September  2018, 23(7): 2775-2801. doi: 10.3934/dcdsb.2018163

Domain-growth-induced patterning for reaction-diffusion systems with linear cross-diffusion

1. 

School of Mathematical and Physical Sciences, Department of Mathematics, University of Sussex, Pevensey Ⅲ, 5C15, Falmer, Brigton, BN1 9QH, England, UK

2. 

Escola Superior de Tecnologia do Barreiro/IPS, Rua Américo da Silva Marinho-Lavradio, 2839-001 Barreiro, Portugal

Corresponding author: a.madzvamuse@sussex.ac.uk

Received  March 2017 Revised  March 2018 Published  June 2018

In this article we present, for the first time, domain-growth induced pattern formation for reaction-diffusion systems with linear cross-diffusion on evolving domains and surfaces. Our major contribution is that by selecting parameter values from spaces induced by domain and surface evolution, patterns emerge only when domain growth is present. Such patterns do not exist in the absence of domain and surface evolution. In order to compute these domain-induced parameter spaces, linear stability theory is employed to establish the necessary conditions for domain-growth induced cross-diffusion-driven instability for reaction-diffusion systems with linear cross-diffusion. Model reaction-kinetic parameter values are then identified from parameter spaces induced by domain-growth only; these exist outside the classical standard Turing space on stationary domains and surfaces. To exhibit these patterns we employ the finite element method for solving reaction-diffusion systems with cross-diffusion on continuously evolving domains and surfaces.

Citation: Anotida Madzvamuse, Raquel Barreira. Domain-growth-induced patterning for reaction-diffusion systems with linear cross-diffusion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2775-2801. doi: 10.3934/dcdsb.2018163
References:
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E. J. CrampinW. W. Hackborn and P. K. Maini, Pattern formation in reaction-diffusion models with nonuniform domain growth, Bull. Math. Biol., 64 (2002), 746-769.   Google Scholar

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J. B. GreerA. L. Bertozzi and G. Sapiro, Fourth order partial differential equations on general geometries, J. Comput. Phys., 216 (2006), 216-246.  doi: 10.1016/j.jcp.2005.11.031.  Google Scholar

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A. MadzvamuseP. 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.  Google Scholar

[29]

A. MadzvamuseA. 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.  Google Scholar

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A. Madzvamuse, Time-stepping schemes for moving grid finite elements applied to reaction-diffusion systems on fixed and growing domains, J. Sci. Phys., 216 (2006), 239-263.  doi: 10.1016/j.jcp.2005.09.012.  Google Scholar

[31]

A. Madzvamuse, A modified backward Euler scheme for advection-reaction-diffusion systems. Mathematical modeling of biological systems, Mathematical Modeling of Biological Systems, 1 (2007), 183-190.   Google Scholar

[32]

A. MadzvamuseE. 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.  Google Scholar

[33]

A. MadzvamuseH. S. Ndakwo and R. Barreira, Cross-diffusion-driven instability for reaction-diffusion systems: Analysis and simulations, Journal of Math. Bio., 70 (2014), 709-743.  doi: 10.1007/s00285-014-0779-6.  Google Scholar

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A. Madzvamuse and R. Barreira, Exhibiting cross-diffusion-induced patterns for reaction-diffusion systems on evolving domains and surfaces, Physical Review E, 90 (2014). 043307-1-043307-14. ISSN 1539-3755. Google Scholar

[35]

A. MadzvamuseH. S. Ndakwo and R. Barreira, Stability analysis of reaction-diffusion models on evolving domains: The effects of cross-diffusion, Discrete and Continuous Dynamical Systems. Series A., 36 (2016), 2133-2170.  doi: 10.3934/dcds.2016.36.2133.  Google Scholar

[36]

P. K. Maini, E. J. Crampin, A. Madzvamuse, A. J. Wathen and R. D. K. Thomas, Implications of Domain Growth in Morphogenesis, Mathematical Modelling and Computing in Biology and Medicine: Proceedings of the 5th European Conference for Mathematics and Theoretical Biology Conference, 2002. Google Scholar

[37]

