# American Institute of Mathematical Sciences

November  2012, 32(11): 3975-4000. doi: 10.3934/dcds.2012.32.3975

## Characterization of turing diffusion-driven instability on evolving domains

 1 Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, United States 2 University of Sussex, School of Mathematical and Physical Sciences, Pevensey III, 5C15, Brighton, BN1 9QH, United Kingdom 3 Department of Mathematics and Statistics, Auburn University, Auburn University, AL 36849-5310

Received  May 2011 Revised  November 2011 Published  June 2012

In this paper we establish a general theoretical framework for Turing diffusion-driven instability for reaction-diffusion systems on time-dependent evolving domains. The main result is that Turing diffusion-driven instability for reaction-diffusion systems on evolving domains is characterised by Lyapunov exponents of the evolution family associated with the linearised system (obtained by linearising the original system along a spatially independent solution). This framework allows for the inclusion of the analysis of the long-time behavior of the solutions of reaction-diffusion systems. Applications to two special types of evolving domains are considered: (i) time-dependent domains which evolve to a final limiting fixed domain and (ii) time-dependent domains which are eventually time periodic. Reaction-diffusion systems have been widely proposed as plausible mechanisms for pattern formation in morphogenesis.
Citation: Georg Hetzer, Anotida Madzvamuse, Wenxian Shen. Characterization of turing diffusion-driven instability on evolving domains. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 3975-4000. doi: 10.3934/dcds.2012.32.3975
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Turing, The chemical basis of morphogenesis,, Phil. Trans. R. Soc. Lond. B, 237 (1952), 37. doi: 10.1098/rstb.1952.0012. Google Scholar [39] C. Varea, J. L. Aragón and R. A. Barrio, Confined Turing patterns in growing systems,, Phys. Rev. E., 60 (1999), 4588. doi: 10.1103/PhysRevE.60.4588. Google Scholar [40] C. Venkataraman, O. Lakkis and A. Madzvamuse, Global existence for semilinear reaction-diffusion systems on evolving domains,, J. Math. Biol., 64 (2010), 41. doi: 10.1007/s00285-011-0404-x. Google Scholar [41] P. Weidemaier, Local existence for parabolic problems with fully nonlinear boundary conditions; an $L_p$-approach,, Annali di Matematica Pura and Applicata (4), 160 (1991), 207. doi: 10.1007/BF01764128. Google Scholar

