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

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.
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

[1]

Piermarco Cannarsa, Giuseppe Da Prato. Invariance for stochastic reaction-diffusion equations. Evolution Equations & Control Theory, 2012, 1 (1) : 43-56. doi: 10.3934/eect.2012.1.43

[2]

Hakima Bessaih, Yalchin Efendiev, Razvan Florian Maris. Stochastic homogenization for a diffusion-reaction model. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5403-5429. doi: 10.3934/dcds.2019221

[3]

Wilhelm Stannat, Lukas Wessels. Deterministic control of stochastic reaction-diffusion equations. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020087

[4]

Hideki Murakawa. Fast reaction limit of reaction-diffusion systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020405

[5]

Wei Wang, Anthony Roberts. Macroscopic discrete modelling of stochastic reaction-diffusion equations on a periodic domain. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 253-273. doi: 10.3934/dcds.2011.31.253

[6]

Parker Childs, James P. Keener. Slow manifold reduction of a stochastic chemical reaction: Exploring Keizer's paradox. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1775-1794. doi: 10.3934/dcdsb.2012.17.1775

[7]

Perla El Kettani, Danielle Hilhorst, Kai Lee. A stochastic mass conserved reaction-diffusion equation with nonlinear diffusion. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5615-5648. doi: 10.3934/dcds.2018246

[8]

Yuncheng You. Random attractors and robustness for stochastic reversible reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 301-333. doi: 10.3934/dcds.2014.34.301

[9]

N. U. Ahmed. Weak solutions of stochastic reaction diffusion equations and their optimal control. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1011-1029. doi: 10.3934/dcdss.2018059

[10]

Jifa Jiang, Junping Shi. Dynamics of a reaction-diffusion system of autocatalytic chemical reaction. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 245-258. doi: 10.3934/dcds.2008.21.245

[11]

Dieter Bothe, Michel Pierre. The instantaneous limit for reaction-diffusion systems with a fast irreversible reaction. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 49-59. doi: 10.3934/dcdss.2012.5.49

[12]

Peter E. Kloeden, Thomas Lorenz, Meihua Yang. Reaction-diffusion equations with a switched--off reaction zone. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1907-1933. doi: 10.3934/cpaa.2014.13.1907

[13]

M. Grasselli, V. Pata. A reaction-diffusion equation with memory. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1079-1088. doi: 10.3934/dcds.2006.15.1079

[14]

Anne Shiu, Timo de Wolff. Nondegenerate multistationarity in small reaction networks. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2683-2700. doi: 10.3934/dcdsb.2018270

[15]

Shangbing Ai, Wenzhang Huang, Zhi-An Wang. Reaction, diffusion and chemotaxis in wave propagation. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 1-21. doi: 10.3934/dcdsb.2015.20.1

[16]

Shouchuan Hu, Nikolaos S. Papageorgiou. Nonlinear Dirichlet problems with a crossing reaction. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2749-2766. doi: 10.3934/cpaa.2014.13.2749

[17]

S.-I. Ei, M. Mimura, M. Nagayama. Interacting spots in reaction diffusion systems. Discrete & Continuous Dynamical Systems - A, 2006, 14 (1) : 31-62. doi: 10.3934/dcds.2006.14.31

[18]

Juan Dávila, Louis Dupaigne, Marcelo Montenegro. The extremal solution of a boundary reaction problem. Communications on Pure & Applied Analysis, 2008, 7 (4) : 795-817. doi: 10.3934/cpaa.2008.7.795

[19]

Lu Yang, Meihua Yang. Long-time behavior of stochastic reaction-diffusion equation with dynamical boundary condition. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2627-2650. doi: 10.3934/dcdsb.2017102

[20]

Dingshi Li, Kening Lu, Bixiang Wang, Xiaohu Wang. Limiting dynamics for non-autonomous stochastic retarded reaction-diffusion equations on thin domains. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3717-3747. doi: 10.3934/dcds.2019151

2019 Impact Factor: 1.233

Metrics

  • PDF downloads (30)
  • HTML views (0)
  • Cited by (2)

[Back to Top]