December  2013, 18(10): 2689-2704. doi: 10.3934/dcdsb.2013.18.2689

Numerical simulation of chemotaxis models on stationary surfaces

1. 

Institut für Angewandte Mathematik, TU Dortmund, 44227 Dortmund, Germany, Germany, Germany

Received  November 2012 Revised  April 2013 Published  October 2013

In this paper we present an implicit finite element method for a class of chemotaxis models, where a new linearized flux-corrected transport (FCT) algorithm is modified in such a way as to keep the density of on-surface living cells nonnegative. Level set techniques are adopted for an implicit description of the surface and for the numerical treatment of the corresponding system of partial differential equations. The presented scheme is able to deliver a robust and accurate solution for a large class of chemotaxis-driven models. The numerical behavior of the proposed scheme is tested on the blow-up model on a sphere and an ellipsoid and on the pattern-forming dynamics model of Escherichia coli on a sphere.
Citation: Andriy Sokolov, Robert Strehl, Stefan Turek. Numerical simulation of chemotaxis models on stationary surfaces. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2689-2704. doi: 10.3934/dcdsb.2013.18.2689
References:
[1]

M. Aida, T. Tsujikawa, M. Efendiev, A. Yagi and M. Mimura, Lower estimate of the attractor dimension for a chemotaxis growth system,, J. London Math. Soc., 74 (2006), 453. doi: 10.1112/S0024610706023015.

[2]

M. Aida and A. Yagi, Target pattern solutions for chemotaxis-growth system,, Scientiae Mathematicae Japonicae, 59 (2004), 577.

[3]

D. Ambrosi, F. Bussolino and L. Preziosi, A review of vasculogenesis models,, J. Theor. Med., 6 (2005), 1. doi: 10.1080/1027366042000327098.

[4]

A. R. A. Anderson, M. A. J. Chaplain, E. L. Newman, R. J. C. Steele and A. M. Thompson, Mathematical modelling of tumour invasion and metastasis,, Journal of Theoretical Medicine, 2 (2000), 129. doi: 10.1080/10273660008833042.

[5]

M. Bergdorf, I. F. Sbalzarini and P. Koumoutsakos, A Lagrangian particle method for reaction-diffusion systems on deforming surfaces,, J. Math. Biol., 61 (2010), 649. doi: 10.1007/s00285-009-0315-2.

[6]

M. A. J. Chaplain, The mathematical modelling of tumour angiogenesis and invasion,, ACTA Biotheoretica, 43 (1995), 387. doi: 10.1007/BF00713561.

[7]

M. A. J. Chaplain, Continuous and discrete mathematical models of tumor-induced angiogenesis,, Bulletin of Mathematical Biology, 60 (1998), 857.

[8]

M. A. J. Chaplain, Mathematical modelling of angiogenesis,, Journal of Neuro-Oncology, 50 (2000), 37.

[9]

M. A. J. Chaplain and A. M. Stuart, A model mechanism for the chemotactic response of endothelial cells to tumour angiogenesis factor,, IMA Journal of Mathematics Applied in Medicine and Biology, 10 (1993), 149. doi: 10.1093/imammb/10.3.149.

[10]

A. Chertock and A. Kurganov, A second-order positivity preserving central-upwind scheme for chemotaxis and haptotaxis models,, Numer. Math., 111 (2008), 169. doi: 10.1007/s00211-008-0188-0.

[11]

L. Corriasa, B. Perthame and H. Zaag, Global solutions of some chemotaxis and angiogenesis systems in high space dimensions,, Milan J. Math., 72 (2004), 1. doi: 10.1007/s00032-003-0026-x.

[12]

L. Corriasa and B. Perthame, Asymptotic decay for the solutions of the parabolic Keller-Segel chemotaxis system in critical spaces,, Mathematical and Computer Modelling, 47 (2008), 755. doi: 10.1016/j.mcm.2007.06.005.

[13]

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

[14]

Y. Epshteyn, Discontinuous Galerkin methods for the chemotaxis and haptotaxis models,, J. Comput. Appl. Math., 224 (2009), 168. doi: 10.1016/j.cam.2008.04.030.

[15]

Y. Epshteyn and A. Kurganov, New interior penalty discontinuous Galerkin methods for the Keller-Segel chemotaxis model,, SIAM J. Numer. Anal., 47 (): 386. doi: 10.1137/07070423X.