M. S. McAfree and O. Annunziata, Cross-diffusion in a colloid-polymer aqueous system, Fluid Phase Equilibria, 356 (2013), 46-55.   Google Scholar

[38]

J. D. Murray, Mathematical Biology. II, volume 18 of Interdisciplinary Applied Mathematics. Springer-Verlag, New York. Third edition. Spatial models and biomedical applications, 2003.  Google Scholar

[39]

S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, volume 153 of Applied Mathematical Sciences. Springer-Verlag, New York, 2003. doi: 10.1007/b98879.  Google Scholar

[40]

I. Prigogine and R. Lefever, Symmetry breaking instabilities in dissipative systems. Ⅱ, J. Chem. Phys., 48 (1968), 1695-1700.   Google Scholar

[41]

F. RossiV. K. VanagE. Tiezzi and I. R. Epstein, Quaternary cross-diffusion in water-in-oil microemulsions loaded with a component of the Belousov-Zhabotinsky reaction, The Journal of Physical Chemistry B, 114 (2010), 8140-8146.   Google Scholar

[42]

R. Ruiz-Baier and C. Tian, Mathematical analysis and numerical simulation of pattern formation under cross-diffusion, Nonlinear Analysis: Real World Applications, 14 (2013), 601-612.  doi: 10.1016/j.nonrwa.2012.07.020.  Google Scholar

[43]

A. Schmidt and K. G. Siebert, Design of Adaptive Element Software - The Finite Element Toolbox ALBERTA, vol. 42 of Lecture Notes in Computational Science and Engineering, Springer, 2005.  Google Scholar

[44]

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.  Google Scholar

[45]

J. A. Sethian, Level Set Methods and Fast Marching Methods, volume 3 of Cam- bridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge, second edition. Evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science, 1999.  Google Scholar

[46]

L. TianZ. Lin and M. Pedersen, Instability induced by cross-diffusion in reaction-diffusion systems, Nonlinear Analysis: Real World Applications, 11 (2010), 1036-1045.  doi: 10.1016/j.nonrwa.2009.01.043.  Google Scholar

[47]

A. Turing, On the chemical basis of morphogenesis, Phil. Trans. Royal Soc. B., 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012.  Google Scholar

[48]

V. K. Vanag and I. R. Epstein, Cross-diffusion and pattern formation in reaction diffusion systems Physical Chemistry Chemical Physics, 17 (2007), 037110, 11 pp. doi: 10.1063/1.2752494.  Google Scholar

[49]

A. VergaraF. CapuanoL. Paduano and R. Sartorio, Lysozyme mutual diffusion in solutions crowded by poly(ethylene glycol), Macromolecules, 39 (2006), 4500-4506.   Google Scholar

[50]

Z. Xie, Cross-diffusion induced Turing instability for a three species food chain model, J. Math. Analy. and Appl., 388 (2012), 539-547.  doi: 10.1016/j.jmaa.2011.10.054.  Google Scholar

[51]

A. M. Zhabotinsky, A history of chemical oscillations and waves, Chaos, 1 (1991), 379-386.   Google Scholar

[52]

J.-F. Zhang, W.-T. Li, Wang, (2011). Turing patterns of a strongly coupled predator-prey system with diffusion effects, Nonlinear Analysis, 74 (2001), 847–858. doi: 10.1016/j.na.2010.09.035.  Google Scholar

[53]

E. P. Zemskov, V. K. Vanag and I. R. Epstein, Amplitude equations for reaction-diffusion systems with cross-diffusion. Phys. Rev. E., 84 (2011), 036216. Google Scholar

show all references

References:
[1]

W. Bangerth, T. Heister, L. Heltai, G. Kanschat, M. Kronbichler, M. Maier, B. Turcksin and T. D. Young, The deal. ii library, version 8. 1, 2013. arXiv: 1312.2266. Google Scholar

[2]

J. Bard and I. Lauder, How well does Turing's theory of morphogenesis work?, J. Theor. Bio., 45 (1974), 501-531.   Google Scholar

[3]

