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Stable spike clusters for the one-dimensional Gierer–Meinhardt system

Published online by Cambridge University Press:  08 November 2016

JUNCHENG WEI  and
MATTHIAS WINTER
JUNCHENG WEI
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, V6T 1Z2, B.C., Canada email: jcwei@math.ubc.ca
MATTHIAS WINTER
Affiliation:
Department of Mathematics, Brunel University London, Uxbridge UB8 3PH, UK email: matthias.winter@brunel.ac.uk
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Abstract

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We consider the Gierer–Meinhardt system with precursor inhomogeneity and two small diffusivities in an interval

$$\begin{equation*} \left\{ \begin{array}{ll} A_t=\epsilon^2 A''- \mu(x) A+\frac{A^2}{H}, &x\in(-1, 1),\,t>0,\\[3mm] \tau H_t=D H'' -H+ A^2, & x\in (-1, 1),\,t>0,\\[3mm] A' (-1)= A' (1)= H' (-1) = H' (1) =0, \end{array} \right. \end{equation*}$$
$$\begin{equation*}\mbox{where } \quad 0<\epsilon \ll\sqrt{D}\ll 1, \quad \end{equation*}$$
$$\begin{equation*} \tau\geq 0 \mbox{ and $\tau$ is independent of $\epsilon$. } \end{equation*}$$
 A spike cluster is the combination of several spikes which all approach the same point in the singular limit. We rigorously prove the existence of a steady-state spike cluster consisting of N spikes near a non-degenerate local minimum point t0 of the smooth positive inhomogeneity μ(x), i.e. we assume that μ′(t0) = 0, μ″(t0) > 0 and we have μ(t0) > 0. Here, N is an arbitrary positive integer. Further, we show that this solution is linearly stable. We explicitly compute all eigenvalues, both large (of order O(1)) and small (of order o(1)). The main features of studying the Gierer–Meinhardt system in this setting are as follows: (i) it is biologically relevant since it models a hierarchical process (pattern formation of small-scale structures induced by a pre-existing large-scale inhomogeneity); (ii) it contains three different spatial scales two of which are small: the O(1) scale of the precursor inhomogeneity μ(x), the $O(\sqrt{D})$ scale of the inhibitor diffusivity and the O(ε) scale of the activator diffusivity; (iii) the expressions can be made explicit and often have a particularly simple form.

Keywords

35B3535J7535K5735K5892C15
Type
Papers
Information
European Journal of Applied Mathematics , Volume 28 , Issue 4 , August 2017 , pp. 576 - 635
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited
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Copyright © Cambridge University Press 2016

