SPATIAL INHOMOGENITY DUE TO TURING BIFURCATION IN A SYSTEM OF GIERER-MEINHARDT TYPE

  • Sandor, Kovacs (Department of Numerical Analysis, Eotvos L. University)
  • 발행 : 2003.01.01

초록

This paper treats the conditions for the existence and stability properties of stationary solutions of reaction-diffusion equations of Gierer-Meinhardt type, subject to Neumann boundary data. The domains in which diffusion takes place are of three types: a regular hexagon, a rectangle and an isosceles rectangular triangle. Considering one of the relevant features of the domains as a bifurcation parameter it will be shown that at a certain critical value a diffusion driven instability occurs and Turing bifurcation takes place: a pattern emerges.

키워드

참고문헌

  1. J. Appl. Math. & Computing v.10 no.1 Stability and bifurcation in a diffusive prey-predator system: non-linear bifurcation analysis R. Bhattacharya;M. Bandyopadhyay;S. Banerjee
  2. SIAM J. Appl. Math. v.33 Stabilitity properties of solutions to systems of reaction diffusion equations R. G. Casten;C. F. Holland
  3. Acta Math. Hunger v.63 Bifurcation in a predator-prey model with momory and diffusion: Ⅱ Turing bifurcation M. Cavani;M. Farkas
  4. Ind. U. Math. J. v.26 Positively invariant regions for systems of nonlinear diffusion equations K. Chueh;C. Conley;J. Smoller
  5. Archive Rat. Mech. anal. v.52 Bifurcation, Perturbation of simple eigenvalues and linearized stability M. G. Crandall;P. H. Rabinowitz
  6. Korean J. Comput. Appl. Math. v.5 no.2 Ratio dependent predation: A bifurcation analyaia Dipak Kesh;Debasis Mukherjee;A. K. Sakar;A. B. Roy
  7. Math. Z. v.194 An initial-boundary-value problem for a certain density-dependent diffusion system P. Deuring
  8. SEA Bull. Math. v.19 no.2 On the distribution of capital and labour in a closed economy M. Farkas
  9. Differential Equations and Dynamical Systems v.7 no.2 Comparison of different ways of modelling cross-diffusion M. Faekas
  10. Partial Differenrial Equation of Parabolic Type A. Friedman
  11. Kybernetik v.12 A theory of biological pattern formation A. Gierer;H. Meinhardt
  12. Quart. J. Appl. Math. v.32 Some mathematical models for population dynamics that lead to segregation M. E. Gurtin
  13. J. Theor Biol. v.65 The diffusive Loka-Votka oscillating system J. Jorne
  14. Bull. Math. Biophys v.21 Further considerations on the statistical mechanics of biological associations E. H. Kerner
  15. Nonlinear Analysis v.21 Smooth solutions to a quasilinear system of diffusion equation for a certain population model J. U. Kim
  16. Annales Univ. Sci. Budapest v.42 Pattern formation in bounded spatial domains S. Kobvacs
  17. Some special case Studia Scienriarum Mathematicarum Hungarica v.11 Complete systems of eigenfunctions of the wave equation E. Makai
  18. Publ. RIMS v.19 Pattern formation in competion-diffusion systems in nonconvex domains H. Matano;M. Mimura
  19. Monographis in Population Biology Stabilitity and Complexity in Model Ecosystems R. May
  20. Hiroshima math J. v.11 Stationary pattern of some deensity-dependent diffusion system with compertitive dynamics M. Mimura
  21. Mathematical Biology, Biomathematics v.19 J. D. Murray
  22. Mathematical Models Diffusion and Ecological Problems A. Okubo
  23. Nonlinear Analysis v.14 Global existence of a strongly coupled puasilinear parabolic systems M. Pozio;A. Tesei
  24. Lecture Notes in Methematics Global solutions of reaction-diffusio systems F. Rothe
  25. Shock Waews and Reaction-Diffusion Equations J. Smoller
  26. Dissipaative Structures and Catastrophes in Ecology Nonlinear Wawes Yu, M. Svirezhev
  27. Mir. Stability of Biological Communities Yu, M. Svirezhev;D. O. Logofet
  28. Physical Review E v.48 no.1 Necessary condition of the Turing instability L. Szill;J. Toth
  29. Tohoke Math. Journ. v.31 Stability of bifurcating solutions of the Gierer-Meinhardt system I. Takagi
  30. Phil. Trans. Roy. soc. v.B237 no.2 A chemical basis of borphogenesis A. Turing
  31. Nonlinear Analysis, Theory, Methods & Applocations v.24 no.9 Global solutions for quasilinear parabolic systems with cross-diffusion effects Y. Yamada