R. BarreiraC. M. Elliott and A. Madzvamuse, The surface finite element method for pattern formation on evolving biological surfaces, Journal of Math. Bio., 63 (2011), 1095-1119.  doi: 10.1007/s00285-011-0401-0.  Google Scholar

[4]

F. J. Blom, Considerations on the spring analogy, Int. J. Numer. Meth. Fluids, 12 (2000), 647-668.   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), 453-463.  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), 510-529.  doi: 10.1006/jmaa.1993.1274.  Google Scholar

[7]

M. A. J. ChaplainM. Ganesh and I. G. Graham, Spatio-temporal pattern formation on spherical surfaces: Numerical simulation and application to solid tumour growth, J. Math. Biol., 42 (2001), 387-423.  doi: 10.1007/s002850000067.  Google Scholar

[8]

E. J. CrampinW. W. Hackborn and P. K. Maini, Pattern formation in reaction-diffusion models with nonuniform domain growth, Bull. Math. Biol., 64 (2002), 746-769.   Google Scholar

[9]

K. DeckelnickG. Dziuk and C. M. Elliott, Computation of geometric partial differential equations and mean curvature flow, Acta Numer., 14 (2005), 139-232.  doi: 10.1017/S0962492904000224.  Google Scholar

[10]

A. Donna and C. Helzel, A finite volume method for solving parabolic equations on logically Cartesian curved surface meshes, SIAM J. Sci. Comput., 31 (2009), 4066-4099.  doi: 10.1137/08073322X.  Google Scholar

[11]

G. Dziuk and C. M. Elliott, Surface finite elements for parabolic equations, J. Comp. Math., 25 (2007), 385-407.   Google Scholar

[12]

G. Dziuk and C. M. Elliott, Eulerian finite element method for parabolic PDEs on implicit surfaces, Interfaces Free Bound, 10 (2008), 119-138.  doi: 10.4171/IFB/182.  Google Scholar

[13]

G. Dziuk and C. M. Elliott, An Eulerian approach to transport and diffusion on evolving implicit surfaces, Comput. Vis. Sci., 13 (2010), 17-28.  doi: 10.1007/s00791-008-0122-0.  Google Scholar

[14]

G. Dziuk and C. M. Elliott, Finite element methods for surface PDEs, Acta Numer., 22 (2013), 289-396.  doi: 10.1017/S0962492913000056.  Google Scholar

[15]

C. M. ElliottB. StinnerV. Styles and R. Welford, Numerical computation of advection and diffusion on evolving diffuse interfaces, IMA J. Numer. Anal., 31 (2011), 786-812.  doi: 10.1093/imanum/drq005.  Google Scholar

[16]

G. GambinoM. C. Lombardo and M. Sammartino, Turing instability and traveling fronts for nonlinear reaction-diffusion system with cross-diffusion, Maths. Comp. in Sim., 82 (2012), 1112-1132.  doi: 10.1016/j.matcom.2011.11.004.  Google Scholar

[17]

G. GambinoM. C. Lombardo and M. Sammartino, Pattern formation driven by cross-diffusion in 2-D domain, Nonlinear Analysis: Real World Applications, 14 (2013), 1755-1779.  doi: 10.1016/j.nonrwa.2012.11.009.  Google Scholar

[18]

A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39.   Google Scholar

[19]

J. B. GreerA. L. Bertozzi and G. Sapiro, Fourth order partial differential equations on general geometries, J. Comput. Phys., 216 (2006), 216-246.  doi: 10.1016/j.jcp.2005.11.031.  Google Scholar

[20]

G. HetzerA. Madzvamuse and W. Shen, Characterization of Turing diffusion-driven instability on evolving domains, Discrete Cont. Dyn. Syst., 32 (2012), 3975-4000.  doi: 10.3934/dcds.2012.32.3975.  Google Scholar

[21]

S. E. Hieber and P. Koumoutsakos, A Lagrangian particle level set method, J. Comput. Phys., 210 (2005), 342-367.  doi: 10.1016/j.jcp.2005.04.013.  Google Scholar

[22]

M. Iida and M. Mimura, Diffusion, cross-diffusion an competitive interaction, J. Math. Biol., 53 (2006), 617-641.  doi: 10.1007/s00285-006-0013-2.  Google Scholar

