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Stability analysis of reaction-diffusion models on evolving domains: the effects of cross-diffusion. (English) Zbl 1332.35173

Summary: 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.

MSC:

35K57 Reaction-diffusion equations
35B35 Stability in context of PDEs
35Q92 PDEs in connection with biology, chemistry and other natural sciences
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
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[1] D. Acheson, <em>Elementary Fluid Dynamics</em>,, Oxford University Press (1990) · Zbl 0719.76001
[2] M. Baines, <em>Moving Finite Elements</em>,, Oxford University Press (1994) · Zbl 0817.65082
[3] J. Bard, How well does Turing’s Theory of morphogenesis work?,, J. Theor. Bio., 45, 501 (1974) · doi:10.1016/0022-5193(74)90128-3
[4] R. Barreira, The surface finite element method for pattern formation on evolving biological surfaces,, J. Math. Bio., 63, 1095 (2011) · Zbl 1234.92007 · doi:10.1007/s00285-011-0401-0
[5] V. Capasso, Asymptotic behaviour of reaction-diffusion systems in population and epidemic models. The role of cross-diffusion,, J. Math. Biol., 32, 453 (1994) · Zbl 0802.35077 · doi:10.1007/BF00160168
[6] V. Capasso, Global attractivity for reaction-diffusion systems. The case of nondiagonal diffusion matrices,, J. Math. Anal. and App., 177, 510 (1993) · Zbl 0794.35077 · doi:10.1006/jmaa.1993.1274
[7] E. J. Crampin, Pattern formation in reaction-diffusion models with nonuniform domain growth,, Bull. Math. Biol., 64, 747 (2002) · Zbl 1334.92035 · doi:10.1006/bulm.2002.0295
[8] G. Gambino, Turing instability and traveling fronts for nonlinear reaction-diffusion system with cross-diffusion,, Maths. Comp. in Sim., 82, 1112 (2012) · Zbl 1320.35170 · doi:10.1016/j.matcom.2011.11.004
[9] G. Gambino, Pattern formation driven by cross-diffusion in 2-D domain,, Non. Anal. Real World Applications, 14, 1755 (2013) · Zbl 1270.35088 · doi:10.1016/j.nonrwa.2012.11.009
[10] A. Gierer, A theory of biological pattern formation,, Kybernetik, 12, 30 (1972) · Zbl 1434.92013 · doi:10.1007/BF00289234
[11] G. Hetzer, Characterization of Turing diffusion-driven instability on evolving domains,, Disc. Con. Dyn. Sys., 32, 3975 (2012) · Zbl 1252.35049 · doi:10.3934/dcds.2012.32.3975
[12] M. Iida, Diffusion, cross-diffusion an competitive interaction,, J. Math. Biol., 53, 617 (2006) · Zbl 1113.92064 · doi:10.1007/s00285-006-0013-2
[13] K. Korvasova, Investigating the Turing conditions for diffusion-driven instability in the presence of binding immobile substrate,, J. Theor. Biol., 367, 286 (2015) · Zbl 1412.92139 · doi:10.1016/j.jtbi.2014.11.024
[14] S. Kovács, Turing bifurcation in a system with cross-diffusion,, Nonlinear Analysis, 59, 567 (2004) · Zbl 1073.35026 · doi:10.1016/S0362-546X(04)00273-1
[15] O. Lakkis, Implicit-explicit timestepping with finite element approximation of reaction-diffusion systems on evolving domains,, SIAM JNA, 51, 2309 (2013) · Zbl 1280.65111 · doi:10.1137/120880112
[16] C. B. Macdonald, Simple computation of reaction- diffusion processes on point clouds,, Proc. Nat. Acad. Sci. USA., 110, 9209 (2013) · Zbl 1292.65093 · doi:10.1073/pnas.1221408110
[17] C. B. Macdonald, The implicit closest point method for the numerical solution of partial differential equations on surfaces,, SIAM J. Sci. Comput., 31, 4330 (2010) · Zbl 1205.65238 · doi:10.1137/080740003
[18] A. Madzvamuse, A numerical approach to the study of spatial pattern formation in the ligaments of arcoid bivalves,, Bulletin of Mathematical Biology, 64, 501 (2002) · Zbl 1334.92039 · doi:10.1006/bulm.2002.0283
[19] A. Madzvamuse, A moving grid finite element method applied to a model biological pattern generator,, J. Comp. Phys., 190, 478 (2003) · Zbl 1029.65113 · doi:10.1016/S0021-9991(03)00294-8
