Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-25T06:26:27.189Z Has data issue: false hasContentIssue false
  • English
  • Français

A Meshfree Technique for Numerical Simulation of Reaction-Diffusion Systems in Developmental Biology

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Mathematical biology in general

Published online by Cambridge University Press:  11 July 2017

Zahra Jannesari  and
Mehdi Tatari
Zahra Jannesari*
Affiliation:
Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, 84156-83111, Iran
Mehdi Tatari*
Affiliation:
Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, 84156-83111, Iran Institute for Research in Fundamental Sciences (IPM), P.O. Box: 19395-5746, Tehran, Iran
*
*Corresponding author. Email:[email protected] (Z. Jannesari), [email protected] (M. Tatari)
*Corresponding author. Email:[email protected] (Z. Jannesari), [email protected] (M. Tatari)
Get access

Abstract

In this work, element free Galerkin (EFG) method is posed for solving nonlinear, reaction-diffusion systems which are often employed in mathematical modeling in developmental biology. A predicator-corrector scheme is applied, to avoid directly solving of coupled nonlinear systems. The EFG method employs the moving least squares (MLS) approximation to construct shape functions. This method uses only a set of nodal points and a geometrical description of the body to discretize the governing equation. No mesh in the classical sense is needed. However a background mesh is used for integration purpose. Numerical solutions for two cases of interest, the Schnakenberg model and the Gierer-Meinhardt model, in various regions is presented to demonstrate the effects of various domain geometries on the resulting biological patterns.

Keywords

Element free Galerkin (EFG) methodreaction-diffusion systemsmeshfree methodsMLS approximationdevelopmental biology

