A positivity-preserving finite element method for chemotaxis problems in 3D

We present an implicit finite element method for a class of chemotaxis models in three spatial dimensions. The proposed algorithm is designed to maintain mass conservation and to guarantee positivity of the cell density. To enforce the discrete maximum principle, the standard Galerkin discretization is constrained using a local extremum diminishing flux limiter. To demonstrate the efficiency and robustness of this approach, we solve blow-up problems in a 3D chemostat domain. To give a flavor of more complex and realistic chemotactic applications, we investigate the pattern dynamics and aggregating behavior of the bacteria Escherichia coli and Salmonella typhimurium. The obtained numerical results are in good qualitative agreement with theoretical studies and experimental data reported in the literature.

[1]  J. Z. Zhu,et al.  The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique , 1992 .

[2]  Petros Koumoutsakos,et al.  Optimization based on bacterial chemotaxis , 2002, IEEE Trans. Evol. Comput..

[3]  L. G. Stern,et al.  Fractional step methods applied to a chemotaxis model , 2000, Journal of mathematical biology.

[4]  Jerome Percus,et al.  Nonlinear aspects of chemotaxis , 1981 .

[5]  K. Painter,et al.  Volume-filling and quorum-sensing in models for chemosensitive movement , 2002 .

[6]  M. Chaplain,et al.  A model mechanism for the chemotactic response of endothelial cells to tumour angiogenesis factor. , 1993, IMA journal of mathematics applied in medicine and biology.

[7]  C. Patlak.  Random walk with persistence and external bias , 1953 .

[8]  John N. Shadid,et al.  Stability of operator splitting methods for systems with indefinite operators: Advection-diffusion-reaction systems , 2009, J. Comput. Phys..

[9]  Mingjun Wang,et al.  A Combined Chemotaxis-haptotaxis System: The Role of Logistic Source , 2009, SIAM J. Math. Anal..

[10]  Michael Winkler,et al.  Boundedness in the Higher-Dimensional Parabolic-Parabolic Chemotaxis System with Logistic Source , 2010 .

[11]  J. Adler.  Chemotaxis in Bacteria , 1966, Science.

[12]  J. Gillis,et al.  Matrix Iterative Analysis , 1961 .

[13]  Dirk Horstmann,et al.  Blow-up in a chemotaxis model without symmetry assumptions , 2001, European Journal of Applied Mathematics.

[14]  Dirk Horstmann,et al.  F ¨ Ur Mathematik in Den Naturwissenschaften Leipzig from 1970 until Present: the Keller-segel Model in Chemotaxis and Its Consequences from 1970 until Present: the Keller-segel Model in Chemotaxis and Its Consequences , 2022 .

[15]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[16]  Dirk Horstmann,et al.  Lyapunov functions and $L^{p}$-estimates for a class of reaction-diffusion systems , 2001 .

[17]  Dirk Horstmann,et al.  Boundedness vs. blow-up in a chemotaxis system , 2005 .

[18]  M. A. Herrero,et al.  A blow-up mechanism for a chemotaxis model , 1997 .

[19]  M. Furi,et al.  Multiplicity of forced oscillations for scalar differential equations , 2001 .

[20]  Masashi Aida,et al.  TARGET PATTERN SOLUTIONS FOR CHEMOTAXIS-GROWTH SYSTEM , 2004 .

[21]  Philip K. Maini,et al.  Applications of mathematical modelling to biological pattern formation , 2001 .

[22]  Christian Schmeiser,et al.  The Keller-Segel Model with Logistic Sensitivity Function and Small Diffusivity , 2005, SIAM J. Appl. Math..

[23]  J. Murray,et al.  Model and analysis of chemotactic bacterial patterns in a liquid medium , 1999, Journal of mathematical biology.

[24]  Dirk Horstmann,et al.  Generalizing the Keller–Segel Model: Lyapunov Functionals, Steady State Analysis, and Blow-Up Results for Multi-species Chemotaxis Models in the Presence of Attraction and Repulsion Between Competitive Interacting Species , 2011, J. Nonlinear Sci..