[16]

F. Filbet, A finite volume scheme for the Patlak-Keller-Segel chemotaxis model,, Numer. Math., 104 (2006), 457. doi: 10.1007/s00211-006-0024-3.

[17]

H. Gajewski, W. Jäger and A. Koshelev, About loss of regularity and 'blow up' of solutions for quasilinear parabolic systems,, R-Report, (1993).

[18]

H. Gajewski and K. Zacharias, Global behavior of a reaction-diffusion system modelling chemotaxis,, Math. Nachr., 195 (1998), 77. doi: 10.1002/mana.19981950106.

[19]

A. Gamba, D. Ambrosi, A. Coniglio, A. de Candia, S. Di Talia, E. Giraudo, G. Serini, L. Preziosi and F. Bussolino, Percolation, morphogenesis, and Burgers dynamics in blood vessels formation,, Phys. Rev. Lett., 90 (2003). doi: 10.1103/PhysRevLett.90.118101.

[20]

D. Horstmann, 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,, J. Nonlinear Sci., 21 (2011), 231. doi: 10.1007/s00332-010-9082-x.

[21]

D. Horstmann and M. Lucia, Uniqueness and symmetry of equilibria in a chemotaxis model,, Journal für die Reine und angewandte Mathematik (Crelle's Journal), 654 (2011), 83. doi: 10.1515/CRELLE.2011.030.

[22]

D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions,, European Journal of Applied Mathematics, 12 (2001), 159. doi: 10.1017/S0956792501004363.

[23]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system,, Journal of Differential Equations, 215 (2005), 52. doi: 10.1016/j.jde.2004.10.022.

[24]

E. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399. doi: 10.1016/0022-5193(70)90092-5.

[25]

E. Keller and L. Segel, Model for chemotaxis,, J. Theor. Biol., 30 (1971), 225. doi: 10.1016/0022-5193(71)90050-6.

[26]

D. Kuzmin, R. Löhner and S. Turek, Flux-Corrected Transport,, Springer, (2012).

[27]

D. Kuzmin and M. Möller, Algebraic flux correction I. Scalar conservation laws,, in Flux-Corrected Transport: Principles, (2005), 155. doi: 10.1007/3-540-27206-2_6.

[28]

D. Kuzmin and S. Turek, Flux correction tools for finite elements,, J. Comput. Phys., 175 (2002), 525. doi: 10.1006/jcph.2001.6955.

[29]

D. Kuzmin, Explicit and implicit FEM-TVD algorithms with flux linearization,, J. Comput. Phys., 228 (2009), 2517. doi: 10.1016/j.jcp.2008.12.011.

[30]

D. Kuzmin, Linearity-preserving flux correction and convergence acceleration for constrained Galerkin schemes,, Journal of Computational and Applied Mathematics, 236 (2012), 2317. doi: 10.1016/j.cam.2011.11.019.

[31]

R. J. LeVeque, Wave propagation algorithms for multi-dimensional hyperbolic systems,, J. Comput. Phys., 131 (1997), 327.

[32]

M. Mimura and T. Tsujikawa, Aggregating pattern dynamics in a chemotaxis model including growth,, Physica A, 230 (1996), 499.

[33]

T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains,, J. Inequal. Appl., 6 (2001), 37. doi: 10.1155/S1025583401000042.

[34]

K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations,, Funkcialaj Ekvacioj, 44 (2001), 441.

[35]

N. Saito, Conservative upwind finite-element method for a simplified Keller-Segel system modelling chemotaxis,, IMA J. Numer. Anal., 27 (2007), 332. doi: 10.1093/imanum/drl018.

[36]

T. Senba and T. Suzuki, Parabolic System of Chemotaxis: Blowup in a Finite and the Infinite Time,, IMS Workshop on Reaction-Diffusion Systems (Shatin, 8 (2001), 349.

[37]

G. Serini, D. Ambrosi, E. Giraudo, A. Gamba, L. Preziosi and F. Bussolino, Modeling the early stages of vascular network assembly,, The EMBO Journal, 22 (2003), 1771. doi: 10.1093/emboj/cdg176.

[38]

R. Strehl, A. Sokolov, D. Kuzmin and S. Turek, A flux-corrected finite element method for chemotaxis problems,, Computational methods in applied mathematics, 10 (2010), 219. doi: 10.2478/cmam-2010-0013.

