Reaction diffusion equation with non-local term arises as a mean field limit of the master equation

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February 2012, 5(1): 115-126. doi: 10.3934/dcdss.2012.5.115

Reaction diffusion equation with non-local term arises as a mean field limit of the master equation

Kazuhisa Ichikawa ^1, , Mahemauti Rouzimaimaiti ^2, and Takashi Suzuki ^3,

1.	The Institute of Medical Science, The University of Tokyo, 4-6-1 Shirokanedai Minato-ku, Tokyo, 108-8639, Japan
2.	Division of Mathematical Science, Department of Systems Innovation, Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyamacho, Toyonakashi, 560-8531, Japan
3.	Japan Science and Technology Agency, CREST 5, Sanbancho, Chiyoda-ku, Tokyo, 102-0075, Japan

Received March 2009 Revised December 2009 Published February 2011

Abstract
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We formulate a reaction diffusion equation with non-local term as a mean field equation of the master equation where the particle density is defined continuously in space and time. In the case of the constant mean waiting time, this limit equation is associated with the diffusion coefficient of A. Einstein, the reaction rate in phenomenology, and the Debye term under the presence of potential.

Keywords: Reaction probability., Stochastic reaction, Reaction radius.

Mathematics Subject Classification: 35K57, 93A3.

Citation: Kazuhisa Ichikawa, Mahemauti Rouzimaimaiti, Takashi Suzuki. Reaction diffusion equation with non-local term arises as a mean field limit of the master equation. Discrete and Continuous Dynamical Systems - S, 2012, 5 (1) : 115-126. doi: 10.3934/dcdss.2012.5.115

References:

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References:

[1]	P. A. Egelstaff, "An Introduction to the Liquid State," Academic Press, London, 1967.
[2]	J. D. Murray, "Mathematical Biology I: An Introduction," 3^rd edition, Interdisciplinary Applied Mathematics, 17, Springer-Verlag, New York, 2001.
[3]	A. Okubo, "Diffusion and Ecological Problems: Modern Perspectives," 2^nd edition edition, Interdisciplinary Applied Mathematics, 14, Springer-Verlag, New York, 2001.
[4]	H. G. Othmer, S. R. Dumber and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298. doi: 10.1007/BF00277392.
[5]	H. G. Othmer and A. Stevens, Aggregation, blowup, and collapse: The ABCs of taxis in reinforced random walks, SIAM J. Appl. Math., 57 (1997), 1044-1081. doi: 10.1137/S0036139995288976.

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