[25]  J. Murray,et al.  A minimal mechanism for bacterial pattern formation , 1999, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[26]  Dmitri Kuzmin,et al.  On the design of general-purpose flux limiters for finite element schemes. I. Scalar convection , 2006, J. Comput. Phys..

[27]  S. Zalesak.  Fully multidimensional flux-corrected transport algorithms for fluids , 1979 .

[28]  Vincent Calvez,et al.  Blow-up, Concentration Phenomenon and Global Existence for the Keller–Segel Model in High Dimension , 2010, 1003.4182.

[29]  Martin Burger,et al.  The Keller-Segel Model for Chemotaxis with Prevention of Overcrowding: Linear vs. Nonlinear Diffusion , 2006, SIAM J. Math. Anal..

[30]  Dmitri Kuzmin,et al.  Algebraic Flux Correction I. Scalar Conservation Laws , 2005 .

[31]  Alexander Kurganov,et al.  New Interior Penalty Discontinuous Galerkin Methods for the Keller-Segel Chemotaxis Model , 2008, SIAM J. Numer. Anal..

[32]  Kevin J. Painter,et al.  Spatio-temporal chaos in a chemotaxis model , 2011 .

[33]  H. Berg,et al.  Dynamics of formation of symmetrical patterns by chemotactic bacteria , 1995, Nature.

[34]  Toshitaka Nagai,et al.  Blowup of nonradial solutions to parabolic–elliptic systems modeling chemotaxis in two-dimensional domains , 2001 .

[35]  Hal L. Smith,et al.  Steady states of models of microbial growth and competition with chemotaxis , 1999 .

[36]  Dirk Horstmann,et al.  Uniqueness and symmetry of equilibria in a chemotaxis model , 2011 .

[37]  A. Jameson.  ANALYSIS AND DESIGN OF NUMERICAL SCHEMES FOR GAS DYNAMICS, 1: ARTIFICIAL DIFFUSION, UPWIND BIASING, LIMITERS AND THEIR EFFECT ON ACCURACY AND MULTIGRID CONVERGENCE , 1995 .

[38]  K. Painter,et al.  A User's Guide to Pde Models for Chemotaxis , 2022 .

[39]  Benoît Perthame,et al.  Global Solutions of Some Chemotaxis and Angiogenesis Systems in High Space Dimensions , 2004 .

[40]  R. A. ANDERSONa,et al.  Mathematical Modelling of Tumour Invasion and Metastasis , 2022 .

[41]  M. A. Herrero.  Asymptotic Properties of Reaction-Diffusion Systems Modeling Chemotaxis , 2000 .

[42]  Eshel Ben-Jacob,et al.  Studies of bacterial branching growth using reaction–diffusion models for colonial development , 1998, Physica A: Statistical Mechanics and its Applications.

[43]  Luigi Preziosi,et al.  A review of vasculogenesis models , 2005 .

[44]  J. V. Hurley,et al.  Chemotaxis , 2005, Infection.

[45]  D. Kuzmin,et al.  Algebraic Flux Correction II , 2012 .

[46]  Herbert Amann,et al.  Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems , 1990, Differential and Integral Equations.

[47]  Herbert Amann,et al.  Nonhomogeneous Linear and Quasilinear Elliptic and Parabolic Boundary Value Problems , 1993 .

[48]  Stephen Childress,et al.  Chemotactic Collapse in Two Dimensions , 1984 .

[49]  Victor A. Galaktionov.  On blow-up , 2009 .

[50]  H. Gajewski,et al.  Global Behaviour of a Reaction‐Diffusion System Modelling Chemotaxis , 1998 .

[51]  Stefan Turek,et al.  A Flux-Corrected Finite Element Method for Chemotaxis Problems , 2010, Comput. Methods Appl. Math..

[52]  D. Kuzmin,et al.  On the design of general-purpose flux limiters for implicit FEM with a consistent mass matrix , 2005 .

