April  2016, 36(4): 2133-2170. doi: 10.3934/dcds.2016.36.2133

Stability analysis of reaction-diffusion models on evolving domains: The effects of cross-diffusion

1. 

University of Sussex, School of Mathematical and Physical Sciences, Pevensey III, 5C15, Brighton, BN1 9QH

2. 

School of Mathematical and Physical Sciences, Department of Mathematics, University of Sussex, Falmer, Brighton, BN1 9QH, England, United Kingdom

3. 

Polytechnic Institute of Setubal, Barreiro School of Technology, Rua Américo da Silva Marinho-Lavradio, 2839-001 Barreiro, Portugal

Received  January 2015 Revised  August 2015 Published  September 2015

This article presents stability analytical results of a two component reaction-diffusion system with linear cross-diffusion posed on continuously evolving domains. First the model system is mapped from a continuously evolving domain to a reference stationary frame resulting in a system of partial differential equations with time-dependent coefficients. Second, by employing appropriately asymptotic theory, we derive and prove cross-diffusion-driven instability conditions for the model system for the case of slow, isotropic domain growth. Our analytical results reveal that unlike the restrictive diffusion-driven instability conditions on stationary domains, in the presence of cross-diffusion coupled with domain evolution, it is no longer necessary to enforce cross nor pure kinetic conditions. The restriction to activator-inhibitor kinetics to induce pattern formation on a growing biological system is no longer a requirement. Reaction-cross-diffusion models with equal diffusion coefficients in the principal components as well as those of the short-range inhibition, long-range activation and activator-activator form can generate patterns only in the presence of cross-diffusion coupled with domain evolution. To confirm our theoretical findings, detailed parameter spaces are exhibited for the special cases of isotropic exponential, linear and logistic growth profiles. In support of our theoretical predictions, we present evolving or moving finite element solutions exhibiting patterns generated by a short-range inhibition, long-range activation reaction-diffusion model with linear cross-diffusion in the presence of domain evolution.
Citation: Anotida Madzvamuse, Hussaini Ndakwo, Raquel Barreira. Stability analysis of reaction-diffusion models on evolving domains: The effects of cross-diffusion. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2133-2170. doi: 10.3934/dcds.2016.36.2133
References:
[1]

D. Acheson, Elementary Fluid Dynamics,, Oxford University Press, (1990).   Google Scholar

[2]

M. Baines, Moving Finite Elements,, Oxford University Press, (1994).   Google Scholar

[3]

J. Bard and I. Lauder, How well does Turing's Theory of morphogenesis work?,, J. Theor. Bio., 45 (1974), 501.  doi: 10.1016/0022-5193(74)90128-3.  Google Scholar

[4]

R. Barreira, C. M. Elliott and A. Madzvamuse, The surface finite element method for pattern formation on evolving biological surfaces,, J. Math. Bio., 63 (2011), 1095.  doi: 10.1007/s00285-011-0401-0.  Google Scholar

[5]

V. Capasso and D. Liddo, Asymptotic behaviour of reaction-diffusion systems in population and epidemic models. The role of cross-diffusion,, J. Math. Biol., 32 (1994), 453.  doi: 10.1007/BF00160168.  Google Scholar

[6]

V. Capasso and D. Liddo, Global attractivity for reaction-diffusion systems. The case of nondiagonal diffusion matrices,, J. Math. Anal. and App., 177 (1993), 510.  doi: 10.1006/jmaa.1993.1274.  Google Scholar

[7]

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), 747.  doi: 10.1006/bulm.2002.0295.  Google Scholar

[8]

G. Gambino, M. C. Lombardo and M. Sammartino, Turing instability and traveling fronts for nonlinear reaction-diffusion system with cross-diffusion,, Maths. Comp. in Sim., 82 (2012), 1112.  doi: 10.1016/j.matcom.2011.11.004.  Google Scholar

[9]

