BIASED RANDOM WALKS, PARTIAL DIFFERENTIAL EQUATIONS AND UPDATE SCHEMES
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-d...
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Published in | The ANZIAM journal Vol. 55; no. 2; pp. 93 - 108 |
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Main Authors | , |
Format | Journal Article |
Language | English |
Published |
Cambridge, UK
Cambridge University Press
01.10.2013
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Subjects | |
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Abstract | 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. |
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AbstractList | 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. |
Author | HYWOOD, JACK D. LANDMAN, KERRY A. |
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CitedBy_id | crossref_primary_10_1098_rsif_2020_0879 crossref_primary_10_1371_journal_pone_0117949 crossref_primary_10_3390_mi12070749 crossref_primary_10_1103_PhysRevE_91_042701 |
Cites_doi | 10.1007/978-3-642-33350-7_42 10.1137/1.9780898718997 10.1103/PhysRevE.78.031912 10.1103/PhysRevE.81.011903 10.2307/3211856 10.1016/j.jtbi.2011.07.013 10.1016/j.plrev.2005.09.001 10.1016/j.physa.2008.10.038 10.1214/aoms/1177693172 10.1016/j.physa.2011.06.034 10.1007/s11538-009-9396-8 10.1016/j.jtbi.2009.04.025 10.1098/rsif.2008.0014 10.1093/oso/9780198537885.001.0001 10.1023/A:1023047703307 10.1080/22054952.2009.11464027 10.1137/S0036139995288976 |
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Keywords | update schemes 60G07 partial differential equations asymmetric random walkers Fokker–Planck |
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References | S1446181113000369_r2 S1446181113000369_r3 Turner (S1446181113000369_r27) 2004; 69 S1446181113000369_r4 S1446181113000369_r5 S1446181113000369_r6 S1446181113000369_r17 S1446181113000369_r8 Soong (S1446181113000369_r25) 1973 S1446181113000369_r9 Hywood (S1446181113000369_r11) 2013; 88 S1446181113000369_r26 S1446181113000369_r21 S1446181113000369_r20 S1446181113000369_r23 S1446181113000369_r22 Karlin (S1446181113000369_r12) 1981 Hughes (S1446181113000369_r10) 1995 Lange (S1446181113000369_r16) 2003 Penington (S1446181113000369_r18) 2011; 84 Alber (S1446181113000369_r1) 2006; 73 Port (S1446181113000369_r19) 1994 S1446181113000369_r14 S1446181113000369_r13 S1446181113000369_r15 Deroulers (S1446181113000369_r7) 2009; 79 Sobczyk (S1446181113000369_r24) 1991 |
References_xml | – volume: 73 year: 2006 ident: S1446181113000369_r1 article-title: Multiscale dynamics of biological cells with chemotactic interactions: from a discrete stochastic model to a continuous description publication-title: Phys. Rev. E contributor: fullname: Alber – ident: S1446181113000369_r14 doi: 10.1007/978-3-642-33350-7_42 – ident: S1446181113000369_r2 doi: 10.1137/1.9780898718997 – volume-title: Random equations in science and engineering year: 1973 ident: S1446181113000369_r25 contributor: fullname: Soong – volume: 69 start-page: 021910 year: 2004 ident: S1446181113000369_r27 article-title: From a discrete to a continuous model of biological cell movement publication-title: Phys. Rev. E contributor: fullname: Turner – ident: S1446181113000369_r4 doi: 10.1103/PhysRevE.78.031912 – ident: S1446181113000369_r8 doi: 10.1103/PhysRevE.81.011903 – ident: S1446181113000369_r13 doi: 10.2307/3211856 – volume-title: Theoretical probability for applications year: 1994 ident: S1446181113000369_r19 contributor: fullname: Port – ident: S1446181113000369_r9 doi: 10.1016/j.jtbi.2011.07.013 – volume: 88 year: 2013 ident: S1446181113000369_r11 article-title: Modelling biological tissue growth: discrete to continuum representations publication-title: Phys. Rev. E contributor: fullname: Hywood – ident: S1446181113000369_r5 doi: 10.1016/j.plrev.2005.09.001 – volume: 84 year: 2011 ident: S1446181113000369_r18 article-title: Building macroscale models from microscale probabilistic models: a general probabilistic approach for nonlinear diffusion and multispecies phenomena publication-title: Phys. Rev. E contributor: fullname: Penington – volume: 79 year: 2009 ident: S1446181113000369_r7 article-title: Modeling tumor cell migration: from microscopic to macroscopic models publication-title: Phys. Rev. E contributor: fullname: Deroulers – ident: S1446181113000369_r23 doi: 10.1016/j.physa.2008.10.038 – ident: S1446181113000369_r21 doi: 10.1214/aoms/1177693172 – volume-title: Applied probability year: 2003 ident: S1446181113000369_r16 contributor: fullname: Lange – ident: S1446181113000369_r15 doi: 10.1016/j.physa.2011.06.034 – ident: S1446181113000369_r17 doi: 10.1007/s11538-009-9396-8 – ident: S1446181113000369_r3 doi: 10.1016/j.jtbi.2009.04.025 – ident: S1446181113000369_r6 doi: 10.1098/rsif.2008.0014 – volume-title: A second course in stochastic processes year: 1981 ident: S1446181113000369_r12 contributor: fullname: Karlin – volume-title: Random walks and random environments, Volume 1 year: 1995 ident: S1446181113000369_r10 doi: 10.1093/oso/9780198537885.001.0001 contributor: fullname: Hughes – ident: S1446181113000369_r20 doi: 10.1023/A:1023047703307 – ident: S1446181113000369_r22 doi: 10.1080/22054952.2009.11464027 – volume-title: Stochastic differential equations, with applications to physics and engineering year: 1991 ident: S1446181113000369_r24 contributor: fullname: Sobczyk – ident: S1446181113000369_r26 doi: 10.1137/S0036139995288976 |
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SubjectTerms | Biology Communities Density Lattices Mathematical analysis Mathematical models Partial differential equations Random walk Stochasticity |
Title | BIASED RANDOM WALKS, PARTIAL DIFFERENTIAL EQUATIONS AND UPDATE SCHEMES |
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