[23]

S. Kovács, Turing bifurcation in a system with cross-diffusion, Nonlinear Analysis, 59 (2004), 567-581.  doi: 10.1016/j.na.2004.07.025.  Google Scholar

[24]

O. LakkisA. Madzvamuse and C. Venkataraman, Implicit-explicit timestepping with finite element approximation of reaction-diffusion systems on evolving domains, SIAM Journal on Numerical Analysis, 51 (2013), 2309-2330.  doi: 10.1137/120880112.  Google Scholar

[25]

C. B. MacdonaldB. Merriman and S. J. Ruuth, Simple computation of reaction- diffusion processes on point clouds, Proc. Nat. Acad. Sci. USA., 110 (2013), 9209-9214.  doi: 10.1073/pnas.1221408110.  Google Scholar

[26]

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), 4330-4350.  doi: 10.1137/080740003.  Google Scholar

[27]

A. MadzvamuseR. K. ThomasP. 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), 501-530.   Google Scholar

[28]

A. MadzvamuseP. 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.  Google Scholar

[29]

A. MadzvamuseA. 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.  Google Scholar

[30]

A. Madzvamuse, Time-stepping schemes for moving grid finite elements applied to reaction-diffusion systems on fixed and growing domains, J. Sci. Phys., 216 (2006), 239-263.  doi: 10.1016/j.jcp.2005.09.012.  Google Scholar

[31]

A. Madzvamuse, A modified backward Euler scheme for advection-reaction-diffusion systems. Mathematical modeling of biological systems, Mathematical Modeling of Biological Systems, 1 (2007), 183-190.   Google Scholar

[32]

A. MadzvamuseE. 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.  Google Scholar

[33]

A. MadzvamuseH. S. Ndakwo and R. Barreira, Cross-diffusion-driven instability for reaction-diffusion systems: Analysis and simulations, Journal of Math. Bio., 70 (2014), 709-743.  doi: 10.1007/s00285-014-0779-6.  Google Scholar

[34]

A. Madzvamuse and R. Barreira, Exhibiting cross-diffusion-induced patterns for reaction-diffusion systems on evolving domains and surfaces, Physical Review E, 90 (2014). 043307-1-043307-14. ISSN 1539-3755. Google Scholar

[35]

A. MadzvamuseH. S. Ndakwo and R. Barreira, Stability analysis of reaction-diffusion models on evolving domains: The effects of cross-diffusion, Discrete and Continuous Dynamical Systems. Series A., 36 (2016), 2133-2170.  doi: 10.3934/dcds.2016.36.2133.  Google Scholar

[36]

P. K. Maini, E. J. Crampin, A. Madzvamuse, A. J. Wathen and R. D. K. Thomas, Implications of Domain Growth in Morphogenesis, Mathematical Modelling and Computing in Biology and Medicine: Proceedings of the 5th European Conference for Mathematics and Theoretical Biology Conference, 2002. Google Scholar

[37]

M. S. McAfree and O. Annunziata, Cross-diffusion in a colloid-polymer aqueous system, Fluid Phase Equilibria, 356 (2013), 46-55.   Google Scholar

[38]

J. D. Murray, Mathematical Biology. II, volume 18 of Interdisciplinary Applied Mathematics. Springer-Verlag, New York. Third edition. Spatial models and biomedical applications, 2003.  Google Scholar

[39]

S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, volume 153 of Applied Mathematical Sciences. Springer-Verlag, New York, 2003. doi: 10.1007/b98879.  Google Scholar

[40]

I. Prigogine and R. Lefever, Symmetry breaking instabilities in dissipative systems. Ⅱ, J. Chem. Phys., 48 (1968), 1695-1700.   Google Scholar

[41]

F. RossiV. K. VanagE. Tiezzi and I. R. Epstein, Quaternary cross-diffusion in water-in-oil microemulsions loaded with a component of the Belousov-Zhabotinsky reaction, The Journal of Physical Chemistry B, 114 (2010), 8140-8146.   Google Scholar