G. Gambino, M. C. Lombardo and M. Sammartino, Pattern formation driven by cross-diffusion in 2-D domain,, Non. Anal. Real World Applications, 14 (2013), 1755.  doi: 10.1016/j.nonrwa.2012.11.009.  Google Scholar

[10]

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

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G. Hetzer, A. Madzvamuse and W. Shen, Characterization of Turing diffusion-driven instability on evolving domains,, Disc. Con. Dyn. Sys., 32 (2012), 3975.  doi: 10.3934/dcds.2012.32.3975.  Google Scholar

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M. Iida and M. Mimura, Diffusion, cross-diffusion an competitive interaction,, J. Math. Biol., 53 (2006), 617.  doi: 10.1007/s00285-006-0013-2.  Google Scholar

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K. Korvasova, E. A. Gaffney, M. P. Maini, M. A. Ferreira and V. Klika, Investigating the Turing conditions for diffusion-driven instability in the presence of binding immobile substrate,, J. Theor. Biol., 367 (2015), 286.  doi: 10.1016/j.jtbi.2014.11.024.  Google Scholar

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S. Kovács, Turing bifurcation in a system with cross-diffusion,, Nonlinear Analysis, 59 (2004), 567.  doi: 10.1016/S0362-546X(04)00273-1.  Google Scholar

[15]

O. Lakkis, A. Madzvamuse and C. Venkataraman, Implicit-explicit timestepping with finite element approximation of reaction-diffusion systems on evolving domains,, SIAM JNA, 51 (2013), 2309.  doi: 10.1137/120880112.  Google Scholar

[16]

C. B. Macdonald, B. Merriman and S. J. Ruuth, Simple computation of reaction- diffusion processes on point clouds,, Proc. Nat. Acad. Sci. USA., 110 (2013), 9209.  doi: 10.1073/pnas.1221408110.  Google Scholar

[17]

C. B. Macdonald and S. J. Ruuth, The implicit closest point method for the numerical solution of partial differential equations on surfaces,, SIAM J. Sci. Comput., 31 (2010), 4330.  doi: 10.1137/080740003.  Google Scholar

[18]

A. Madzvamuse, R. D. K. Thomas, P. K. Maini and A. J. Wathen, A numerical approach to the study of spatial pattern formation in the ligaments of arcoid bivalves,, Bulletin of Mathematical Biology, 64 (2002), 501.  doi: 10.1006/bulm.2002.0283.  Google Scholar

[19]

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

[20]

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

[21]

A. Madzvamuse, Time-stepping schemes for moving grid finite elements applied to reaction-diffusion systems on fixed and growing domains,, J. Sci. Phys., 214 (2006), 239.  doi: 10.1016/j.jcp.2005.09.012.  Google Scholar

[22]

A. Madzvamuse and M. 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

[23]

A. Madzvamuse, Diffusion-driven instability for growing domains with divergence free mesh velocity,, Nonlinear Analysis: Theory, 17 (2009).   Google Scholar

[24]

A. Madzvamuse, E. A. Gaffney and M. 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

[25]

A. Madzvamuse and R. Barreira, Exhibiting cross-diffusion-induced patterns for reaction-diffusion systems on evolving domains and surfaces,, Physical Review E, 90 (2014).  doi: 10.1103/PhysRevE.90.043307.  Google Scholar

[26]

A. Madzvamuse, H. S. Ndakwo and R. Barreira, Cross-diffusion-driven instability for reaction-diffusion systems: Analysis and simulations,, Journal of Math. Bio., 70 (2015), 709.  doi: 10.1007/s00285-014-0779-6.  Google Scholar

[27]

P. K. Maini, E. J. Crampin, A. Madzvamuse, A. J. Wathen and R. D. K. Thomas, Implications of domain growth in morphogenesis,, in Mathematical Modelling and Computing in Biology and Medicine, 1 (2003), 67.   Google Scholar

[28]

M. S. McAfree and O. Annunziata, Cross-diffusion in a colloid-polymer aqueous system,, Fluid Phase Equilibria, 356 (2013), 46.   Google Scholar

