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Stability transitions and dynamics of mesa patterns near the shadow limit of reaction-diffusion systems in one space dimension
1. | Department of Mathematics and Statistics, Dalhousie University, Halifax, N.S. B3H 3J5, Canada |
2. | Department of Mathematics, Dalhousie University, Halifax, N.S. B3H 3J5, Canada |
References:
[1] |
Y. Nishiura and H. Fujii, Stability of singularly perturbed solutions to systems of reaction-diffusion equations, SIAM J. Math. Anal., 18 (1987), 1726-1770.
doi: 10.1137/0518124. |
[2] |
Y. Nishiura and M. Mimura, Layer oscillations in reaction-diffusion systems, SIAM J.Appl. Math., 49 (1989), 481-514.
doi: 10.1137/0149029. |
[3] |
C. B. Muratov and V. Osipov, Scenarios of domain pattern formation in a reaction-diffusion system, Phys. Rev. E., 54 (1996), 4860-4879.
doi: 10.1103/PhysRevE.54.4860. |
[4] |
J. Rubinstein, P. Sternberg and J. B. Keller, Fast reaction, slow diffusion, and curve shortening, SIAM J. Appl. Math., 49 (1989), 116-133.
doi: 10.1137/0149007. |
[5] |
R. E. Goldstein, D. J. Muraki and D. M. Petrich, Interface proliferation and the growth of labyrinths in a reaction-diffusion system, Phys. Rev. E., 53 (1996), 3933-3957.
doi: 10.1103/PhysRevE.53.3933. |
[6] |
G. Nicolis and I. Prigogine, "Self-organization in Non-Equilibrium Systems," Wiley Interscience, New York, 1977. |
[7] |
J. D. Murray, "Mathematical Biology," Springer-Verlag, Berlin, 1989. |
[8] |
B. S. Kerner and V. V. Osipov, "Autosolitons: A New Approach to Problems of Self-Organization and Turbulence," Kluwer, Dordrecht, 1994. |
[9] |
R. Kapral and K. Showalter, eds., "Chemical Waves and Patterns," Kluwer, Dordrecht, 1995. |
[10] |
M. Taki, M. Tlidi and T. Kolokolnikov, eds., "Dissipative Localized Structures in Extended Systems," Chaos Focus issue, 17, 2007. |
[11] |
T. Kolokolnikov, M. Ward and J. Wei, Self-replication of mesa patterns in reaction-diffusion models, Physica D, 236 (2007), 104-122.
doi: 10.1016/j.physd.2007.07.014. |
[12] |
W.-M. Ni, P. Poláčik and E. Yanagida, Monotonicity of stable solutions in shadow systems, Trans. A.M.S., 352 (2001), 5057-5069.
doi: 10.1090/S0002-9947-01-02880-X. |
[13] |
C. Muratov and V. V. Osipov, General theory of instabilities for patterns with sharp interfaces in reaction-diffusion systems, Phys, Rev. E., 53 (1996), 3101-3116.
doi: 10.1103/PhysRevE.53.3101. |
[14] |
A. Kaminaga, V. K. Vanag and I. R. Epstein, A reaction-diffusion memory device, Angewandte chemie, 45 (2006), 3087-3089.
doi: 10.1002/anie.200600400. |
[15] |
A. Kaminaga, V. K. Vanag and I. R. Epstein, Black spots in a surfactant-rich Belousov-Zhabotinsky reaction dispersed in a water-in-oil microemulsion system, J. Chem. Phys., 122 (2005), 061747.
doi: 10.1063/1.1888386. |
[16] |
T. Kolokolnikov and M. Tlidi, Spot deformation and replication in the two-dimensional Belousov-Zhabotinski reaction in a water-in-oil microemulsion system, Phys. Rev. Lett., 98 (2007), 188-303.
doi: 10.1103/PhysRevLett.98.188303. |
[17] |
T. Kolokolnikov, T. Erneux and J. Wei, Mesa-type patterns in the one-dimensional Brusselator and their stability, Physica D, 214 (2006), 63-77.
