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

[2]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[14]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

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

[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.

[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.

[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.

[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.

[28]

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

[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.

[30]

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

[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.

[32]

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

[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.

[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.

[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.

[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.

[37]

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

[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.

[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.

[40]

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

[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.

[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.

[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.

show all references

References:
[1]

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

[2]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[14]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

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

[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.

[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.

[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.

[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.

[28]

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

[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.

[30]

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

[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.

[32]

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

[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.

[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.

[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.

[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.

[37]

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

[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.

[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.

[40]

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

[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.

[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.

[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.

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