[29]

C. C. McCluskey, A strategy for constructing Lyapunov functions for non-autonomous linear differential equations,, Linear Algebra and its Applications, 409 (2005), 100.  doi: 10.1016/j.laa.2005.04.006.  Google Scholar

[30]

J. D. Murray, Mathematical Biology. II,, Volume 18 of Interdisciplinary Applied Mathematics. Springer-Verlag, (2003).   Google Scholar

[31]

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

[32]

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

[33]

F. Rossi, V. K. Vanag, E. Tiezzi and I. R. Epstein, Quaternary cross-diffusion in water-in-oil microemulsions loaded with a component of the Belousov-Zhabotinsky reaction,, J. Phys. Chem. B, 114 (2010), 8140.  doi: 10.1021/jp102753b.  Google Scholar

[34]

R. Ruiz-Baier and C. Tian, Mathematical analysis and numerical simulation of pattern formation under cross-diffusion,, Non. Anal. Real World Applications, 14 (2013), 601.  doi: 10.1016/j.nonrwa.2012.07.020.  Google Scholar

[35]

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

[36]

L. Z. Tian and M. Pedersen, Instability induced by cross-diffusion in reaction-diffusion systems,, Non. Anal.: Real World Applications, 11 (2010), 1036.  doi: 10.1016/j.nonrwa.2009.01.043.  Google Scholar

[37]

A. Turing, On the chemical basis of morphogenesis,, Phil. Trans. Royal Soc. B, 237 (1952), 37.   Google Scholar

[38]

V. K. Vanag and I. R. Epstein, Cross-diffusion and pattern formation in reaction diffusion systems,, Phys. Chem. Chem. Phys., 11 (2009), 897.  doi: 10.1039/B813825G.  Google Scholar

[39]

C. Venkataraman, O. Lakkis and A. Madzvamuse, Global existence for semilinear reaction-diffusion systems on evolving domains,, Journal of Mathematical Biology, 64 (2012), 41.  doi: 10.1007/s00285-011-0404-x.  Google Scholar

[40]

A. Vergara. F. Capuano, L. Paduano and R. Sartorio, Lysozyme mutual diffusion in solutions crowded by poly(ethylene glycol),, Macromolecules, 39 (2006), 4500.   Google Scholar

[41]

Z. Xie, Cross-diffusion induced Turing instability for a three species food chain model,, J. Math. Analy. and Appl., 388 (2012), 539.  doi: 10.1016/j.jmaa.2011.10.054.  Google Scholar

[42]

J. F. Zhang, W. T. Li and Y. X. Wang, Turing patterns of a strongly coupled predator-prey system with diffusion effects,, Non. Anal., 74 (2011), 847.  doi: 10.1016/j.na.2010.09.035.  Google Scholar

[43]

E. P. Zemskov, V. K. Vanag and I. R. Epstein, Amplitude equations for reaction-diffusion systems with cross-diffusion,, Phys. Rev. E., 84 (2011).  doi: 10.1103/PhysRevE.84.036216.  Google Scholar

show all references

References:
[1]

D. Acheson, Elementary Fluid Dynamics,, Oxford University Press, (1990).   Google Scholar

[2]

M. Baines, Moving Finite Elements,, Oxford University Press, (1994).   Google Scholar

[3]

J. Bard and I. Lauder, How well does Turing's Theory of morphogenesis work?,, J. Theor. Bio., 45 (1974), 501.  doi: 10.1016/0022-5193(74)90128-3.  Google Scholar

[4]

R. Barreira, C. M. Elliott and A. Madzvamuse, The surface finite element method for pattern formation on evolving biological surfaces,, J. Math. Bio., 63 (2011), 1095.  doi: 10.1007/s00285-011-0401-0.  Google Scholar

[5]

V. Capasso and D. Liddo, Asymptotic behaviour of reaction-diffusion systems in population and epidemic models. The role of cross-diffusion,, J. Math. Biol., 32 (1994), 453.  doi: 10.1007/BF00160168.  Google Scholar

