Microbiological experiments show that the colonies of the bacterium bacillus subtilis placed on a dish filled with an agar medium and nutrient form varied spatial patterns while the individual cells grow, reproduce and migrate on the dish in clumps. In this paper, we discuss a system of reaction-diffusion equations that can be used with a view to modelling this phenomenon and we solve it numerically by means of the method of lines. For the spatial discretization, we use the finite difference method and Galerkin finite element method. We present how the spatial patterns obtained depend on the spatial discretization employed and we measure the experimental order of convergence of the numerical schemes used. Further, we present the numerical results obtained by solving the model in a cubic domain.
References
[1]
J. Wakita, H. Shimada, H. Itoh, T. Matsuyama and M. Matsushita, “Periodic Colony Formation by Bacterial Species Bacillus Subtilis,” Journal of the Physical Society of Japan, Vol. 70, No. 3, 2001, pp. 911-919.
doi:10.1143/JPSJ.70.911
[2]
M. Mimura, H. Sakaguchi and M. Matsushita, “Reaction-Diffusion Modelling of Bacterial Colony Patterns,” Physica A, Vol. 282, No. 1-2, 2000, pp. 283-303.
doi:10.1016/S0378-4371(00)00085-6
[3]
J. D. Murray, “Mathematical Biology,” 3rd Edition, Springer, Berlin, 2002.
[4]
T. Vicsek, “Pattern Formation in Diffusion-Limited Aggregation,” Physical Review Letters, Vol. 53, No.24, 1984, pp. 2281-2284. doi:10.1103/PhysRevLett.53.2281
[5]
I. Golding, Y. Kozlovsky, I. Cohen and E. Ben-Jacob, “Studies of Bacterial Branching Growth Using Reaction-Diffusion Models for Colonial Development,” Physica A, Vol. 260, No. 3-4, 1998, pp. 510-554.
doi:10.1016/S0378-4371(98)00345-8
[6]
Y. Kozlovsky, I. Cohen, I. Golding and E. Ben-Jacob, “Lubricating Bacteria Model for Branching Growth of Bacterial Colonies,” Physical Review E, Vol. 59, No.6, 1999, pp. 7025-7035. doi:10.1103/PhysRevE.59.7025
[7]
R. Straka and Z. Culík, “Numerical Aspects of a Bacteria Growth Model,” Proceedings of Algoritmy, Vysoké Tatry-Podbanské, 13-18 March 2005, pp. 175-184.
[8]
E. Feireisl, D. Hilhorst, M. Mimura and R. Weidenfeld, “On a Nonlinear Diffusion System with Resource-Consumer Interaction,” Hiroshima Mathematical Journal, Vol. 33, No. 2, 2003, pp. 253-295.
[9]
S. Brenner and R. Scott, “The Mathematical Theory of Finite Element Methods,” 3rd Edition, Springer, New York, 2008. doi:10.1007/978-0-387-75934-0
[10]
H. Fujita, N. Saito and T. Suzuki, “Operator Theory and Numerical Methods,” Elsevier, Amsterdam, 2001.
[11]
V. Thomée, “Galerkin Finite Element Methods for Parabolic Problems,” Springer, Berlin, 1997.
doi:10.1007/978-3-662-03359-3
[12]
M. Benes, “Mathematical Analysis of Phase-Field Equations with Numerically Efficient Coupling Terms,” Interfaces and Free Boundaries, Vol. 3, No.2, 2001, pp. 201-221. doi:10.4171/IFB/38
[13]
T. Oberhuber, “Finite Difference Scheme for the Willmore Flow of Graphs,” Kybernetika, Vol. 43, No. 6, 2007, pp. 855-867.
[14]
J. Sembera and M. Benes, “Nonlinear Galerkin Method for Reaction-Diffusion Systems Admitting Invariant Regions,” Journal of Computational and Applied Mathematics, Vol. 136, No. 1-2, 2001, pp. 163-176.
doi:10.1016/S0377-0427(00)00582-3
[15]
W. E. Schiesser, “The Numerical Method of Lines,” Academic, New York, 1991.
[16]
G. Dziuk, “Convergence of a Semi-Discrete Scheme for the Curve Shortening Flow,” Mathematical Models and Methods in Applied Sciences, Vol. 4. No. 4, 1994, pp. 589-606. doi:10.1142/S0218202594000339
[17]
O. Pártl, “Reaction-Diffusion Systems in Mathematical Biology,” Master Thesis, Czech Technical University in Prague, Prague, 2012.