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##### References:
 [1] H. Amann, Existence and regularity for semilinear parabolic evolution equations,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 11 (1984), 593. Google Scholar [2] V. Castets, E. Dulos, J. Boissonade and P. De Kepper, Experimental evidence of a sustained Turing-type equilibrium chemical pattern,, Phys. Rev. Lett., 64 (1990), 2953. doi: 10.1103/PhysRevLett.64.2953. Google Scholar [3] M. Chaplain, A. J. Ganesh and I. G. Graham, Spatio-temporal pattern formation on spherical surfaces:Numerical simulation and application to solid tumor growth,, J. Math. Biol., 42 (2001), 387. doi: 10.1007/s002850000067. Google Scholar [4] C. Chicone and Y. Latushkin, "Evolution Semigroups in Dynamical Systems and Differential Equations,'', Mathematical Surveys and Monographs, 70 (1999). Google Scholar [5] E. J. Crampin, E. A. Gaffney and P. K. Maini, Reaction and diffusion on growing domains: Scenarios for robust pattern formation,, Bull. Math. Biol., 61 (1999), 1093. doi: 10.1006/bulm.1999.0131. Google Scholar [6] E. J. Crampin, W. W. Hackborn and P. K. Maini, Pattern formation in reaction-diffusion models with nonuniform domain growth,, Bull. Math. Biol., 64 (2002), 746. doi: 10.1006/bulm.2002.0295. Google Scholar [7] D. Daners, Perturbation of semi-linear evolution equations under weak assumptions at initial time,, J. Diff. Eq., 210 (2005), 352. doi: 10.1016/j.jde.2004.08.004. Google Scholar [8] R. Denk, M. Hieber and J. Prüss, "$R$-Boundedness, Fourier Multipliers and Problems of Elliptic and Parabolic Type,'', Mem. Amer. Math. Soc., 166 (2003). Google Scholar [9] R. Denk, M. Hieber and J. Prüss, "Optimal Lp-Lq-Regularity for Parabolic Problems with Inhomogeneous Boundary Data,'', Konstanzer Schriften in Mathematik und Informatik, (2005). Google Scholar [10] L. Edelstein-Keshet, "Mathematical Models in Biology,'', The Random House/Birkh\, (1988). Google Scholar [11] A. Gierer and H. Meinhardt, A theory of biological pattern formation,, Kybernetik, 12 (1972), 30. doi: 10.1007/BF00289234. Google Scholar [12] G. H. Golub and C. F. Van Loan, "Matrix Computations,'', JHU Press, (1996). Google Scholar [13] S. Kondo and R. Asai, A reaction-diffusion wave on the skin of the marine anglefish, Pomacanthus,, Nature, 376 (1995), 765. doi: 10.1038/376765a0. Google Scholar [14] S. S. Liaw, C. C. Yang, R. T. Liu and J. T. Hong, Turing model for the patterns of lady beetles,, Phys. Rev. E., 64 (2001), 041909. doi: 10.1103/PhysRevE.64.041909. Google Scholar [15] Y. Latushkin, J. Prüss and R. Schnaubelt, Stable and unstable manifolds for quasilinear parabolic systems with fully nonlinear boundary conditions,, J. Evol. Eq., 6 (2006), 537. Google Scholar [16] A. Madzvamuse, E. A. Gaffney and P. K. Maini, Stability analysis of non-autonomous reaction-diffusion systems: The effects of growing domains,, J. Math. Biol., 61 (2010), 133. doi: 10.1007/s00285-009-0293-4. Google Scholar [17] A. Madzvamuse, Turing instability conditions for growing domains with divergence free mesh velocity,, Nonlinear Analysis: Theory, (2009). Google Scholar [18] A. Madzvamuse, Stability analysis of reaction-diffusion systems with constant coefficients on growing domains,, Int J. of Dynamical and Differential Equations, 1 (2008), 250. Google Scholar [19] A. Madzvamuse and P. K. Maini, Velocity-induced numerical solutions of reaction-diffusion systems on fixed and growing domains,, J. Comp. Phys., 225 (2007), 100. doi: 10.1016/j.jcp.2006.11.022. Google Scholar [20] A. Madzvamuse, Time-stepping schemes for moving grid finite elements applied to reaction-diffusion systems on fixed and growing domains,, J. Comp. Phys., 214 (2006), 239. doi: 10.1016/j.jcp.2005.09.012. Google Scholar [21] A. Madzvamuse, A. J. Wathen and P. K. Maini, A moving grid finite element method for the simulation of pattern generation by Turing models on growing domains,, J. Sci. Comp., 24 (2005), 247. doi: 10.1007/s10915-004-4617-7. Google Scholar [22] A. Madzvamuse, P. K. Maini and A. J. Wathen, A moving grid finite element method applied to a model biological pattern generator,, J. Comp. Phys., 190 (2003), 478. doi: 10.1016/S0021-9991(03)00294-8. Google Scholar [23] P. K. Maini, R. E. Baker and C. M. Chong, The Turing model comes of molecular age, (Invited Perspective), Science, 314 (2006), 1397. doi: 10.1126/science.1136396. Google Scholar [24] J. Mierczynski and W. Shen, Spectral theory for forward nonautonomous parabolic equations and applications,, to appear in Fields Inst. Commun., (). Google Scholar [25] J. Mierczynski and W. Shen, Persistence in forward nonautonomous competitive systems of parabolic equations,, submitted., (). Google Scholar [26] J. Mierczynksi and W. Shen, "Spectral Theory for Random and Nonautonomous Parabolic Equations and Applications,'', Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 139 (2008). Google Scholar [27] J. D. Murray, "Mathematical Biology I and II,'' 3$^rd$ edition,, Springer-Verlag, (2002). Google Scholar [28] K. J. Painter, H. G. Othmer and P. K. Maini, Stripe formation in juvenile Pomacanthus explained by a generalized Turing mechanism with chemotaxis,, Proc. Natl. Acad. Sci., 96 (1999). doi: 10.1073/pnas.96.10.5549. Google Scholar [29] R. G. Plaza, F. Sánchez-Garduño, P. Padilla, R. A. Barrio, and P. K. Maini, The effect of growth and curvature on pattern formation,, J. Dynam. and Diff. Eqs., 16 (2004), 1093. doi: 10.1007/s10884-004-7834-8. Google Scholar [30] I. Prigogine and R. Lefever, Symmetry breaking instabilities in dissipative systems II,, J. Chem. Phys., 48 (1968), 1695. doi: 10.1063/1.1668896. Google Scholar [31] A. M. Oster and P. C. Bressloff, A developmental model of ocular dominance column formation on a growing cortex,, Bull. of Math. Biol., 68 (2006), 73. doi: 10.1007/s11538-005-9055-7. Google Scholar [32] Q. Ouyang and H. L. Swinney, Transition from a uniform state to hexagonal and striped Turing patterns,, Nature, 352 (1991), 610. doi: 10.1038/352610a0. Google Scholar [33] J. Schnakenberg, Simple chemical reaction systems with limit cycle behaviour,, J. Theor. Biol., 81 (1979), 389. doi: 10.1016/0022-5193(79)90042-0. Google Scholar [34] R. Schnaubelt, Asymptotic behavior of parabolic nonautonomous evolution equations,, in, 1855 (2004), 401. Google Scholar [35] Y. Shiferaw and A. Karma, Turing instability mediated by voltage and calcium diffusion in paced cardiac cells,, PNAS, 103 (2006), 5670. doi: 10.1073/pnas.0511061103. Google Scholar [36] S. Sick, S. Reinker, J. Timmer and T. Schlake, WNT and DKK determine hair follicle spacing through a reaction-diffusion mechanism,, Science, 314 (2006), 1447. doi: 10.1126/science.1130088. Google Scholar [37] L. Solnica-Krezel, Vertebrate development: Taming the nodal waves,, Curr Biol., 13 (2003), 7. doi: 10.1016/S0960-9822(02)01378-7. Google Scholar [38] A. Turing, The chemical basis of morphogenesis,, Phil. Trans. R. Soc. Lond. B, 237 (1952), 37. doi: 10.1098/rstb.1952.0012. Google Scholar [39] C. Varea, J. L. Aragón and R. A. Barrio, Confined Turing patterns in growing systems,, Phys. Rev. E., 60 (1999), 4588. doi: 10.1103/PhysRevE.60.4588. Google Scholar [40] C. Venkataraman, O. Lakkis and A. Madzvamuse, Global existence for semilinear reaction-diffusion systems on evolving domains,, J. Math. Biol., 64 (2010), 41. doi: 10.1007/s00285-011-0404-x. Google Scholar [41] P. Weidemaier, Local existence for parabolic problems with fully nonlinear boundary conditions; an $L_p$-approach,, Annali di Matematica Pura and Applicata (4), 160 (1991), 207. doi: 10.1007/BF01764128. Google Scholar
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