[20] A. Madzvamuse, A moving grid finite element method for the simulation of pattern generation by Turing models on growing domains,, J. Sci. Comp., 24, 247 (2005) · Zbl 1080.65091 · 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, 239 (2006) · Zbl 1089.65098 · doi:10.1016/j.jcp.2005.09.012
[22] A. Madzvamuse, Velocity-induced numerical solutions of reaction-diffusion systems on fixed and growing domains,, J. Comp. Phys., 225, 100 (2007) · Zbl 1122.65076 · 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) · Zbl 1239.35167
[24] A. Madzvamuse, Stability analysis of non-autonomous reaction-diffusion systems: the effects of growing domains,, J. Math. Biol., 61, 133 (2010) · Zbl 1202.92010 · doi:10.1007/s00285-009-0293-4
[25] A. Madzvamuse, 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, Cross-diffusion-driven instability for reaction-diffusion systems: Analysis and simulations,, Journal of Math. Bio., 70, 709 (2015) · Zbl 1335.92035 · doi:10.1007/s00285-014-0779-6
[27] P. K. Maini, Implications of domain growth in morphogenesis,, in Mathematical Modelling and Computing in Biology and Medicine, 1, 67 (2003)
[28] M. S. McAfree, Cross-diffusion in a colloid-polymer aqueous system,, Fluid Phase Equilibria, 356, 46 (2013)
[29] C. C. McCluskey, A strategy for constructing Lyapunov functions for non-autonomous linear differential equations,, Linear Algebra and its Applications, 409, 100 (2005) · Zbl 1076.37012 · doi:10.1016/j.laa.2005.04.006
[30] J. D. Murray, <em>Mathematical Biology. II</em>,, Volume 18 of Interdisciplinary Applied Mathematics. Springer-Verlag (2003)
[31] R. G. Plaza, The effect of growth and curvature on pattern formation,, J. Dynam. and Diff. Eqs., 16, 1093 (2004) · Zbl 1073.35117 · doi:10.1007/s10884-004-7834-8
[32] I. Prigogine, Symmetry breaking instabilities in dissipative systems. II,, J. Chem. Phys., 48, 1695 (1968) · doi:10.1063/1.1668896
[33] F. Rossi, Quaternary cross-diffusion in water-in-oil microemulsions loaded with a component of the Belousov-Zhabotinsky reaction,, J. Phys. Chem. B, 114, 8140 (2010) · doi:10.1021/jp102753b
[34] R. Ruiz-Baier, Mathematical analysis and numerical simulation of pattern formation under cross-diffusion,, Non. Anal. Real World Applications, 14, 601 (2013) · Zbl 1263.92007 · doi:10.1016/j.nonrwa.2012.07.020
[35] J. Schnakenberg, Simple chemical reaction systems with limit cycle behaviour,, J. Theor. Biol., 81, 389 (1979) · doi:10.1016/0022-5193(79)90042-0
[36] L. Z. Tian, Instability induced by cross-diffusion in reaction-diffusion systems,, Non. Anal.: Real World Applications, 11, 1036 (2010) · Zbl 1180.35098 · doi:10.1016/j.nonrwa.2009.01.043
[37] A. Turing, On the chemical basis of morphogenesis,, Phil. Trans. Royal Soc. B, 237, 37 (1952) · Zbl 1403.92034
[38] V. K. Vanag, Cross-diffusion and pattern formation in reaction diffusion systems,, Phys. Chem. Chem. Phys., 11, 897 (2009) · doi:10.1039/B813825G
[39] C. Venkataraman, Global existence for semilinear reaction-diffusion systems on evolving domains,, Journal of Mathematical Biology, 64, 41 (2012) · Zbl 1284.35232 · doi:10.1007/s00285-011-0404-x
[40] A. Vergara. F. Capuano, Lysozyme mutual diffusion in solutions crowded by poly(ethylene glycol),, Macromolecules, 39, 4500 (2006)
[41] Z. Xie, Cross-diffusion induced Turing instability for a three species food chain model,, J. Math. Analy. and Appl., 388, 539 (2012) · Zbl 1243.35093 · doi:10.1016/j.jmaa.2011.10.054
[42] J. F. Zhang, Turing patterns of a strongly coupled predator-prey system with diffusion effects,, Non. Anal., 74, 847 (2011) · Zbl 1204.35167 · doi:10.1016/j.na.2010.09.035
[43] E. P. Zemskov, Amplitude equations for reaction-diffusion systems with cross-diffusion,, Phys. Rev. E., 84 (2011) · doi:10.1103/PhysRevE.84.036216
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