[39]

R. Strehl, A. Sokolov, D. Kuzmin, D. Horstmann and S. Turek, A positivity-preserving finite element method for chemotaxis problems in 3D,, Journal of Computational and Applied Mathematics, 239 (2013), 290. doi: 10.1016/j.cam.2012.09.041.

[40]

R. Strehl, A. Sokolov and S. Turek, Efficient, accurate and flexible Finite Element solvers for Chemotaxis problems,, Computers and Mathematics with Applications, 64 (2012), 175. doi: 10.1016/j.camwa.2011.12.040.

[41]

S. Turek, Efficient Solvers for Incompressible Flow Problems: An Algorithmic and Computational Approach,, Springer, (1999). doi: 10.1007/3-540-48092-7.

[42]

R. Tyson, S. R. Lubkin and J. D. Murray, A minimal mechanism for bacterial pattern formation,, Proc. R. Soc. Lond. B, 266 (1999), 299. doi: 10.1098/rspb.1999.0637.

[43]

R. Tyson, S. R. Lubkin and J. D. Murray, Model and analysis of chemotactic bacterial patterns in a liquid medium,, J. Math. Biol., 38 (1999), 359. doi: 10.1007/s002850050153.

[44]

R. Tyson, L. G. Stern and R. J. LeVeque, Fractional step methods applied to a chemotaxis model,, J. Math. Biol., 41 (2000), 455. doi: 10.1007/s002850000038.

[45]

C. M. Elliott, B. Stinner and C. Venkataraman, Modelling cell motility and chemotaxis with evolving surface finite elements,, J. R. Soc. Interface, 9 (2012), 3027. doi: 10.1098/rsif.2012.0276.

[46]

C. Landsberg, F. Stenger, A. Deutsch, M. Gelinsky, A. Rösen-Wolff and A. Voigt, Chemotaxis of mesenchymal stem cells within 3D biomimetic scaffolds-a modeling approach,, J. Biomech, 44 (2011), 359.

[47]

S. T. Zalesak, Fully multidimensional flux-corrected transport algorithms for fluids,, J. Comput. Phys., 31 (1979), 335. doi: 10.1016/0021-9991(79)90051-2.

show all references

References:
[1]

M. Aida, T. Tsujikawa, M. Efendiev, A. Yagi and M. Mimura, Lower estimate of the attractor dimension for a chemotaxis growth system,, J. London Math. Soc., 74 (2006), 453. doi: 10.1112/S0024610706023015.

[2]

M. Aida and A. Yagi, Target pattern solutions for chemotaxis-growth system,, Scientiae Mathematicae Japonicae, 59 (2004), 577.

[3]

D. Ambrosi, F. Bussolino and L. Preziosi, A review of vasculogenesis models,, J. Theor. Med., 6 (2005), 1. doi: 10.1080/1027366042000327098.

[4]

A. R. A. Anderson, M. A. J. Chaplain, E. L. Newman, R. J. C. Steele and A. M. Thompson, Mathematical modelling of tumour invasion and metastasis,, Journal of Theoretical Medicine, 2 (2000), 129. doi: 10.1080/10273660008833042.

[5]

M. Bergdorf, I. F. Sbalzarini and P. Koumoutsakos, A Lagrangian particle method for reaction-diffusion systems on deforming surfaces,, J. Math. Biol., 61 (2010), 649. doi: 10.1007/s00285-009-0315-2.

[6]

M. A. J. Chaplain, The mathematical modelling of tumour angiogenesis and invasion,, ACTA Biotheoretica, 43 (1995), 387. doi: 10.1007/BF00713561.

[7]

M. A. J. Chaplain, Continuous and discrete mathematical models of tumor-induced angiogenesis,, Bulletin of Mathematical Biology, 60 (1998), 857.

[8]

M. A. J. Chaplain, Mathematical modelling of angiogenesis,, Journal of Neuro-Oncology, 50 (2000), 37.

[9]

M. A. J. Chaplain and A. M. Stuart, A model mechanism for the chemotactic response of endothelial cells to tumour angiogenesis factor,, IMA Journal of Mathematics Applied in Medicine and Biology, 10 (1993), 149. doi: 10.1093/imammb/10.3.149.

[10]

A. Chertock and A. Kurganov, A second-order positivity preserving central-upwind scheme for chemotaxis and haptotaxis models,, Numer. Math., 111 (2008), 169. doi: 10.1007/s00211-008-0188-0.