References

[1] Alberti, G. & Müller, S. (2001) A new approach to variational problems with multiple scales. Commun. Pure Appl. Math. 54 (7), 761–825.CrossRefGoogle Scholar
[2] Benson, D. L., Maini, P. K. & Sherratt, J. A. (1998) Unravelling the turing bifurcation using spatially varying diffusion coefficients. J. Math. Biol. 37 (5), 381–417.CrossRefGoogle Scholar
[3] Bode, P. M., Awad, T. A., Koizumi, O., Nakashima, Y., Grimmelikhuijzen, C. J. P. & Bode, H. R. (1988) Development of the two-part pattern during regeneration of the head in hydra . Development 102, 223–235.Google Scholar
[4] Broun, M., Gee, L., Reinhardt, B. & Bode, H. R. (2005) Formation of the head organizer in hydra involves the canonical Wnt pathway . Development 132, 2907–2916.CrossRefGoogle ScholarPubMed
[5] Coen, E. S. & Meyerowitz, E. M. (1991) The war of the whorls: Genetic interactions controlling flower development. Nature 353 (6339), 31–37.Google Scholar
[6] Dancer, E. N. (2001) On stability and Hopf bifurcations for chemotaxis systems. Methods Appl. Anal. 8 (2), 245–256.Google Scholar
[7] Doelman, A., Gardner, R. A. & Kaper, T. J. (2001) Large stable pulse solutions in reaction-diffusion equations. Indiana Univ. Math. J. 49 (1), 443–507.Google Scholar
[8] Doelman, A., Kaper, T. J. & van der Ploeg, H. (2001) Spatially periodic and aperiodic multi-pulse patterns in the one-dimensional Gierer-Meinhardt equation. Methods Appl. Anal. 8 (3), 387–414.Google Scholar
[9] Ei, S.-I. & Wei, J. (2002) Dynamics of metastable localized patterns and its application to the interaction of spike solutions for the Gierer-Meinhardt systems in two spatial dimensions. Japan J. Ind. Appl. Math. 19 (2), 181–226.Google Scholar
[10] Floer, A. & Weinstein, A. (1986) Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal. 69 (3), 397–408.Google Scholar
[11] Gierer, A. & Meinhardt, H. (1972) A theory of biological pattern formation, Kybernetik (Berlin) 12 (1), 30–39.CrossRefGoogle ScholarPubMed
[12] Gui, C. & Wei, J. (1999) Multiple interior peak solutions for some singular perturbation problems. J. Differ. Equ. 158 (1), 1–27.Google Scholar
[13] Gui, C., Wei, J. & Winter, M. (2000) Multiple boundary peak solutions for some singularly perturbed Neumann problems. Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (1), 47–82.Google Scholar
[14] Herschowitz-Kaufman, M. (1975) Bifurcation analysis of nonlinear reaction-diffusion equations II, steady-state solutions and comparison with numerical simulations. Bull. Math. Biol. 37 (6), 589–636.Google Scholar
[15] Iron, D., Ward, M. J. & Wei, J. (2001) The stability of spike solutions to the one-dimensional Gierer-Meinhardt model. Physica D. 50 (1–2), 25–62.Google Scholar
[16] Meinhardt, H. (1982) Models of Biological Pattern Formation, Academic Press, London.Google Scholar
[17] Meinhardt, H. (1993) A model for pattern-formation of hypostome, tentacles, & foot in hydra: How to form structures close to each other, how to form them at a distance. Dev. Biol. 157 (2), 321–333.CrossRefGoogle Scholar
[18] Meinhardt, H. (1998) The Algorithmic Beauty of Sea Shells, 2nd ed. Springer, Berlin, Heidelberg.Google Scholar
[19] Meinhardt, H. (2012) Modeling pattern formation in hydra: A route to understanding essential steps in development. Int. J. Dev. Biol. 56 (6–8), 447–462.Google Scholar
[20] Müller, S. (1993) Singular perturbations as a selection criterion for periodic minimizing sequences. Calc. Var. Partial Differ. Equ. 1 (2), 169–204.Google Scholar
[21] Ni, W.-M. & Takagi, I. (1991) On the shape of least energy solution to a semilinear Neumann problem. Commun. Pure Appl. Math. 41 (7), 819–851.Google Scholar
[22] Ni, W.-M. & Takagi, I. (1993) Locating the peaks of least energy solutions to a semilinear Neumann problem. Duke Math. J. 70 (2), 247–281.CrossRefGoogle Scholar
[23] Nishiura, Y. (1995) Coexistence of infinitely many stable solutions to reaction-diffusion equation in the singular limit. In: Jones, C. K. R. T. & Kirchgraber, U. (editors), Dynamics Reported: Expositions in Dynamical Systems, Vol. 3, Springer–Verlag, New York, pp. 25–103.Google Scholar
[24] Nishiura, Y., Teramoto, T., Yuan, X. & Ueda, K. (2007) Dynamics of traveling pulses in heterogeneous media. Chaos 17 (3), 037104.Google Scholar