[53]  H. Berg,et al.  Complex patterns formed by motile cells of Escherichia coli , 1991, Nature.

[54]  Lenya Ryzhik,et al.  Traveling waves for the Keller–Segel system with Fisher birth terms , 2008 .

[55]  Atsushi Yagi,et al.  NORM BEHAVIOR OF SOLUTIONS TO A PARABOLIC SYSTEM OF CHEMOTAXIS , 1997 .

[56]  Yekaterina Epshteyn,et al.  Discontinuous Galerkin methods for the chemotaxis and haptotaxis models , 2009 .

[57]  V. Nanjundiah,et al.  Chemotaxis, signal relaying and aggregation morphology. , 1973, Journal of theoretical biology.

[58]  T. Hillen.  M5 mesoscopic and macroscopic models for mesenchymal motion , 2006, Journal of mathematical biology.

[59]  Renato Spigler.  Applied and Industrial Mathematics, Venice-2, 1998 , 2000 .

[60]  Michael Winkler,et al.  A Chemotaxis System with Logistic Source , 2007 .

[61]  Tohru Tsujikawa,et al.  Aggregating pattern dynamics in a chemotaxis model including growth , 1996 .

[62]  Benoit Perthame,et al.  Asymptotic decay for the solutions of the parabolic-parabolic Keller-Segel chemotaxis system in critical spaces , 2008, Math. Comput. Model..

[63]  Francis Filbet,et al.  A finite volume scheme for the Patlak–Keller–Segel chemotaxis model , 2006, Numerische Mathematik.

[64]  L. Segel,et al.  Initiation of slime mold aggregation viewed as an instability. , 1970, Journal of theoretical biology.

[65]  J. Peraire,et al.  TVD ALGORITHMS FOR THE SOLUTION OF THE COMPRESSIBLE EULER EQUATIONS ON UNSTRUCTURED MESHES , 1994 .

[66]  Takashi Suzuki,et al.  Parabolic System of Chemotaxis: Blowup in a Finite and the Infinite Time , 2001 .

[67]  L A Segel,et al.  A numerical study of the formation and propagation of traveling bands of chemotactic bacteria. , 1974, Journal of theoretical biology.

[68]  Luigi Preziosi,et al.  Modeling cell movement in anisotropic and heterogeneous network tissues , 2007, Networks Heterog. Media.

[69]  Samuel Finestone.  Elementany Survey Analysis. Prentice-Hall Methods of Social Science Series. James A. Davis. Englewood Cliffs, N. J: Prentice-Hall, 195 Pages, $3.95. Paperbound , 1972 .

[70]  Miguel A. Herrero,et al.  Self-similar blow-up for a reaction-diffusion system , 1998 .

[71]  L. Segel,et al.  Model for chemotaxis. , 1971, Journal of theoretical biology.

[72]  Norikazu Saito,et al.  Conservative upwind finite-element method for a simplified Keller–Segel system modelling chemotaxis , 2007 .

[73]  A. Boy,et al.  Analysis for a system of coupled reaction-diffusion parabolic equations arising in biology , 1996 .

[74]  Thomas Hillen,et al.  Metastability in Chemotaxis Models , 2005 .

[75]  A. Krall.  Applied Analysis , 1986 .

[76]  Alexander Kurganov,et al.  A second-order positivity preserving central-upwind scheme for chemotaxis and haptotaxis models , 2008, Numerische Mathematik.

[77]  Inon Cohen,et al.  Modeling branching and chiral colonial patterning of lubricating bacteria , 1999, cond-mat/9903382.

[78]  B. Perthame.  Transport Equations in Biology , 2006 .

[79]  Tohru Tsujikawa,et al.  Lower Estimate of the Attractor Dimension for a Chemotaxis Growth System , 2006 .

[80]  Miguel A. Herrero,et al.  Finite-time aggregation into a single point in a reaction - diffusion system , 1997 .

[81]  Hendrik J. Kuiper,et al.  A priori bounds and global existence for a strongly coupled quasilinear parabolic system modeling chemotaxis , 2001 .