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
References:
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H. Amann, Existence and regularity for semilinear parabolic evolution equations,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 11 (1984), 593.   Google Scholar

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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

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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

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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

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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

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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

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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

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L. Edelstein-Keshet, "Mathematical Models in Biology,'', The Random House/Birkh\, (1988).   Google Scholar

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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

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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

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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

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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

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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

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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

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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

show all references

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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