American Institute of Mathematical Sciences

January  2012, 17(1): 191-220. doi: 10.3934/dcdsb.2012.17.191

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

Received  December 2010 Revised  March 2011 Published  October 2011

We consider a class of one-dimensional reaction-diffusion systems, $\left\{ \begin{array} [ u_{t}=\varepsilon^{2}u_{xx}+f(u,w)\\ \tau w_{t}=Dw_{xx}+g(u,w) \end{array} \right.$ with homogeneous Neumann boundary conditions on a one dimensional interval. Under some generic conditions on the nonlinearities $f,g$ and in the singular limit $\varepsilon\rightarrow0,$ such a system admits a steady state for which $u$ consists of sharp back-to-back interfaces. For a sufficiently large $D$ and for sufficiently small $\tau$, such a steady state is known to be stable in time. On the other hand, it is also known that in the so-called shadow limit $D\rightarrow\infty,$ patterns having more than one interface are unstable. In this paper we analyse in detail the transition between the stable patterns when $D=O(1)$ and the shadow system when $D\rightarrow\infty$. We show that this transition occurs when $D$ is exponentially large in $\varepsilon$ and we derive instability thresholds $D_{1}\gg D_{2}\gg D_{3}\gg\ldots$ such that a periodic pattern with $2K$ interfaces is stable if $D < D_{K}$ and is unstable when $D > D_{K}$. We also study the dynamics of the interfaces when $D$ is exponentially large; this allows us to describe in detail the mechanism leading to the instability. Direct numerical computations of stability and dynamics are performed, and these results are in excellent agreement with corresponding results as predicted by the asymptotic theory.
Citation: Rebecca McKay, Theodore Kolokolnikov. Stability transitions and dynamics of mesa patterns near the shadow limit of reaction-diffusion systems in one space dimension. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 191-220. doi: 10.3934/dcdsb.2012.17.191
References:
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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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [27] R. Choksi and X. 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J., 54 (2005), 1219-1301. doi: 10.1512/iumj.2005.54.2792.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [38] C. Muratov, Theory of domain patterns in systems with long-range interactions of Coulomb type, Phys. Rev. E., 66 (2002), 066108, 25 pp.  Google Scholar [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.  Google Scholar

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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.  Google Scholar [2] Y. Nishiura and M. Mimura, Layer oscillations in reaction-diffusion systems, SIAM J.Appl. Math., 49 (1989), 481-514. doi: 10.1137/0149029.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] G. Nicolis and I. Prigogine, "Self-organization in Non-Equilibrium Systems," Wiley Interscience, New York, 1977. Google Scholar [7] J. D. Murray, "Mathematical Biology," Springer-Verlag, Berlin, 1989. Google Scholar [8] B. S. Kerner and V. V. Osipov, "Autosolitons: A New Approach to Problems of Self-Organization and Turbulence," Kluwer, Dordrecht, 1994. Google Scholar [9] R. Kapral and K. Showalter, eds., "Chemical Waves and Patterns," Kluwer, Dordrecht, 1995. Google Scholar [10] M. Taki, M. Tlidi and T. Kolokolnikov, eds., "Dissipative Localized Structures in Extended Systems," Chaos Focus issue, 17, 2007. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] H. Meinhardt, "Models of Biological Pattern Formation," Academic Press, London, 1982. Google Scholar [19] H. Meinhardt, "The Algorithmic Beauty of Sea Shells," Springer-verlag, Berlin, 1995. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [28] R. McKay and T. Kolokolnikov, Interface oscillations in reaction-diffusion systems beyond the Hopf bifurcation,, submitted., ().   Google Scholar [29] , FlexPDE software., Available from: \url{http://www.pdesolutions.com}., ().   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [38] C. Muratov, Theory of domain patterns in systems with long-range interactions of Coulomb type, Phys. Rev. E., 66 (2002), 066108, 25 pp.  Google Scholar [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.  Google Scholar
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