MSC classification

Secondary: 65M99: None of the above, but in this section 92B99: None of the above, but in this section
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 9 , Issue 5 , October 2017 , pp. 1225 - 1249
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Atluri, S. N. and Zhu, T., A modified collocation method and a penalty formulation for enforcing the essential boundary conditions in the element free Galerkin method, Comput. Mech., 21 (1998), pp. 211–222.Google Scholar
[2] Atluri, S. N. and Zhu, T., A new meshless local petrov-Galerkin (MLPG) approach in computational mechanics, Comput. Mech. 22 (1998), pp. 117–127.Google Scholar
[3] Atluri, S. N. and Shen, S., The Meshless Local Petrov-Galerkin (MLPG) Method, Tech Science Press, 2002.Google Scholar
[4] Barrio, S. R., Varea, C., Aragon, J. and Maini, P., A two-dimensional numerical study of spatial pattern formation in interacting systems, Bull. Math. Biol., 61 (1999), pp. 483–505.Google Scholar
[5] Baurmanna, M., Gross, T. and Feudel, U., Instabilities in spatially extended predator-prey systems: spatio-temporal patterns in the neighborhood of Turing-Hopf bifurcations, J. Theor. Biol., 245 (2007), pp. 220–229.Google Scholar
[6] Belytschko, T., Lu, Y. and Gu, L., Element free Galerkin methods, Int. J. Num. Meth. Eng., 37 (1994), pp. 229–256.Google Scholar
[7] Chaplain, M., Ganesh, A. and Graham, I., Spatio-temporal pattern formation on spherical surfaces: numerical simulation and application to solid tumor growth, J. Math. Biol., 42 (2001), pp. 387–423.Google Scholar
[8] Crampin, E., Gaffney, E. and Maini, P., Reaction and diffusion on growing domains: Scenarios for robust pattern formation, Bull. Math. Biol., 61 (1999), pp. 1093–1120.CrossRefGoogle ScholarPubMed
[9] Dolbow, J. and Belytschko, T., Numerical integration of the Galerkin weak form in meshfree methods, Comput. Mech., 23 (1999), pp. 219–230.Google Scholar
[10] Dolbow, J. and Belytschko, T., An introduction to programming the meshless element free Galerkin method, Comput. Meth. Eng., 5(3) (1998), pp. 207–241.Google Scholar
[11] Duarte, C. A. and Oden, J. T., An hp adaptive method using clouds, Comput. Methods Appl. Mech. Eng., 139 (1996), pp. 237–262.Google Scholar
[12] Ferreira, S., Martins, M. and Vilela, M., Reaction-diffusion model for the growth of avascular tumor, Phys. Rev., 65(2) (2002), pp. 1467–1476.Google ScholarPubMed
[13] Frederik, H., Maini, P., Madzvamuse, A., Wathen, A. and Sekimura, T., Pigmentation pattern formation in butterflies: experiments and models, C. R. Biol., 326 (2003), pp. 717–727.Google Scholar
[14] García-Aznar, J., Kuiper, J., Gómez-Benito, M., Doblaré, M. and Richardson, J., Computational simulation of fracture healing: influence of interfragmentary movement on the callus growth, J. Biomech., 40 (2007), pp. 1467–1476.Google Scholar
[15] Garzón-Alvarado, D. A., Galeano, C. H. and Mantilla, J. M., Turing pattern formation for reaction-convection-diffusion systems in fixed domains submitted to toroidal velocity fields, Appl. Math. Model., 35 (2011), pp. 4913–4925.Google Scholar
[16] Gierer, A. and Meinhardt, H., A theory of biological pattern formation, Kybernetik, 12 (1972), pp. 30–39.Google Scholar
[17] Gockenbach, M. S., Understanding and Implementing the Finite Element Method, SIAM, 2006.Google Scholar
[18] Hundsdorfer, W. and Verwer, J., Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations, Springer, Berlin, 2003.Google Scholar
[19] Kassam, A. and Trefethen, L., Solving reaction-diffusion equations 10 times faster, Oxford University: Numerical Analysis Group Research, Report 16, 2003.Google Scholar
[20] Kondo, S. and Asai, R., A reaction-diffusion wave on the skin of the marine anglefish Pomacanthus, Nature, 376 (1995), pp. 765–768.Google Scholar
[21] Lancaster, P. and Salkauskas, K., Surface generated by moving least squares methods, Math. Comput., 37 (1981), pp. 141–158.Google Scholar
[22] Liu, W. K., Jun, S., Li, S., Adee, J. and Belytschko, T., Reproducing kernel particle methods for structural dynamics, Int. J. Num. Meth. Eng., 5(3) (1995), pp. 1655–1679.Google Scholar
[23] Liu, W. K., Chen, Y., Jun, S., Chen, J. S., Belytschko, T., Pan, C., Uras, R. A. and Chang, C. T., Overview and applications of the reproducing kernel particle method, Arch. Comput. Meth. Eng., 3(1) (1996), pp. 3–80.CrossRefGoogle Scholar
[24] Madzvamuse, A., A Numerical Approach to the Study of Spatial Pattern Formation, Ph.D. Thesis, University of Oxford, 2000.Google Scholar
[25] Madzvamuse, A., Wathen, A. and Maini, P., A moving grid finite element method for the simulation of pattern generation by Turing models on growing domains, J. Sci. Comput., 24 (2005), pp. 247–262.Google Scholar
[26] Madzvamuse, A., Wathen, A. and Maini, P., A moving grid finite element method applied to a model biological pattern generator, J. Comput. Phys., 190 (2003), pp. 478–500.Google Scholar
[27] Madzvamuse, A. and Maini, P., Velocity-induced numerical solution of reaction-diffusion systems on continuously growing domains, J. Comput. Phys., 225 (2007), pp. 100–119.Google Scholar
[28] Meinhardt, H., Models of Biological Pattern Formation, Academic Press, New York, 1982.Google Scholar
[29] Melenk, J. M. and Babuska, I., The partition of unity finite element method: Basic theory and applications, Comput. Methods Appl. Mech. Eng., 139 (1996), pp. 289–314.Google Scholar
[30] Monaghan, J. J., Smoothed particle hydrodynamics: Some recent improvement and applications, Annu. Rev. Astron. Phys., 30 (1992), pp. 543–574.CrossRefGoogle Scholar
[31] Murray, J. D., A prepattern formation mechanism for animal coat markings, J. Theor. Biol., 88 (1981), pp. 161–199.Google Scholar
[32] Murray, J. D., Mathematical Biology, Springer, Heidelberg, New York, 1993.Google Scholar
[33] Nayroles, B., Touzot, G. and Villon, P., Generalizing the FEM: Diffuse approximation and diffuse elements, Comput. Mech., 10 (1992), pp. 307–318.Google Scholar
[34] Oñate, E., Idelsohn, S., Zienkiewicz, O. C. and Fisher, T., Finite point method for analysis of fluid flow problems, Proceedings of the 9th Int. Conference on Finite Element Methods in Fluids. Venize, Italy, October, 1995, 15-21.Google Scholar
[35] Oñate, E., Idelsohn, S., Zienkiewicz, O. C. and Taylor, R. L., A finite point method in computational mechanics, Applications to convective transport and fluid flow, Int. J. Num. Meth. Eng., 39 (1996), pp. 3839–3866.Google Scholar
[36] Randles, P. W. and Libersky, L. D., Smoothed particle hydrodynamics: Some recent improvement and applications, Comput. Methods Appl. Mech. Eng., 139 (1996), pp. 375–408.CrossRefGoogle Scholar
[37] Shakeri, F. and Dehghan, M., The finite volume spectral element method to solve Turing models in the biological pattern formation, J. Comput. Math. Appl., 32 (2011), pp. 4322–4336.Google Scholar
[38] Schnakenberg, J., Simple chemical reaction systems with limit cycle behavior, J. Theoret. Biol., 81 (1979), pp. 389–400.Google Scholar
[39] Tatari, M., Kamranian, M. and Dehghan, M., The finite point method for reaction-diffusion systems in developmental biology, CMES., 82(1) (2011), pp. 1–27.Google Scholar
[40] Thomas, D., Artificial enzyme membrane, transport, memory and oscillatory phenomena, in: Thomas, D., Kervenez, J.-P. (Eds.), Analysis and Control of Immobilised Enzyme Systems, Springer, Berlin, Heidelberg, New York, (1975), pp. 115–150.Google Scholar
[41] Turing, A., The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London, 237 (1952), pp. 37–72.Google Scholar
[42] Yi, F., Wei, J. and Shi, J., Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Differ. Equation, 246(5) (2009), pp. 1944–1977.Google Scholar
[43] Yaw, L. L., Co-Rotational Meshfree Formulation for Large Deformation Inelastic Analysis of Two Dimensional Structural Systems, Ph.D. Thesis, University of California, 2008.Google Scholar
[44] Zhu, T., D Zhang, J. and Atluri, S. N., A local boundary integral equation (lbie) method in computational mechanics, and a meshless discretization approach, Comput. Mech., 21 (1998), pp. 223–235.Google Scholar
[45] Zhu, T., Zhang, J. D. and Atluri, S. N., A meshless local boundary integral equation (lbie) method for solving nonlinear problems, Comput. Mech., 22 (1998), pp. 174–186.Google Scholar