The analysis of spatial pattern formation in reaction-diffusion system near bifurcation
S.E. Kurushina

S. P. Korolyov Samara State Aerospace University

Full text of article: Russian language.

Abstract:
Generalized two-component of reaction-diffusion system in external fluctuated environment is considered. Ginzburg-Landau equations under condition of soft mode instability are proposed. The properties of this model in neighborhood of Turing bifurcation point were analyzed. Numerical investigation of specific reaction-diffusion systems was conducted. Agreement of the analytical and numerical results is shown.

Key words:
reaction-diffusion system, multiplicative fluctuations of parameters, spatial pattern, unstable modes, Ginzburg-Landau equation, noise-induced parametrical excitation, simulation.

References:

  1. Bray, Mark-Anthony. Experimental and Theoretical Analysis of Phase Singularity Dynamics in Cardiac Tissue / Bray Mark-Anthony, Shien-Fong Lin, Rubin R. Aliev, Bradley J. Roth, John P. Wikswo // Journal of cardiovascular electrophysiology. – 2001. –  Vol. 12,  6. – P. 716-722.
  2. Castets, V. Experimental evidence of a sustained standing Turing-type nonequilibrium chemical pattern / V. Castets, E. Dulos, J. Boissonade, P. De Kepper // Phys. Rev. Lett. – 1990. – Vol. 64. – P. 2953-2956.
  3. Efimov, I.R. Dynamics of Rotating Vortices in the Beeler-Reuter Model of Cardiac Tissue / I.R. Efimov, V.I. Krinsky, J. Jalife // Chaos, Solitons & Fractals. – 1995. – Vol. 5. –  3/4. – P. 513-526.
  4. Meinhardt, H. Applications of a theory of biological pattern formation based on lateral inhibition / H. Meinhardt, A. Gierer // Journ. Cell. Sci. – 1974. – Vol. 15. – P. 321.
  5. Meinhardt, H. The Algorithmic Beauty of Sea Shells – Berlin, Heidelberg, New York. Springer-Verlag, 1999. – 280 p.
  6. Sun, G.-Q. Dynamical complexity of a spatial predator–prey model with migration / Gui-Quan Sun, Zhen Jin, Quan-Xing Liu, Li Li // Ecological modeling. – 2008. – Vol. 219. – P. 248-255.
  7. Murray, J.D. Mathematical Biology – Berlin Heidelberg: Springer-Verlag, 2002. – 576 p.
  8. Malchow, H. Spatiotemporal patterns in an excitable plankton system with lysogenic viral infection / H. Mal­chow, F.M. Hilker, R.R. Sarkar, K. Brauer // Mathematical and Computer Modeling. – 2005. – Vol. 42. – P. 1035-1048.
  9. Romanovsky, Yu.M. Mathematical modeling in biophy­sics (The introduction in theoretical biophysics) / Yu.. Ro­manovsky, N.V. Stepanova, D.S. Chernavcky – Moscow-Izhevsk: ICI, 2004. – 472 p. – (In Russian).
  10. Scheffer, M. Fish and nutrients interplay determines algal biomass: a minimal model // OIKOS. – 1991. – Vol. 62. – P. 271-282.
  11. Malchow, H. Motional instabilities in prey-predator systems / H. Malchow // J. Theor. Biol. – 2000. –  Vol. 204. – P. 639-647.
  12. Klyatskin, V.I. Stochastic equations through the eye of the physicist – oscow: “Fizmatlit” Publisher, 2001. – 528 p. – (In Russian).
  13.  Haken, H. Synergetics – oscow: “Mir” Publisher, 1980. – 406 p. – (In Russian).
  14. Horsthemke, W. Noise-induced Transitions: theory and application in physics, chemistry, and biology / W. Horsthemke, R. Lefevr – oscow: “Mir” Publisher, 1987. – 400 p. – (In Russian).
  15. Kurushina, S.E. Dissipative structures of reaction-diffusion system simulation in multiplicative fluctuation phone / S.. Kurushina, .. Ivanov // Izvestiya VUZ. Allied nonlinear dynamics. – 2010. –  3. – P. 85-103. – (In Russian).
  16. Kurushina, S.E. Spatiotemporal pattern of prey-predator system simulation in external fluctuate environment / S.. Kurushina, Yu.V. Zhelnov, I.P. Zavershinsky, .. Iva­nov, V.V. Maximov // Matematicheskoe Modelirovanie. –2010. – Vol. 22, N 10. – P. 3-17. – (In Russian).
  17. Stratonovich, R.L. Nonlinear nonequilibrium thermodyna­mics – .: “Nauka” Publisher, 1985. – 480 p.

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