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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

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 & Continuous Dynamical Systems - S, 2012, 5 (1) : 115-126. doi: 10.3934/dcdss.2012.5.115
References:
[1]

P. A. Egelstaff, "An Introduction to the Liquid State,", Academic Press, (1967).   Google Scholar

[2]

J. D. Murray, "Mathematical Biology I: An Introduction,", 3rd edition, (2001).   Google Scholar

[3]

A. Okubo, "Diffusion and Ecological Problems: Modern Perspectives,", 2nd, (2001).   Google Scholar

[4]

H. G. Othmer, S. R. Dumber and W. Alt, Models of dispersal in biological systems,, J. Math. Biol., 26 (1988), 263.  doi: 10.1007/BF00277392.  Google Scholar

[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.  doi: 10.1137/S0036139995288976.  Google Scholar

show all references

References:
[1]

P. A. Egelstaff, "An Introduction to the Liquid State,", Academic Press, (1967).   Google Scholar

[2]

J. D. Murray, "Mathematical Biology I: An Introduction,", 3rd edition, (2001).   Google Scholar

[3]

A. Okubo, "Diffusion and Ecological Problems: Modern Perspectives,", 2nd, (2001).   Google Scholar

[4]

H. G. Othmer, S. R. Dumber and W. Alt, Models of dispersal in biological systems,, J. Math. Biol., 26 (1988), 263.  doi: 10.1007/BF00277392.  Google Scholar

[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.  doi: 10.1137/S0036139995288976.  Google Scholar

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