[42]

R. Ruiz-Baier and C. Tian, Mathematical analysis and numerical simulation of pattern formation under cross-diffusion, Nonlinear Analysis: Real World Applications, 14 (2013), 601-612.  doi: 10.1016/j.nonrwa.2012.07.020.  Google Scholar

[43]

A. Schmidt and K. G. Siebert, Design of Adaptive Element Software - The Finite Element Toolbox ALBERTA, vol. 42 of Lecture Notes in Computational Science and Engineering, Springer, 2005.  Google Scholar

[44]

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.  Google Scholar

[45]

J. A. Sethian, Level Set Methods and Fast Marching Methods, volume 3 of Cam- bridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge, second edition. Evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science, 1999.  Google Scholar

[46]

L. TianZ. Lin and M. Pedersen, Instability induced by cross-diffusion in reaction-diffusion systems, Nonlinear Analysis: Real World Applications, 11 (2010), 1036-1045.  doi: 10.1016/j.nonrwa.2009.01.043.  Google Scholar

[47]

A. Turing, On the chemical basis of morphogenesis, Phil. Trans. Royal Soc. B., 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012.  Google Scholar

[48]

V. K. Vanag and I. R. Epstein, Cross-diffusion and pattern formation in reaction diffusion systems Physical Chemistry Chemical Physics, 17 (2007), 037110, 11 pp. doi: 10.1063/1.2752494.  Google Scholar

[49]

A. VergaraF. CapuanoL. Paduano and R. Sartorio, Lysozyme mutual diffusion in solutions crowded by poly(ethylene glycol), Macromolecules, 39 (2006), 4500-4506.   Google Scholar

[50]

Z. Xie, Cross-diffusion induced Turing instability for a three species food chain model, J. Math. Analy. and Appl., 388 (2012), 539-547.  doi: 10.1016/j.jmaa.2011.10.054.  Google Scholar

[51]

A. M. Zhabotinsky, A history of chemical oscillations and waves, Chaos, 1 (1991), 379-386.   Google Scholar

[52]

J.-F. Zhang, W.-T. Li, Wang, (2011). Turing patterns of a strongly coupled predator-prey system with diffusion effects, Nonlinear Analysis, 74 (2001), 847–858. doi: 10.1016/j.na.2010.09.035.  Google Scholar

[53]

E. P. Zemskov, V. K. Vanag and I. R. Epstein, Amplitude equations for reaction-diffusion systems with cross-diffusion. Phys. Rev. E., 84 (2011), 036216. Google Scholar

