{"title":"Transient Population Dynamics of Phase Singularities in 2D Beeler-Reuter Model","authors":" Hidetoshi Konno, Akio Suzuki","volume":80,"journal":"International Journal of Industrial and Manufacturing Engineering","pagesStart":1647,"pagesEnd":1653,"ISSN":"1307-6892","URL":"https:\/\/publications.waset.org\/pdf\/16095","abstract":"<p>The paper presented a transient population dynamics of phase singularities in 2D Beeler-Reuter model. Two stochastic modelings are examined: (i) the Master equation approach with the transition rate (i.e., λ(n, t) = λ(t)n and μ(n, t) = μ(t)n) and (ii) the nonlinear Langevin equation approach with a multiplicative noise. The exact general solution of the Master equation with arbitrary time-dependent transition rate is given. Then, the exact solution of the mean field equation for the nonlinear Langevin equation is also given. It is demonstrated that transient population dynamics is successfully identified by the generalized Logistic equation with fractional higher order nonlinear term. It is also demonstrated the necessity of introducing time-dependent transition rate in the master equation approach to incorporate the effect of nonlinearity.<\/p>\r\n","references":"<p>[1] R. H. Clayton, et al. , Progr. Biophys. Molecul. Biol. 104 (2011) pp. 22-48. [2] R. Harada and H. Konno, Pacific Science Review, 12 (2011) pp. 208-213. [3] R. R. Aliev and A. V. Panfilov, Chaos, Soliton & Fractals, 7 (1996) pp. 293-301: u\u02d9 = −Ku(u − a)(u − 1) − uv + D∇2u and v\u02d9 = (\u0002 + μ1v\/(μ2+u))(−ν−Ku(u−b−1)) with the parameters a = 0.05,b = 0.1,K = 8.0,\u0002 = 0.01,μ1 = 0.07, μ2 = 0.3 and D = 1.0. [4] A. Suzuki and H. Konno, AIP Advances, 1 (2011) 032103 1-13. [5] G. Beeler and H. Reuter, Reconstruction of the action potential of ventricular myocardial fibres, J. Physiol. 268, 177 (1977) pp. 177-210. [6] H. Konno and P. Lomdahl, Stochastic Processes Having Fractional Order Nonlinearity Associated with Hyper Gamma Distribution, J. Phys. Soc. Jpn., 73 (2004) pp. 573-579. [7] H. Konno, Advances in Mathematical Physics, 2010 (2010) 504267 1-12. [8] W. Feller, Introduction to Probability Theory and Its Applications, Vol. I and II (Wiley, NY, 1967). [9] Y. Ogata, J. Amer. Statist. Assoc., 83 (1988) pp. 9-27. [10] C. W. Gardiner, Handbook of Stochastic Methods, (Springer, Berlin, 1983). [11] L. Gil, J. Lega and J. M. Meunier, Phys. Rev. A41 (1990) pp.1138-1141. [12] K. E. Daniels and E. Bodenschatz, Phys. Rev. Lett. 88 (2002) 034501 1-4. [13] C. Beta, A. S. Mikailov, H. H. Rotemund and G. Ertl, Europhys. Lett. 75 (2006) pp. 868-874. [14] D. T. Gillespie, J. Phys. Chem. 81 (1977) pp.2341-2361.<\/p>\r\n","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 80, 2013"}

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