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 and Continuous Dynamical Systems, 2012, 32 (11) : 3975-4000. doi: 10.3934/dcds.2012.32.3975
References:
[1]

H. Amann, Existence and regularity for semilinear parabolic evolution equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 11 (1984), 593-676.

[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), 2953-2956. doi: 10.1103/PhysRevLett.64.2953.

[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), 387-423. doi: 10.1007/s002850000067.

[4]

C. Chicone and Y. Latushkin, "Evolution Semigroups in Dynamical Systems and Differential Equations,'' Mathematical Surveys and Monographs, 70, American Mathematical Society, Providence, RI, 1999.

[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), 1093-1120. doi: 10.1006/bulm.1999.0131.

[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), 746-769. doi: 10.1006/bulm.2002.0295.

[7]

D. Daners, Perturbation of semi-linear evolution equations under weak assumptions at initial time, J. Diff. Eq., 210 (2005), 352-382. doi: 10.1016/j.jde.2004.08.004.

[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), viii+114 pp.

[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, Nr. 205, April 2005.

[10]

L. Edelstein-Keshet, "Mathematical Models in Biology,'' The Random House/Birkhäuser Mathematics Series, Random House, Inc., New York, 1988.

[11]

A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39. doi: 10.1007/BF00289234.

[12]

G. H. Golub and C. F. Van Loan, "Matrix Computations,'' JHU Press, 1996.

[13]

S. Kondo and R. Asai, A reaction-diffusion wave on the skin of the marine anglefish, Pomacanthus, Nature, 376 (1995), 765-768. doi: 10.1038/376765a0.

[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), 041909-1-041909-5. doi: 10.1103/PhysRevE.64.041909.

[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), 537-576.

[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), 133-164. doi: 10.1007/s00285-009-0293-4.

[17]

A. Madzvamuse, Turing instability conditions for growing domains with divergence free mesh velocity, Nonlinear Analysis: Theory, Methods and Applications, (2009), e2250-e2257.

[18]

A. Madzvamuse, Stability analysis of reaction-diffusion systems with constant coefficients on growing domains, Int J. of Dynamical and Differential Equations, 1 (2008), 250-262.

[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), 100-119. doi: 10.1016/j.jcp.2006.11.022.

[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), 239-263. doi: 10.1016/j.jcp.2005.09.012.

[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), 247-262. doi: 10.1007/s10915-004-4617-7.

[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), 478-500. doi: 10.1016/S0021-9991(03)00294-8.

[23]

P. K. Maini, R. E. Baker and C. M. Chong, The Turing model comes of molecular age, (Invited Perspective) Science, 314 (2006), 1397-1398. doi: 10.1126/science.1136396.

[24]

J. Mierczynski and W. Shen, Spectral theory for forward nonautonomous parabolic equations and applications,, to appear in Fields Inst. Commun., (). 

[25]

J. Mierczynski and W. Shen, Persistence in forward nonautonomous competitive systems of parabolic equations,, submitted., (). 

[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, CRC Press, Boca Raton, FL, 2008.

[27]

J. D. Murray, "Mathematical Biology I and II,'' 3$^rd$ edition, Springer-Verlag, Berlin, 2002.

[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), 5549. doi: 10.1073/pnas.96.10.5549.

[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), 1093-1121. doi: 10.1007/s10884-004-7834-8.

[30]

I. Prigogine and R. Lefever, Symmetry breaking instabilities in dissipative systems II, J. Chem. Phys., 48 (1968), 1695-1700. doi: 10.1063/1.1668896.

[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), 73-98. doi: 10.1007/s11538-005-9055-7.

[32]

Q. Ouyang and H. L. Swinney, Transition from a uniform state to hexagonal and striped Turing patterns, Nature, 352 (1991), 610-612. doi: 10.1038/352610a0.

[33]

J. Schnakenberg, Simple chemical reaction systems with limit cycle behaviour, J. Theor. Biol., 81 (1979), 389-400. doi: 10.1016/0022-5193(79)90042-0.