[25] Oh, Y. G. (1988) Existence of semi-classical bound states of nonlinear Schrödinger equations with potentials of the class (V) a . Comm. PDE 13 (12), 1499–1519.Google Scholar
[26] Oh, Y. G. (1990) On positive multi-bump bound states of nonlinear Schrödinger equations under multiple-well potentials. Comm. Math. Phys. 131 (2), 223–253.Google Scholar
[27] Page, K., Maini, P. K. & Monk, N. A. M. (2003) Pattern formation in spatially heterogeneous turing reaction-diffusion models. Phys. D 181 (1–2), 80–101.Google Scholar
[28] Page, K., Maini, P. K. & Monk, N. A. M. (2005) Complex pattern formation in reaction-diffusion systems with spatially varying parameters. Phys. D 202 (1–2), 95–115.Google Scholar
[29] Sandstede, B. (1998) Stability of multiple-pulse solutions, Trans. Am. Math. Soc. 350 (2), 429–472.CrossRefGoogle Scholar
[30] Takagi, I. (1986) Point-condensation for a reaction-diffusion system. J. Differ. Equ. 61 (2), 208–249.Google Scholar
[31] Turing, A. M. (1952) The chemical basis of morphogenesis. Phil. Trans. Roy. Soc. Lond. B 237 (641), 37–72.Google Scholar
[32] Ward, M. J., McInerney, D., Houston, P., Gavaghan, D. & Maini, P. K. (2002) The dynamics and pinning of a spike for a reaction-diffusion system. SIAM J. Appl. Math. 62 (4), 1297–1328.Google Scholar
[33] Ward, M. J. & Wei, J. (2002) Asymmetric spike patterns for the one-dimensional Gierer-Meinhardt model: Equilibria and stability, Eur. J. Appl. Math. 13 (3), 283–320.Google Scholar
[34] Ward, M. J. & Wei, J. (2003) Hopf bifurcation of spike solutions for the shadow Gierer-Meinhardt system. Eur. J. Appl. Math. 14 (6), 677–711.CrossRefGoogle Scholar
[35] Ward, M. J. & Wei, J. (2003) Hopf bifurcations and oscillatory instabilities of solutions for the one-dimensional Gierer-Meinhardt model. J. Nonlinear Sci. 13 (2), 209–264.Google Scholar
[36] Wei, J. (1999) On single interior spike solutions of Gierer-Meinhardt system: Uniqueness and spectrum estimates. Eur. J. Appl. Math. 10 (4), 353–378.CrossRefGoogle Scholar
[37] Wei, J. & Winter, M. (1998) Stationary solutions for the Cahn-Hilliard equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (4), 459–492.CrossRefGoogle Scholar
[38] Wei, J. & Winter, M. (1999) On the two-dimensional Gierer-Meinhardt system with strong coupling, SIAM J. Math. Anal. 30 (6), 1241–1263.CrossRefGoogle Scholar
[39] Wei, J. & Winter, M. (2002) Spikes for the two-dimensional Gierer-Meinhardt system: The strong coupling case. J. Differ. Equ. 178 (2), 478–518.Google Scholar
[40] Wei, J. & Winter, M. (2001) Spikes for the two-dimensional Gierer-Meinhardt system: The weak coupling case. J. Nonlinear Sci. 6 (6), 415–458.CrossRefGoogle Scholar
[41] Wei, J. & Winter, M. (2004) Existence and stability analysis of asymmetric patterns for the Gierer-Meinhardt system, J. Math. Pures Appl. 83 (4), 433–476.Google Scholar
[42] Wei, J. & Winter, M. (2007) Symmetric and asymmetric multiple clusters in a reaction-diffusion system. NoDEA Nonlinear Differ. Equ. Appl. 14 (5–6), 787–823.CrossRefGoogle Scholar
[43] Wei, J. & Winter, M. (2007) Existence, classification and stability analysis of multiple-peaked solutions for the Gierer-Meinhardt system in ℝ. Methods Appl. Anal. 14 (2), 119–163.CrossRefGoogle Scholar
[44] Wei, J. & Winter, M. (2009) Spikes for the Gierer-Meinhardt system with discontinuous diffusion coefficients. J. Nonlinear Sci. 19 (3), 301–339.Google Scholar
[45] Wei, J. & Winter, M. (2009) On the Gierer-Meinhardt system with precursors. Discrete Contin. Dyn. Syst. 25 (1), 363–398.Google Scholar
[46] Wei, J. & Winter, M. (2014) Stability of cluster solutions in a cooperative consumer chain model. J. Math. Biol. 68 (1–2), 1–39.Google Scholar
[47] Wei, J. & Winter, M. (2014) Mathematical Aspects of Pattern Formation in Biological Systems, Applied Mathematical Sciences, Vol. 189, Springer, London.CrossRefGoogle Scholar
[48] Yip, N. K. (2006) Structure of stable solutions of a one-dimensional variational problem. ESAIM Control Optim. Calc. Var. 12 (4), 721–751.Google Scholar
[49] Yuan, X., Teramoto, T. & Nishiura, Y. (2007) Heterogeneity-induced defect bifurcation and pulse dynamics for a three-component reaction-diffusion system. Phys. Rev. E 75 (3), 036220.CrossRefGoogle ScholarPubMed