder, B. J., Landman, K. A., Simpson, M. J., Mariani, M. and Newgreen, D. F., “Modeling proliferative tissue growth: a general approach and an avian case study”, Phys. Rev. E 78 (2008) ; doi:10.1103/PhysRevE.78.031912.CrossRefGoogle Scholar
Chowdhury, D., Schadschneider, A. and Nishinari, K., “Physics of transport and traffic phenomena in biology: from molecular motors and cells to organisms”, Phys. Life Rev. 2 (2005) 318–352 ; doi:10.1016/j.plrev.2005.09.001.CrossRefGoogle Scholar
Codling, E. A., Plank, M. J. and Benhamou, S., “Random walk models in biology”, J. R. Soc. Interface 5 (2008) 813–834 ; doi:10.1098/rsif.2008.0014.CrossRefGoogle ScholarPubMed
Deroulers, C., Aubert, M., Badoual, M. and Grammaticos, B., “Modeling tumor cell migration: from microscopic to macroscopic models”, Phys. Rev. E 79 (2009) ; doi:10.1103/PhysRevE.79.031917.Google ScholarPubMed
Fernando, A. E., Landman, K. A. and Simpson, M. J., “Nonlinear diffusion and exclusion processes with contact interactions”, Phys. Rev. E 81 (2010) ; doi:10.1103/PhysRevE.81.011903.CrossRefGoogle ScholarPubMed
Hackett-Jones, E. J., Landman, K. A., Newgreen, D. F. and Zhang, D., “On the role of differential adhesion in gangliogenesis in the enteric nervous system”, J. Theor. Biol. 287 (2011) 148–159 ; doi:10.1016/j.jtbi.2011.07.013.CrossRefGoogle ScholarPubMed
Hughes, B. D., Random walks and random environments, Volume 1 (Oxford University Press, Oxford, 1995).CrossRefGoogle Scholar
Hywood, J. D., Hackett-Jones, E. J. and Landman, K. A., “Modelling biological tissue growth: discrete to continuum representations”, Phys. Rev. E 88 (2013) ; doi:10.1103/PhysRevE.88.032704.Google ScholarPubMed
Karlin, S. and Taylor, H. M., A second course in stochastic processes (Academic Press, Orlando, 1981).Google Scholar
Kimura, M., “Diffusion models in population genetics”, J. Appl. Prob. 1 (1964) 177–232 ; doi:10.2307/3211856.CrossRefGoogle Scholar
Landman, K. A., Binder, B. J. and Newgreen, D. F., “Modeling development and disease in our ‘second’ brain”, in: Cellular automata, Volume 7495 of Lecture Notes in Computer Science (eds G. C. Sirakoulis and S. Bandini), (Springer, Berlin-Heidelberg, 2012), 405–414; doi:10.1007/978-3-642-33350-7_42.CrossRefGoogle Scholar
Landman, K. A. and Fernando, A. E., “Myopic random walkers and exclusion processes: single and multispecies”, Phys. A 390 (2011) 3742–3753 ; doi:10.1016/j.physa.2011.06.034.CrossRefGoogle Scholar
Lange, K., Applied probability (Springer, New York, 2003).Google Scholar
Painter, K. J., “Continuous models for cell migration in tissues and applications to cell sorting via differential chemotaxis”, Bull. Math. Biol 71 (2009) 1117–1147 ; doi:10.1007/s11538-009-9396-8.CrossRefGoogle Scholar
Penington, C. J., Hughes, B. D. and Landman, K. A., “Building macroscale models from microscale probabilistic models: a general probabilistic approach for nonlinear diffusion and multispecies phenomena”, Phys. Rev. E 84 (2011) ; doi:10.1103/PhysRevE.84.041120.Google ScholarPubMed
Port, S. C., Theoretical probability for applications (Wiley & Sons, New York, 1994).Google Scholar
Rajewsky, N., Santen, L., Schadschneider, A. and Schreckenberg, M., “The asymmetric exclusion process: comparison of update procedures”, J. Stat. Phys. 92 (1998) 151–194 ; doi:10.1023/A:1023047703307.CrossRefGoogle Scholar
Simons, G. and Johnson, N. L., “On the convergence of binomial to Poisson distributions”, Ann. Math. Stat. 42 (1971) 1735–1736 ; doi:10.1214/aoms/1177693172.CrossRefGoogle Scholar
Simpson, M. J., Landman, K. A. and Hughes, B. D., “Diffusing populations: ghosts or folks?”, Australasian J. Engrg. Ed. 15 (2009) 59–67.CrossRefGoogle Scholar
Simpson, M. J., Landman, K. A. and Hughes, B. D., “Multi-species simple exclusion processes”, Phys. A 388 (2009) 399–406 ; doi:10.1016/j.physa.2008.10.038.CrossRefGoogle Scholar
Sobczyk, K., Stochastic differential equations, with applications to physics and engineering (Kluwer Academic Publishers, Dordrecht, 1991).Google Scholar
Soong, T. T., Random equations in science and engineering (Academic Press, New York, 1973).Google Scholar
Stevens, A. and Othmer, H. G., “Aggregation, blowup and collapse: the ABC’s of taxis in reinforced random walks”, SIAM J. Appl. Math. 57 (1997) 1044–1081 ; doi:10.1137/S0036139995288976.CrossRefGoogle Scholar
Turner, S., Sherratt, J. A., Painter, K. J. and Savill, N. J., “From a discrete to a continuous model of biological cell movement”, Phys. Rev. E 69 (2004) 021910; doi:10.1103/PhysRevE.69.021910.Google ScholarPubMed