[34]

R. Schnaubelt, Asymptotic behavior of parabolic nonautonomous evolution equations, in "Functional Analytic Methodsfor Evolution Equations," Lecture Notes in Mathematics, 1855, Springer, Berlin, (2004), 401-472.

[35]

Y. Shiferaw and A. Karma, Turing instability mediated by voltage and calcium diffusion in paced cardiac cells, PNAS, 103 (2006), 5670-5675. doi: 10.1073/pnas.0511061103.

[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), 1447-1450. doi: 10.1126/science.1130088.

[37]

L. Solnica-Krezel, Vertebrate development: Taming the nodal waves, Curr Biol., 13 (2003), R7-9. doi: 10.1016/S0960-9822(02)01378-7.

[38]

A. Turing, The chemical basis of morphogenesis, Phil. Trans. R. Soc. Lond. B, 237 (1952), 37-72. doi: 10.1098/rstb.1952.0012.

[39]

C. Varea, J. L. Aragón and R. A. Barrio, Confined Turing patterns in growing systems, Phys. Rev. E., 60 (1999), 4588-4592. doi: 10.1103/PhysRevE.60.4588.

[40]

C. Venkataraman, O. Lakkis and A. Madzvamuse, Global existence for semilinear reaction-diffusion systems on evolving domains, J. Math. Biol., 64 (2010), 41-67. doi: 10.1007/s00285-011-0404-x.

[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), 207-222. doi: 10.1007/BF01764128.

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), 593-676.

[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), 2953-2956. doi: 10.1103/PhysRevLett.64.2953.

[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), 387-423. doi: 10.1007/s002850000067.

[4]

C. Chicone and Y. Latushkin, "Evolution Semigroups in Dynamical Systems and Differential Equations,'' Mathematical Surveys and Monographs, 70, American Mathematical Society, Providence, RI, 1999.

[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), 1093-1120. doi: 10.1006/bulm.1999.0131.

[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), 746-769. doi: 10.1006/bulm.2002.0295.

[7]

D. Daners, Perturbation of semi-linear evolution equations under weak assumptions at initial time, J. Diff. Eq., 210 (2005), 352-382. doi: 10.1016/j.jde.2004.08.004.

[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), viii+114 pp.

[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, Nr. 205, April 2005.

[10]

L. Edelstein-Keshet, "Mathematical Models in Biology,'' The Random House/Birkhäuser Mathematics Series, Random House, Inc., New York, 1988.

[11]

A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39. doi: 10.1007/BF00289234.

[12]

G. H. Golub and C. F. Van Loan, "Matrix Computations,'' JHU Press, 1996.

[13]

S. Kondo and R. Asai, A reaction-diffusion wave on the skin of the marine anglefish, Pomacanthus, Nature, 376 (1995), 765-768. doi: 10.1038/376765a0.

[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), 041909-1-041909-5. doi: 10.1103/PhysRevE.64.041909.

[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), 537-576.

[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), 133-164. doi: 10.1007/s00285-009-0293-4.

[17]

A. Madzvamuse, Turing instability conditions for growing domains with divergence free mesh velocity, Nonlinear Analysis: Theory, Methods and Applications, (2009), e2250-e2257.

[18]

A. Madzvamuse, Stability analysis of reaction-diffusion systems with constant coefficients on growing domains, Int J. of Dynamical and Differential Equations, 1 (2008), 250-262.

[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), 100-119. doi: 10.1016/j.jcp.2006.11.022.

[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), 239-263. doi: 10.1016/j.jcp.2005.09.012.

[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), 247-262. doi: 10.1007/s10915-004-4617-7.

[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), 478-500. doi: 10.1016/S0021-9991(03)00294-8.

[23]

P. K. Maini, R. E. Baker and C. M. Chong, The Turing model comes of molecular age, (Invited Perspective) Science, 314 (2006), 1397-1398. doi: 10.1126/science.1136396.