Figure 1.  Phase-diagrams corresponding to the non-autonomous system of ordinary differential equations (3.1)-(3.2) with exponential growth. A stable limit cycle or spiral point exists depending on the choice of the parameter values $a$ and $b$. (a) $a = 0.1$, $b = 0.75$, (b) $a = 0.15$, $b = 0.6$, (c) $a = 0.15$, $b = 0.5$, (d) $a = 0.1$, $b = 0.1$, (e) $a = 0.1$, $b = 0.75$, and (f) $a = 0.12$, $b = 0.5$. Other parameter values are fixed as follows: $r = 0.01$, $\gamma = 200$, $m = 2$, $\kappa = 2$ and $A = 1$
Figure 5.  (a) Snap-shots of the evolving parameter space for exponential growth rate with diffusion coefficient $d = 10, $ linear cross-diffusion coefficients $d_u = d_v = 0$, with $\gamma = 200$ and growth rate $r = 0.01$. We select the model kinetic parameter values $a = 0.1$ and $b = 0.75$ from the green parameter space in (a). (b)- (e) Finite element numerical simulations exhibiting the formation of spatial structure corresponding to the chemical specie $u$ during growth development. In the absence of domain growth, these patterns are non-existent
Figure 6.  (a) Snap-shots of the evolving parameter space for exponential growth rate with diffusion coefficient $d = 10, $ linear cross-diffusion coefficients $d_u = 1, d_v = 0$, with $\gamma = 200$ and growth rate $r = 0.01$. We select the model kinetic parameter values $a = 0.1$ and $b = 0.75$ from the green parameter space in (a). (b)- (h) Finite element numerical simulations exhibiting the formation of spatial structure corresponding to the chemical specie $u$ during growth development. Note that when the domain is not sufficiently large enough, no patterns are observed
Figure 7.  (a) Snap-shots of the evolving parameter space for exponential growth rate with diffusion coefficient $d = 10, $ linear cross-diffusion coefficients $d_u = 0, d_v = 1$, with $\gamma = 200$ and growth rate $r = 0.01$. We select the model kinetic parameter values $a = 0.15$ and $b = 0.6$ from the green parameter space in (a). (b)- (e) Finite element numerical simulations exhibiting the formation of spatial structure corresponding to the chemical specie $u$ during growth development
Figure 8.  (a) Snap-shots of the evolving parameter space for exponential growth rate with diffusion coefficient $d = 10, $ linear cross-diffusion coefficients $d_u = 1, d_v = 1$, with $\gamma = 200$ and growth rate $r = 0.01$. We select the model kinetic parameter values $a = 0.15$ and $b = 0.5$ from the green parameter space in (a). (b)- (f) Finite element numerical simulations exhibiting the formation of spatial structure corresponding to the chemical specie $u$ during growth development
Figure 9.  (a) Snap-shots of the evolving parameter space for exponential growth rate with diffusion coefficient $d = 1, $ linear cross-diffusion coefficients $d_u = 1, d_v = 0.5$, with $\gamma = 200$ and growth rate $r = 0.01$. We select the model kinetic parameter values $a = 0.1$ and $b = 0.1$ from the green parameter space in (a). (b)- (e) Finite element numerical simulations exhibiting the formation of spatial structure corresponding to the chemical specie $u$ during growth development
Figure 10.  (a) Snap-shots of the evolving parameter space for exponential growth rate with diffusion coefficient $d = 1, $ linear cross-diffusion coefficients $d_u = 1, d_v = 0.5$, with $\gamma = 200$ and growth rate $r = 0.01$. We select the model kinetic parameter values $a = 0.1$ and $b = 0.75$ from the green parameter space in (a). (b) - (f) Finite element numerical simulations exhibiting the formation of spatial structure corresponding to the chemical specie $u$ during growth development
Figure 11.  (a) Snap-shots of the evolving parameter space for exponential growth rate with diffusion coefficient $d = 1, $ linear cross-diffusion coefficients $d_u = -0.8, d_v = 1$, with $\gamma = 200$ and growth rate $r = 0.01$. We select the model kinetic parameter values $a = 0.15$ and $b = 0.25$ from the green parameter space in (a). (b) - (f) Finite element numerical simulations exhibiting the formation of spatial structure corresponding to the chemical specie $u$ during growth development