[6]

V. Capasso and D. Liddo, Global attractivity for reaction-diffusion systems. The case of nondiagonal diffusion matrices,, J. Math. Anal. and App., 177 (1993), 510.  doi: 10.1006/jmaa.1993.1274.  Google Scholar

[7]

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), 747.  doi: 10.1006/bulm.2002.0295.  Google Scholar

[8]

G. Gambino, M. C. Lombardo and M. Sammartino, Turing instability and traveling fronts for nonlinear reaction-diffusion system with cross-diffusion,, Maths. Comp. in Sim., 82 (2012), 1112.  doi: 10.1016/j.matcom.2011.11.004.  Google Scholar

[9]

G. Gambino, M. C. Lombardo and M. Sammartino, Pattern formation driven by cross-diffusion in 2-D domain,, Non. Anal. Real World Applications, 14 (2013), 1755.  doi: 10.1016/j.nonrwa.2012.11.009.  Google Scholar

[10]

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

[11]

G. Hetzer, A. Madzvamuse and W. Shen, Characterization of Turing diffusion-driven instability on evolving domains,, Disc. Con. Dyn. Sys., 32 (2012), 3975.  doi: 10.3934/dcds.2012.32.3975.  Google Scholar

[12]

M. Iida and M. Mimura, Diffusion, cross-diffusion an competitive interaction,, J. Math. Biol., 53 (2006), 617.  doi: 10.1007/s00285-006-0013-2.  Google Scholar

[13]

K. Korvasova, E. A. Gaffney, M. P. Maini, M. A. Ferreira and V. Klika, Investigating the Turing conditions for diffusion-driven instability in the presence of binding immobile substrate,, J. Theor. Biol., 367 (2015), 286.  doi: 10.1016/j.jtbi.2014.11.024.  Google Scholar

[14]

S. Kovács, Turing bifurcation in a system with cross-diffusion,, Nonlinear Analysis, 59 (2004), 567.  doi: 10.1016/S0362-546X(04)00273-1.  Google Scholar

[15]

O. Lakkis, A. Madzvamuse and C. Venkataraman, Implicit-explicit timestepping with finite element approximation of reaction-diffusion systems on evolving domains,, SIAM JNA, 51 (2013), 2309.  doi: 10.1137/120880112.  Google Scholar

[16]

C. B. Macdonald, B. Merriman and S. J. Ruuth, Simple computation of reaction- diffusion processes on point clouds,, Proc. Nat. Acad. Sci. USA., 110 (2013), 9209.  doi: 10.1073/pnas.1221408110.  Google Scholar

[17]

C. B. Macdonald and S. J. Ruuth, The implicit closest point method for the numerical solution of partial differential equations on surfaces,, SIAM J. Sci. Comput., 31 (2010), 4330.  doi: 10.1137/080740003.  Google Scholar

[18]

A. Madzvamuse, R. D. K. Thomas, P. K. Maini and A. J. Wathen, A numerical approach to the study of spatial pattern formation in the ligaments of arcoid bivalves,, Bulletin of Mathematical Biology, 64 (2002), 501.  doi: 10.1006/bulm.2002.0283.  Google Scholar

[19]

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

[20]

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

[21]

A. Madzvamuse, Time-stepping schemes for moving grid finite elements applied to reaction-diffusion systems on fixed and growing domains,, J. Sci. Phys., 214 (2006), 239.  doi: 10.1016/j.jcp.2005.09.012.  Google Scholar

[22]

A. Madzvamuse and M. 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

[23]

A. Madzvamuse, Diffusion-driven instability for growing domains with divergence free mesh velocity,, Nonlinear Analysis: Theory, 17 (2009).   Google Scholar

[24]

A. Madzvamuse, E. A. Gaffney and M. 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

[25]