[24]

J. Mierczynski and W. Shen, Spectral theory for forward nonautonomous parabolic equations and applications,, to appear in Fields Inst. Commun., (). 

[25]

J. Mierczynski and W. Shen, Persistence in forward nonautonomous competitive systems of parabolic equations,, submitted., (). 

[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, CRC Press, Boca Raton, FL, 2008.

[27]

J. D. Murray, "Mathematical Biology I and II,'' 3$^rd$ edition, Springer-Verlag, Berlin, 2002.

[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), 5549. doi: 10.1073/pnas.96.10.5549.

[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), 1093-1121. doi: 10.1007/s10884-004-7834-8.

[30]

I. Prigogine and R. Lefever, Symmetry breaking instabilities in dissipative systems II, J. Chem. Phys., 48 (1968), 1695-1700. doi: 10.1063/1.1668896.

[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), 73-98. doi: 10.1007/s11538-005-9055-7.

[32]

Q. Ouyang and H. L. Swinney, Transition from a uniform state to hexagonal and striped Turing patterns, Nature, 352 (1991), 610-612. doi: 10.1038/352610a0.

[33]

J. Schnakenberg, Simple chemical reaction systems with limit cycle behaviour, J. Theor. Biol., 81 (1979), 389-400. doi: 10.1016/0022-5193(79)90042-0.

[34]

R. Schnaubelt, Asymptotic behavior of parabolic nonautonomous evolution equations, in "Functional Analytic Methodsfor Evolution Equations," Lecture Notes in Mathematics, 1855, Springer, Berlin, (2004), 401-472.

[35]

Y. Shiferaw and A. Karma, Turing instability mediated by voltage and calcium diffusion in paced cardiac cells, PNAS, 103 (2006), 5670-5675. doi: 10.1073/pnas.0511061103.

[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), 1447-1450. doi: 10.1126/science.1130088.

[37]

L. Solnica-Krezel, Vertebrate development: Taming the nodal waves, Curr Biol., 13 (2003), R7-9. doi: 10.1016/S0960-9822(02)01378-7.

[38]

A. Turing, The chemical basis of morphogenesis, Phil. Trans. R. Soc. Lond. B, 237 (1952), 37-72. doi: 10.1098/rstb.1952.0012.

[39]

C. Varea, J. L. Aragón and R. A. Barrio, Confined Turing patterns in growing systems, Phys. Rev. E., 60 (1999), 4588-4592. doi: 10.1103/PhysRevE.60.4588.

[40]

C. Venkataraman, O. Lakkis and A. Madzvamuse, Global existence for semilinear reaction-diffusion systems on evolving domains, J. Math. Biol., 64 (2010), 41-67. doi: 10.1007/s00285-011-0404-x.

[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), 207-222. doi: 10.1007/BF01764128.

[1]

Siqing Li, Zhonghua Qiao. A meshless collocation method with a global refinement strategy for reaction-diffusion systems on evolving domains. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 601-617. doi: 10.3934/dcdsb.2021057

[2]

Weihua Jiang, Xun Cao, Chuncheng Wang. Turing instability and pattern formations for reaction-diffusion systems on 2D bounded domain. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 1163-1178. doi: 10.3934/dcdsb.2021085

[3]

Anotida Madzvamuse, Hussaini Ndakwo, Raquel Barreira. Stability analysis of reaction-diffusion models on evolving domains: The effects of cross-diffusion. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 2133-2170. doi: 10.3934/dcds.2016.36.2133

[4]

Masaharu Taniguchi. Instability of planar traveling waves in bistable reaction-diffusion systems. Discrete and Continuous Dynamical Systems - B, 2003, 3 (1) : 21-44. doi: 10.3934/dcdsb.2003.3.21

[5]

Martino Prizzi. A remark on reaction-diffusion equations in unbounded domains. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 281-286. doi: 10.3934/dcds.2003.9.281

[6]