Figure 12.  (a) Snap-shots of the evolving parameter space for exponential growth rate with diffusion coefficient $d = 1, $ linear cross-diffusion coefficients $d_u = 1, d_v = 0.8$, with $\gamma = 200$ and growth rate $r = 0.01$. We select the model kinetic parameter values $a = 0.12$ and $b = 0.5$ from parameter space in $(a)$. (b) - (g) Finite element numerical simulations exhibiting the formation of spatial structure corresponding to the chemical specie $u$ during growth development
Figure 13.  (a)-(d) Snap-shots of the finite element numerical simulations on the evolving unit sphere under exponential growth rate with diffusion coefficient $d = 1, $ linear cross-diffusion coefficients $d_u = -0.8, d_v = 1$, with $\gamma = 200$ and growth rate $r = 0.01$. We select the model kinetic parameter values $a = 0.15$ and $b = 0.2$ from the domain-induced parameter space shown in Figure 11(a)
Figure 14.  (a)-(d) Snap-shots of the finite element numerical simulations on the evolving saddle like surface under exponential growth rate with diffusion coefficient $d = 1, $ linear cross-diffusion coefficients $d_u = 1, d_v = 0.5$, with $\gamma = 200$ and growth rate $r = 0.01$. We select the model kinetic parameter values $a = 0.1$ and $b = 0.1$ from the domain-induced parameter space shown in Figure 9(a)
Figure 2.  Snap-shots of continuously evolving parameter spaces for an exponential evolution of the domain with diffusion coefficient $d = 10$ with varying linear cross-diffusion coefficients $d_u$ and $d_v$ and the growth rate $r$. Row-wise: (a)-(c) $d_u = d_v = 0$, (d)-(f) $d_u = 0$ and $d_v = 1$, (g)-(i) $d_u = 1$ and $d_v = 0$, (j)-(l) $d_u = d_v = 1$. Column-wise: First column: $r = 0.01$, second column, $r = 0.04$ and third column: $r = 0.08$. All plots are exhibited at times $t = 0$ (no growth), $t = 0.1$ and $t = 0.3$
Figure 3.  Snap-shots of continuously evolving parameter spaces for an exponential evolution of the domain with diffusion coefficient $d = 1$ with variable linear cross-diffusion coefficients and growth rates: Row-wise: (a)-(c) $d_u = 1$, $d_v = 0.8$, (d)-(f) $d_u = 0.8$, $d_v = 1$, and (g)-(i) $d_u = -0.8$, $d_v = 1$. Column-wise: First column: $r = 0.01$, second column, $r = 0.04$ and third column: $r = 0.08$. All plots are exhibited at times $t = 0$ (no growth), $t = 0.1$ and $t = 0.3$. Such spaces do not exist if linear cross-diffusion is absent (with or without domain growth)
Figure 4.  Super imposed parameter spaces with no growth, linear, logistic and exponential growth profiles of the domain with diffusion coefficient $d = 1$ and linear cross-diffusion coefficients $d_u = 1$ and $d_v = 0.8$. The growth rate $r$ is varied row-wise accordingly: (a)-(b) $r = 0.01$, (c)-(d) $r = 0.04$, and (e)-(f) $r = 0.08$. Snap-shots of the continuously evolving parameter spaces at times $t = 0$ and $t = 0.1$ (first column) and $t = 0$ and $t = 0.3$ (second column), respectively. Without linear cross-diffusion, these spaces do not exist
Table 1.  Table illustrating the function $h(t)$ for linear, exponential and logistic growth functions on evolving domains. Similar functions can be obtained for evolving surfaces. $\kappa$ is the carrying capacity (final domain size) corresponding to the logistic growth function. In the above $h (t)$ as defined by (2.6) (or (2.7)) on planar domains (or on surfaces)
Type of growthGrowth Function $\rho (t)$$h(t)= \nabla \cdot \mathit{\boldsymbol{v}}$$q(t)=e^{-\int_{t_0}^t h(\tau) d\tau }$
Linear$\rho(t)=rt+1$ $h(t)=\frac{m\, r}{rt+1}$$q(t)= \left(\frac{1}{rt+1}\right)^m$
Exponential$\rho(t)=e^{rt}$$h(t)=m\, r$$q(t)=e^{-mrt}$
Logistic$\rho (t) = \frac{\kappa Ae^{\kappa rt}}{1+Ae^{\kappa rt}}, \, A = \frac{1}{\kappa-1}$$h(t)= m\, r\, \Big(\kappa-\rho (t)\Big)$$q(t) = \left(\frac{e^{-\kappa r t}+A}{1+A}\right)^m$
Type of growthGrowth Function $\rho (t)$$h(t)= \nabla \cdot \mathit{\boldsymbol{v}}$$q(t)=e^{-\int_{t_0}^t h(\tau) d\tau }$
Linear$\rho(t)=rt+1$ $h(t)=\frac{m\, r}{rt+1}$$q(t)= \left(\frac{1}{rt+1}\right)^m$
Exponential$\rho(t)=e^{rt}$$h(t)=m\, r$$q(t)=e^{-mrt}$
Logistic$\rho (t) = \frac{\kappa Ae^{\kappa rt}}{1+Ae^{\kappa rt}}, \, A = \frac{1}{\kappa-1}$$h(t)= m\, r\, \Big(\kappa-\rho (t)\Big)$$q(t) = \left(\frac{e^{-\kappa r t}+A}{1+A}\right)^m$
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