doi: 10.1016/j.physd.2005.12.005. |
[18] |
H. Meinhardt, "Models of Biological Pattern Formation," Academic Press, London, 1982. |
[19] |
H. Meinhardt, "The Algorithmic Beauty of Sea Shells," Springer-verlag, Berlin, 1995. |
[20] |
A. J. Koch and H. Meinhardt, Biological pattern formation: From basic mechanisms to complex structures, Rev. Modern Physics, 66 (1994), 1481-1507.
doi: 10.1103/RevModPhys.66.1481. |
[21] |
T. Kolokolnikov, M. Ward, W. Sun and J. Wei, The stability of a stripe for the Gierer-Meinhardt model and the effect of saturation, SIAM J. Appl. Dyn. Systems, 5 (2006), 313-363.
doi: 10.1137/050635080. |
[22] |
I. Lengyel and I. R. Epstein, Modeling of Turing structures in the chlorite-iodide-malonic acid-starch reaction system, Science, 251 (1991), 650-652.
doi: 10.1126/science.251.4994.650. |
[23] |
W. Ni and M. Tang, Turing patterns in the Lengyel-Epstein system for the CIMA reaction, Trans. AMS, 357 (2005), 3953-3969.
doi: 10.1090/S0002-9947-05-04010-9. |
[24] |
M. Mimura, Y. Nishiura, A. Tesei and T. Tsujikawa, Coexistence problem for two competing species models with density-dependent diffusion, Hiroshima Math J., 14 (1984), 425-449. |
[25] |
E. Meron, H. Yizhaq and E. Gilad, Localized structures in dryland vegetation: Forms and functions, Chaos, 17 (2007), 037109.
doi: 10.1063/1.2767246. |
[26] |
T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[27] |
R. Choksi and X. Ren, On the derivation of a density functional theory for microphase separation of diblock copolymers, J. Stat. Phys., 113 (2003), 151-176.
doi: 10.1023/A:1025722804873. |
[28] |
R. McKay and T. Kolokolnikov, Interface oscillations in reaction-diffusion systems beyond the Hopf bifurcation, submitted. |
[29] |
, FlexPDE software. Available from: http://www.pdesolutions.com. |
[30] |
U. Ascher, R. Christiansen and R. Russell, Collocation software for boundary value ode's, Math. Comp., 33 (1979), 659-579.
doi: 10.1090/S0025-5718-1979-0521281-7. |
[31] |
D. Iron, M. J. Ward and J. Wei, The stability of spike solutions to the one-dimensional Gierer-Meinhardt model, Physica D, 150 (2001), 25-62.
doi: 10.1016/S0167-2789(00)00206-2. |
[32] |
H. van der Ploeg and A. Doelman, Stability of spatially periodic pulse patterns in a class of singularly perturbed reaction-diffusion equations, Indiana Univ. Math. J., 54 (2005), 1219-1301.
doi: 10.1512/iumj.2005.54.2792. |
[33] |
K. B. Glasner and T. P. Witelski, Collision versus collapse of droplets in coarsening of dewetting thin films, Physica D, 209 (2005), 80-104. |
[34] |
F. Otto, T. Rump and D. Slepčev, Coarsening rates for a droplet model: Rigorous upper bounds, SIAM J. Math. Analysis, 38 (2007), 503-529.
doi: 10.1137/050630192. |
[35] |
K. Promislow, A renormalization method for modulational stability of quasi-steady patterns in dispersive systems, SIAM J. Math. Anal., 33 (2002), 1455-1482.
doi: 10.1137/S0036141000377547. |
[36] |
A. Doelman, T. J. Kaper and K. Promislow, Nonlinear asymptotic stability of the semistrong pulse dynamics in a regularized Gierer-Meinhardt model, SIAM J. Math. Anal., 38 (2007), 1760-1787.
doi: 10.1137/050646883. |
[37] |
J. Wei and M. Winter, Multi-peak solutions for a wide class of singular perturbation problems, J. London Math. Soc., 59 (1999), 585-606.
doi: 10.1112/S002461079900719X. |
[38] |
C. Muratov, Theory of domain patterns in systems with long-range interactions of Coulomb type, Phys. Rev. E., 66 (2002), 066108, 25 pp. |
[39] |
T. Kolokolnikov and M. J. Ward, Bifurcation of spike equilibria in the near-shadow Gierer-Meinhardt model, Discrete and Continuous Dynamical Systems B, 4 (2004), 1033-1064.