Hideki Murakawa. Fast reaction limit of reaction-diffusion systems. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1047-1062. doi: 10.3934/dcdss.2020405

[7]

Peter E. Kloeden, Meihua Yang. Forward attracting sets of reaction-diffusion equations on variable domains. Discrete and Continuous Dynamical Systems - B, 2019, 24 (3) : 1259-1271. doi: 10.3934/dcdsb.2019015

[8]

Shin-Ichiro Ei, Kota Ikeda, Eiji Yanagida. Instability of multi-spot patterns in shadow systems of reaction-diffusion equations. Communications on Pure and Applied Analysis, 2015, 14 (2) : 717-736. doi: 10.3934/cpaa.2015.14.717

[9]

Alexey Cheskidov, Songsong Lu. The existence and the structure of uniform global attractors for nonautonomous Reaction-Diffusion systems without uniqueness. Discrete and Continuous Dynamical Systems - S, 2009, 2 (1) : 55-66. doi: 10.3934/dcdss.2009.2.55

[10]

Ching-Shan Chou, Yong-Tao Zhang, Rui Zhao, Qing Nie. Numerical methods for stiff reaction-diffusion systems. Discrete and Continuous Dynamical Systems - B, 2007, 7 (3) : 515-525. doi: 10.3934/dcdsb.2007.7.515

[11]

Laurent Desvillettes, Klemens Fellner. Entropy methods for reaction-diffusion systems. Conference Publications, 2007, 2007 (Special) : 304-312. doi: 10.3934/proc.2007.2007.304

[12]

A. Dall'Acqua. Positive solutions for a class of reaction-diffusion systems. Communications on Pure and Applied Analysis, 2003, 2 (1) : 65-76. doi: 10.3934/cpaa.2003.2.65

[13]

Ciprian G. Gal, Mahamadi Warma. Reaction-diffusion equations with fractional diffusion on non-smooth domains with various boundary conditions. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1279-1319. doi: 10.3934/dcds.2016.36.1279

[14]

Dieter Bothe, Michel Pierre. The instantaneous limit for reaction-diffusion systems with a fast irreversible reaction. Discrete and Continuous Dynamical Systems - S, 2012, 5 (1) : 49-59. doi: 10.3934/dcdss.2012.5.49

[15]

Jacson Simsen, Mariza Stefanello Simsen, Marcos Roberto Teixeira Primo. Reaction-Diffusion equations with spatially variable exponents and large diffusion. Communications on Pure and Applied Analysis, 2016, 15 (2) : 495-506. doi: 10.3934/cpaa.2016.15.495

[16]

Alexandra Köthe, Anna Marciniak-Czochra, Izumi Takagi. Hysteresis-driven pattern formation in reaction-diffusion-ODE systems. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3595-3627. doi: 10.3934/dcds.2020170

[17]

Vandana Sharma, Jyotshana V. Prajapat. Global existence of solutions to reaction diffusion systems with mass transport type boundary conditions on an evolving domain. Discrete and Continuous Dynamical Systems, 2022, 42 (1) : 109-135. doi: 10.3934/dcds.2021109

[18]

Dingshi Li, Kening Lu, Bixiang Wang, Xiaohu Wang. Limiting behavior of dynamics for stochastic reaction-diffusion equations with additive noise on thin domains. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 187-208. doi: 10.3934/dcds.2018009

[19]

Dingshi Li, Kening Lu, Bixiang Wang, Xiaohu Wang. Limiting dynamics for non-autonomous stochastic retarded reaction-diffusion equations on thin domains. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 3717-3747. doi: 10.3934/dcds.2019151

[20]

Henri Berestycki, Luca Rossi. Reaction-diffusion equations for population dynamics with forced speed II - cylindrical-type domains. Discrete and Continuous Dynamical Systems, 2009, 25 (1) : 19-61. doi: 10.3934/dcds.2009.25.19

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (119)
  • HTML views (0)
  • Cited by (13)

Other articles
by authors

[Back to Top]