A. Madzvamuse and R. Barreira, Exhibiting cross-diffusion-induced patterns for reaction-diffusion systems on evolving domains and surfaces,, Physical Review E, 90 (2014).  doi: 10.1103/PhysRevE.90.043307.  Google Scholar

[26]

A. Madzvamuse, H. S. Ndakwo and R. Barreira, Cross-diffusion-driven instability for reaction-diffusion systems: Analysis and simulations,, Journal of Math. Bio., 70 (2015), 709.  doi: 10.1007/s00285-014-0779-6.  Google Scholar

[27]

P. K. Maini, E. J. Crampin, A. Madzvamuse, A. J. Wathen and R. D. K. Thomas, Implications of domain growth in morphogenesis,, in Mathematical Modelling and Computing in Biology and Medicine, 1 (2003), 67.   Google Scholar

[28]

M. S. McAfree and O. Annunziata, Cross-diffusion in a colloid-polymer aqueous system,, Fluid Phase Equilibria, 356 (2013), 46.   Google Scholar

[29]

C. C. McCluskey, A strategy for constructing Lyapunov functions for non-autonomous linear differential equations,, Linear Algebra and its Applications, 409 (2005), 100.  doi: 10.1016/j.laa.2005.04.006.  Google Scholar

[30]

J. D. Murray, Mathematical Biology. II,, Volume 18 of Interdisciplinary Applied Mathematics. Springer-Verlag, (2003).   Google Scholar

[31]

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

[32]

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

[33]

F. Rossi, V. K. Vanag, E. Tiezzi and I. R. Epstein, Quaternary cross-diffusion in water-in-oil microemulsions loaded with a component of the Belousov-Zhabotinsky reaction,, J. Phys. Chem. B, 114 (2010), 8140.  doi: 10.1021/jp102753b.  Google Scholar

[34]

R. Ruiz-Baier and C. Tian, Mathematical analysis and numerical simulation of pattern formation under cross-diffusion,, Non. Anal. Real World Applications, 14 (2013), 601.  doi: 10.1016/j.nonrwa.2012.07.020.  Google Scholar

[35]

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

[36]

L. Z. Tian and M. Pedersen, Instability induced by cross-diffusion in reaction-diffusion systems,, Non. Anal.: Real World Applications, 11 (2010), 1036.  doi: 10.1016/j.nonrwa.2009.01.043.  Google Scholar

[37]

A. Turing, On the chemical basis of morphogenesis,, Phil. Trans. Royal Soc. B, 237 (1952), 37.   Google Scholar

[38]

V. K. Vanag and I. R. Epstein, Cross-diffusion and pattern formation in reaction diffusion systems,, Phys. Chem. Chem. Phys., 11 (2009), 897.  doi: 10.1039/B813825G.  Google Scholar

[39]

C. Venkataraman, O. Lakkis and A. Madzvamuse, Global existence for semilinear reaction-diffusion systems on evolving domains,, Journal of Mathematical Biology, 64 (2012), 41.  doi: 10.1007/s00285-011-0404-x.  Google Scholar

[40]

A. Vergara. F. Capuano, L. Paduano and R. Sartorio, Lysozyme mutual diffusion in solutions crowded by poly(ethylene glycol),, Macromolecules, 39 (2006), 4500.   Google Scholar

[41]

Z. Xie, Cross-diffusion induced Turing instability for a three species food chain model,, J. Math. Analy. and Appl., 388 (2012), 539.  doi: 10.1016/j.jmaa.2011.10.054.  Google Scholar

[42]

J. F. Zhang, W. T. Li and Y. X. Wang, Turing patterns of a strongly coupled predator-prey system with diffusion effects,, Non. Anal., 74 (2011), 847.  doi: 10.1016/j.na.2010.09.035.  Google Scholar

[43]

E. P. Zemskov, V. K. Vanag and I. R. Epstein, Amplitude equations for reaction-diffusion systems with cross-diffusion,, Phys. Rev. E., 84 (2011).  doi: 10.1103/PhysRevE.84.036216.  Google Scholar

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