[11]

L. Corriasa, B. Perthame and H. Zaag, Global solutions of some chemotaxis and angiogenesis systems in high space dimensions,, Milan J. Math., 72 (2004), 1. doi: 10.1007/s00032-003-0026-x.

[12]

L. Corriasa and B. Perthame, Asymptotic decay for the solutions of the parabolic Keller-Segel chemotaxis system in critical spaces,, Mathematical and Computer Modelling, 47 (2008), 755. doi: 10.1016/j.mcm.2007.06.005.

[13]

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

[14]

Y. Epshteyn, Discontinuous Galerkin methods for the chemotaxis and haptotaxis models,, J. Comput. Appl. Math., 224 (2009), 168. doi: 10.1016/j.cam.2008.04.030.

[15]

Y. Epshteyn and A. Kurganov, New interior penalty discontinuous Galerkin methods for the Keller-Segel chemotaxis model,, SIAM J. Numer. Anal., 47 (): 386. doi: 10.1137/07070423X.

[16]

F. Filbet, A finite volume scheme for the Patlak-Keller-Segel chemotaxis model,, Numer. Math., 104 (2006), 457. doi: 10.1007/s00211-006-0024-3.

[17]

H. Gajewski, W. Jäger and A. Koshelev, About loss of regularity and 'blow up' of solutions for quasilinear parabolic systems,, R-Report, (1993).

[18]

H. Gajewski and K. Zacharias, Global behavior of a reaction-diffusion system modelling chemotaxis,, Math. Nachr., 195 (1998), 77. doi: 10.1002/mana.19981950106.

[19]

A. Gamba, D. Ambrosi, A. Coniglio, A. de Candia, S. Di Talia, E. Giraudo, G. Serini, L. Preziosi and F. Bussolino, Percolation, morphogenesis, and Burgers dynamics in blood vessels formation,, Phys. Rev. Lett., 90 (2003). doi: 10.1103/PhysRevLett.90.118101.

[20]

D. Horstmann, 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,, J. Nonlinear Sci., 21 (2011), 231. doi: 10.1007/s00332-010-9082-x.

[21]

D. Horstmann and M. Lucia, Uniqueness and symmetry of equilibria in a chemotaxis model,, Journal für die Reine und angewandte Mathematik (Crelle's Journal), 654 (2011), 83. doi: 10.1515/CRELLE.2011.030.

[22]

D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions,, European Journal of Applied Mathematics, 12 (2001), 159. doi: 10.1017/S0956792501004363.

[23]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system,, Journal of Differential Equations, 215 (2005), 52. doi: 10.1016/j.jde.2004.10.022.

[24]

E. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399. doi: 10.1016/0022-5193(70)90092-5.

[25]

E. Keller and L. Segel, Model for chemotaxis,, J. Theor. Biol., 30 (1971), 225. doi: 10.1016/0022-5193(71)90050-6.

[26]

D. Kuzmin, R. Löhner and S. Turek, Flux-Corrected Transport,, Springer, (2012).

[27]

D. Kuzmin and M. Möller, Algebraic flux correction I. Scalar conservation laws,, in Flux-Corrected Transport: Principles, (2005), 155. doi: 10.1007/3-540-27206-2_6.

[28]

D. Kuzmin and S. Turek, Flux correction tools for finite elements,, J. Comput. Phys., 175 (2002), 525. doi: 10.1006/jcph.2001.6955.

[29]

D. Kuzmin, Explicit and implicit FEM-TVD algorithms with flux linearization,, J. Comput. Phys., 228 (2009), 2517. doi: 10.1016/j.jcp.2008.12.011.

[30]

D. Kuzmin, Linearity-preserving flux correction and convergence acceleration for constrained Galerkin schemes,, Journal of Computational and Applied Mathematics, 236 (2012), 2317. doi: 10.1016/j.cam.2011.11.019.

[31]

R. J. LeVeque, Wave propagation algorithms for multi-dimensional hyperbolic systems,, J. Comput. Phys., 131 (1997), 327.

[32]

M. Mimura and T. Tsujikawa, Aggregating pattern dynamics in a chemotaxis model including growth,, Physica A, 230 (1996), 499.

[33]

T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains,, J. Inequal. Appl., 6 (2001), 37. doi: 10.1155/S1025583401000042.

[34]

K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations,, Funkcialaj Ekvacioj, 44 (2001), 441.

