ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)


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Kurushina S. E. Analytical research and numerical simulation of contrast dissipative structures in the field of fluctuations of dynamical variables. Izvestiya VUZ. Applied Nonlinear Dynamics, 2009, vol. 17, iss. 6, pp. 125-138. DOI: 10.18500/0869-6632-2009-17-6-125-138

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
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Language: 
Russian
Article type: 
Article
UDC: 
536.75, 519.6

Analytical research and numerical simulation of contrast dissipative structures in the field of fluctuations of dynamical variables

Autors: 
Kurushina Svetlana Evgenevna, Samara State University
Abstract: 

The influence of additive homogeneous isotropic field of Gauss fluctuations of dynamical variables of Gierer–Meinhardt model to formation of dissipative structures in soft mode regime was investigated. The system of equations for description of undamped modes interaction was received. It was shown that fluctuations of dynamical variables are widening the instability region. The numerical simulation of considered model with different boundary condition was performed. It was shown that far from Turing bifurcation fluctuations further to acceleration of dissipative structures forming and change the order of location of separate peaks. 

Reference: 
  1. Vasiliev VA, Romanovsky YuM, Yakhno VG. Avtovwave processes. Moscow: Nauka; 1987. 240 p. (In Russian).
  2. Krinsky VI, Mikhailov AS. Avtovolny. Moscow: Znanie; 1984. 64 p. (In Russian).
  3. Ivanitsky GR, Krinsky VI, Selkov EE. Mathematical biophysics of a cell. Moscow: Nauka; 1978. 310 p. (In Russian).
  4. Gerisch G. Wilhelm Roux Archiv Entwickelungsmech Organizmen, 1965;156:127.
  5. Belousov BP. Collection of abstracts on radiation medicine for 1958. Moscow: Medgiz; 1959, 145 p. (In Russian).
  6. Zhabotinsky AM. Concentration self-oscillations. Moscow: Nauka; 1974. 178 p. (In Russian).
  7. Buzdin AI, Mikhailov AS. Auto-waves in magnetic superconductors. JETP. 1986;63(1):169–171.
  8. Scott A. Active and nonlinear wave propagation in electronics. New York: Wiley; 1970. 368 p.(In Russian).
  9. Cohen D, White A. Sharp fronts due to diffusion and viscoelastic relaxation in polymers. SIAMJ. Appl. Math. 1991;51(2):472--483.
  10. Romanovsky YuM, Stepanova NV, Chernavsky DS. Mathematical modeling in biophysics (Introduction to theoretical biophysics). Moscow-Izhevsk: ICR; 2004. 472 p. (In Russian).
  11. Ebeling V. Formation of structures in irreversible processes. Introduction to the theory of dissipative structures. Moscow-Izhevsk: ICR, RCD; 2004. 255 p. (In Russian).
  12. Haken H. Synergetics. Moscow: Mir; 1980. 405 p. (In Russian).
  13. Belintsev BN. Dynamic collective properties of developing systems. Dissertation for the degree of candidate of physical and mathematical sciences. Moscow: MIPT; 1979.
  14. Meinhardt H, Gierer A. Generation and regeneration of sequence of structures during morphogenesis. J Theor Biol. 1980;85(3):429–450. DOI: 10.1016/0022-5193(80)90318-5.
  15. Keener IP. Activaters and ingibitors in pattern formation. Stadies and Applied Mathematics. 1978;59(1):1–23.
  16. Gierer A, Meinhardt H. Biological pattern formation involving lateral inhibition. Lectures on Mathematics in the Life Sciences. 1974;7:163–183.
  17. Meinhardt H, Gierer A. Applications of a theory of biological pattern formation based on lateral inhibition. J Cell Sci. 1974;15(2):321–346.
  18. Horsthemke V, Lefebvre R. Noise-induced transitions: theory and application. Moscow: Mir; 1987. 397 p. (In Russian).
  19. Mikhailov AS, Uporov IV. Critical phenomena in media with breeding, decay, and diffusion. Phys. Usp. 1984;27(9):695–714. DOI: 10.3367/UFNr.0144.198409c.0079.
  20. Solyanik G.I., Chernavsky D.S. Mathematical models of morphogenesis. Preprint. Moscow: FIAN; 1980, No 8. 16 p. (In Russian).
  21. Gauze GF. Struggle for existence. Moscow-Izhevsk: RCD; 2000. 234 p. (In Russian).
  22. Belintsev BN. Physical foundations of biological formation. Moscow: Nauka; 1991. 251 p. (In Russian).
  23. Belintsev BN. Dissipative structures and the problem of biological pattern formation. Phys. Usp. 1983;26(9):775–800. DOI: 10.3367/UFNr.0141.198309b.0055.
  24. Belintsev BN. Elementary processes of formation of the supracellular organization during morphogenesis. Autoreferat dissertation for the degree of doctor of physical and mathematical sciences. Moscow: MIPT; 1986. (In Russian).
  25. Meinhardt H. The Algorithmic Beauty of Sea Shells. Berlin, Heidelberg, New York: Springer-Verlag; 1999. 269 p.
  26. Klyatskin VI. Stochastic equations through the eyes of a physicist. Moscow: Fizmatlit; 2001. 527 p. (In Russian).
  27. Abramov EI, Kurushina SE. The influence of fluctuations in dynamic variables on the formation of dissipative structures in the Girer-Main-hardt morphogenesis model. Materials of the international interdisciplinary scientific conference "III Kurdyumov Readings. Synergies in the Natural Sciences." Tver. 2007. P. 48.
  28. Svirizhev YuM, Logofet DO. Sustainability of biological communities. Moscow: Nauka; 1978. 352 p. (In Russian).
Received: 
27.10.2008
Accepted: 
17.06.2009
Published: 
31.12.2009
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