About the distance between random walkers on some graphs

AbstractThere 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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Analysis of Transmission of Infection in Epidemics: Confined Random Walkers in Dimensions Higher Than One

Bulletin of Mathematical Biology ◽

10.1007/s11538-018-0507-2 ◽

2018 ◽

Vol 80 (12) ◽

pp. 3106-3126 ◽

Cited By ~ 2

Author(s):

S. Sugaya ◽

V. M. Kenkre

Keyword(s):

Random Walkers

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A Reaction-Diffusion Model for Interference in Meiotic Crossing Over

Genetics ◽

10.1093/genetics/161.1.365 ◽

2002 ◽

Vol 161 (1) ◽

pp. 365-372 ◽

Cited By ~ 1

Author(s):

Youhei Fujitani ◽

Shintaro Mori ◽

Ichizo Kobayashi

Keyword(s):

Genetic Distance ◽

Diffusion Model ◽

Genetic Model ◽

Initial Density ◽

Reaction Diffusion ◽

Crossover Point ◽

Random Walker ◽

Random Walkers ◽

Reaction Diffusion Model ◽

Two Parameters

Abstract One crossover point between a pair of homologous chromosomes in meiosis appears to interfere with occurrence of another in the neighborhood. It has been revealed that Drosophila and Neurospora, in spite of their large difference in the frequency of crossover points, show very similar plots of coincidence—a measure of the interference—against the genetic distance of the interval, defined as one-half the average number of crossover points within the interval. We here propose a simple reaction-diffusion model, where a “randomly walking” precursor becomes immobilized and matures into a crossover point. The interference is caused by pair-annihilation of the random walkers due to their collision and by annihilation of a random walker due to its collision with an immobilized point. This model has two parameters—the initial density of the random walkers and the rate of its processing into a crossover point. We show numerically that, as the former increases and/or the latter decreases, plotted curves of the coincidence vs. the genetic distance converge on a unique curve. Thus, our model explains the similarity between Drosophila and Neurospora without parameter values adjusted finely, although it is not a “genetic model” but is a “physical model,” specifying explicitly what happens physically.

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IODINE FRACTALS IN ION CONDUCTING POLYMER MATRIX

Fractals ◽

10.1142/s0218348x01000622 ◽

2001 ◽

Vol 09 (02) ◽

pp. 171-175 ◽

Cited By ~ 1

Author(s):

AMITA CHANDRA

Keyword(s):

Conducting Polymer ◽

Polymer Matrix ◽

Strain Field ◽

Polymeric System ◽

Fluorescence Studies ◽

Random Walkers ◽

Diffusion Limited ◽

Large Size ◽

Ion Conducting ◽

Ion Conducting Polymer

Large size diffusion limited aggregates (DLA) in ion conducting polymeric system, PEO:NH 4 I (+ Al 2 O 3), have been obtained. The species responsible for the formation of the DLA have been identified as [Formula: see text] by fluorescence studies. It is also shown that if I 2 gets formed (instead of [Formula: see text] aggregates) at selected regions, then "strain field" is established which forbids further approach of ionic random walkers and hence no fractals grow around these regions.

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