doi: 10.3934/dcdsb.2004.4.1033. |
show all references
References:
[1] |
Y. Nishiura and H. Fujii, Stability of singularly perturbed solutions to systems of reaction-diffusion equations, SIAM J. Math. Anal., 18 (1987), 1726-1770.
doi: 10.1137/0518124. |
[2] |
Y. Nishiura and M. Mimura, Layer oscillations in reaction-diffusion systems, SIAM J.Appl. Math., 49 (1989), 481-514.
doi: 10.1137/0149029. |
[3] |
C. B. Muratov and V. Osipov, Scenarios of domain pattern formation in a reaction-diffusion system, Phys. Rev. E., 54 (1996), 4860-4879.
doi: 10.1103/PhysRevE.54.4860. |
[4] |
J. Rubinstein, P. Sternberg and J. B. Keller, Fast reaction, slow diffusion, and curve shortening, SIAM J. Appl. Math., 49 (1989), 116-133.
doi: 10.1137/0149007. |
[5] |
R. E. Goldstein, D. J. Muraki and D. M. Petrich, Interface proliferation and the growth of labyrinths in a reaction-diffusion system, Phys. Rev. E., 53 (1996), 3933-3957.
doi: 10.1103/PhysRevE.53.3933. |
[6] |
G. Nicolis and I. Prigogine, "Self-organization in Non-Equilibrium Systems," Wiley Interscience, New York, 1977. |
[7] |
J. D. Murray, "Mathematical Biology," Springer-Verlag, Berlin, 1989. |
[8] |
B. S. Kerner and V. V. Osipov, "Autosolitons: A New Approach to Problems of Self-Organization and Turbulence," Kluwer, Dordrecht, 1994. |
[9] |
R. Kapral and K. Showalter, eds., "Chemical Waves and Patterns," Kluwer, Dordrecht, 1995. |
[10] |
M. Taki, M. Tlidi and T. Kolokolnikov, eds., "Dissipative Localized Structures in Extended Systems," Chaos Focus issue, 17, 2007. |
[11] |
T. Kolokolnikov, M. Ward and J. Wei, Self-replication of mesa patterns in reaction-diffusion models, Physica D, 236 (2007), 104-122.
doi: 10.1016/j.physd.2007.07.014. |
[12] |
W.-M. Ni, P. Poláčik and E. Yanagida, Monotonicity of stable solutions in shadow systems, Trans. A.M.S., 352 (2001), 5057-5069.
doi: 10.1090/S0002-9947-01-02880-X. |
[13] |
C. Muratov and V. V. Osipov, General theory of instabilities for patterns with sharp interfaces in reaction-diffusion systems, Phys, Rev. E., 53 (1996), 3101-3116.
doi: 10.1103/PhysRevE.53.3101. |
[14] |
A. Kaminaga, V. K. Vanag and I. R. Epstein, A reaction-diffusion memory device, Angewandte chemie, 45 (2006), 3087-3089.
doi: 10.1002/anie.200600400. |
[15] |
A. Kaminaga, V. K. Vanag and I. R. Epstein, Black spots in a surfactant-rich Belousov-Zhabotinsky reaction dispersed in a water-in-oil microemulsion system, J. Chem. Phys., 122 (2005), 061747.
doi: 10.1063/1.1888386. |
[16] |
T. Kolokolnikov and M. Tlidi, Spot deformation and replication in the two-dimensional Belousov-Zhabotinski reaction in a water-in-oil microemulsion system, Phys. Rev. Lett., 98 (2007), 188-303.
doi: 10.1103/PhysRevLett.98.188303. |
[17] |
T. Kolokolnikov, T. Erneux and J. Wei, Mesa-type patterns in the one-dimensional Brusselator and their stability, Physica D, 214 (2006), 63-77.
doi: 10.1016/j.physd.2005.12.005. |
[18] |
H. Meinhardt, "Models of Biological Pattern Formation," Academic Press, London, 1982. |
[19] |
H. Meinhardt, "The Algorithmic Beauty of Sea Shells," Springer-verlag, Berlin, 1995. |
[20] |
A. J. Koch and H. Meinhardt, Biological pattern formation: From basic mechanisms to complex structures, Rev. Modern Physics, 66 (1994), 1481-1507.
doi: 10.1103/RevModPhys.66.1481. |
[21] |
T. Kolokolnikov, M. Ward, W. Sun and J. Wei, The stability of a stripe for the Gierer-Meinhardt model and the effect of saturation, SIAM J. Appl. Dyn. Systems, 5 (2006), 313-363.
doi: 10.1137/050635080. |
[22] |
I. Lengyel and I. R. Epstein, Modeling of Turing structures in the chlorite-iodide-malonic acid-starch reaction system, Science, 251 (1991), 650-652.
doi: 10.1126/science.251.4994.650. |
[23] |
W. Ni and M. Tang, Turing patterns in the Lengyel-Epstein system for the CIMA reaction, Trans. AMS, 357 (2005), 3953-3969.
doi: 10.1090/S0002-9947-05-04010-9. |
[24] |
M. Mimura, Y. Nishiura, A. Tesei and T. Tsujikawa, Coexistence problem for two competing species models with density-dependent diffusion, Hiroshima Math J., 14 (1984), 425-449. |
[25] |
E. Meron, H. Yizhaq and E. Gilad, Localized structures in dryland vegetation: Forms and functions, Chaos, 17 (2007), 037109.
doi: 10.1063/1.2767246. |
[26] |
T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[27] |
R. Choksi and X. Ren, On the derivation of a density functional theory for microphase separation of diblock copolymers, J. Stat. Phys., 113 (2003), 151-176.
doi: 10.1023/A:1025722804873. |
[28] |
R. McKay and T. Kolokolnikov, Interface oscillations in reaction-diffusion systems beyond the Hopf bifurcation, submitted. |
[29] |
, FlexPDE software. Available from: http://www.pdesolutions.com. |
[30] |
U. Ascher, R. Christiansen and R. Russell, Collocation software for boundary value ode's, Math. Comp., 33 (1979), 659-579.
doi: 10.1090/S0025-5718-1979-0521281-7. |
[31] |
D. Iron, M. J. Ward and J. Wei, The stability of spike solutions to the one-dimensional Gierer-Meinhardt model, Physica D, 150 (2001), 25-62.
doi: 10.1016/S0167-2789(00)00206-2. |
[32] |
H. van der Ploeg and A. Doelman, Stability of spatially periodic pulse patterns in a class of singularly perturbed reaction-diffusion equations, Indiana Univ. Math. J., 54 (2005), 1219-1301.
doi: 10.1512/iumj.2005.54.2792. |
[33] |
K. B. Glasner and T. P. Witelski, Collision versus collapse of droplets in coarsening of dewetting thin films, Physica D, 209 (2005), 80-104. |
[34] |
F. Otto, T. Rump and D. Slepčev, Coarsening rates for a droplet model: Rigorous upper bounds, SIAM J. Math. Analysis, 38 (2007), 503-529.
doi: 10.1137/050630192. |
[35] |
K. Promislow, A renormalization method for modulational stability of quasi-steady patterns in dispersive systems, SIAM J. Math. Anal., 33 (2002), 1455-1482.
doi: 10.1137/S0036141000377547. |
[36] |
A. Doelman, T. J. Kaper and K. Promislow, Nonlinear asymptotic stability of the semistrong pulse dynamics in a regularized Gierer-Meinhardt model, SIAM J. Math. Anal., 38 (2007), 1760-1787.
doi: 10.1137/050646883. |
[37] |
J. Wei and M. Winter, Multi-peak solutions for a wide class of singular perturbation problems, J. London Math. Soc., 59 (1999), 585-606.
doi: 10.1112/S002461079900719X. |
[38] |
C. Muratov, Theory of domain patterns in systems with long-range interactions of Coulomb type, Phys. Rev. E., 66 (2002), 066108, 25 pp. |
[39] |
T. Kolokolnikov and M. J. Ward, Bifurcation of spike equilibria in the near-shadow Gierer-Meinhardt model, Discrete and Continuous Dynamical Systems B, 4 (2004), 1033-1064.
doi: 10.3934/dcdsb.2004.4.1033. |
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