[35]

N. Saito, Conservative upwind finite-element method for a simplified Keller-Segel system modelling chemotaxis,, IMA J. Numer. Anal., 27 (2007), 332. doi: 10.1093/imanum/drl018.

[36]

T. Senba and T. Suzuki, Parabolic System of Chemotaxis: Blowup in a Finite and the Infinite Time,, IMS Workshop on Reaction-Diffusion Systems (Shatin, 8 (2001), 349.

[37]

G. Serini, D. Ambrosi, E. Giraudo, A. Gamba, L. Preziosi and F. Bussolino, Modeling the early stages of vascular network assembly,, The EMBO Journal, 22 (2003), 1771. doi: 10.1093/emboj/cdg176.

[38]

R. Strehl, A. Sokolov, D. Kuzmin and S. Turek, A flux-corrected finite element method for chemotaxis problems,, Computational methods in applied mathematics, 10 (2010), 219. doi: 10.2478/cmam-2010-0013.

[39]

R. Strehl, A. Sokolov, D. Kuzmin, D. Horstmann and S. Turek, A positivity-preserving finite element method for chemotaxis problems in 3D,, Journal of Computational and Applied Mathematics, 239 (2013), 290. doi: 10.1016/j.cam.2012.09.041.

[40]

R. Strehl, A. Sokolov and S. Turek, Efficient, accurate and flexible Finite Element solvers for Chemotaxis problems,, Computers and Mathematics with Applications, 64 (2012), 175. doi: 10.1016/j.camwa.2011.12.040.

[41]

S. Turek, Efficient Solvers for Incompressible Flow Problems: An Algorithmic and Computational Approach,, Springer, (1999). doi: 10.1007/3-540-48092-7.

[42]

R. Tyson, S. R. Lubkin and J. D. Murray, A minimal mechanism for bacterial pattern formation,, Proc. R. Soc. Lond. B, 266 (1999), 299. doi: 10.1098/rspb.1999.0637.

[43]

R. Tyson, S. R. Lubkin and J. D. Murray, Model and analysis of chemotactic bacterial patterns in a liquid medium,, J. Math. Biol., 38 (1999), 359. doi: 10.1007/s002850050153.

[44]

R. Tyson, L. G. Stern and R. J. LeVeque, Fractional step methods applied to a chemotaxis model,, J. Math. Biol., 41 (2000), 455. doi: 10.1007/s002850000038.

[45]

C. M. Elliott, B. Stinner and C. Venkataraman, Modelling cell motility and chemotaxis with evolving surface finite elements,, J. R. Soc. Interface, 9 (2012), 3027. doi: 10.1098/rsif.2012.0276.

[46]

C. Landsberg, F. Stenger, A. Deutsch, M. Gelinsky, A. Rösen-Wolff and A. Voigt, Chemotaxis of mesenchymal stem cells within 3D biomimetic scaffolds-a modeling approach,, J. Biomech, 44 (2011), 359.

[47]

S. T. Zalesak, Fully multidimensional flux-corrected transport algorithms for fluids,, J. Comput. Phys., 31 (1979), 335. doi: 10.1016/0021-9991(79)90051-2.

[1]

Peter Rashkov. Remarks on pattern formation in a model for hair follicle spacing. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1555-1572. doi: 10.3934/dcdsb.2015.20.1555

[2]

Rui Peng, Fengqi Yi. On spatiotemporal pattern formation in a diffusive bimolecular model. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 217-230. doi: 10.3934/dcdsb.2011.15.217

[3]

Artem Dudko. Computability of the Julia set. Nonrecurrent critical orbits. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2751-2778. doi: 10.3934/dcds.2014.34.2751

[4]

R.A. Satnoianu, Philip K. Maini, F.S. Garduno, J.P. Armitage. Travelling waves in a nonlinear degenerate diffusion model for bacterial pattern formation. Discrete & Continuous Dynamical Systems - B, 2001, 1 (3) : 339-362. doi: 10.3934/dcdsb.2001.1.339

[5]

Guanqi Liu, Yuwen Wang. Pattern formation of a coupled two-cell Schnakenberg model. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1051-1062. doi: 10.3934/dcdss.2017056

[6]

Qingyan Shi, Junping Shi, Yongli Song. Hopf bifurcation and pattern formation in a delayed diffusive logistic model with spatial heterogeneity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 467-486. doi: 10.3934/dcdsb.2018182

[7]

Wonlyul Ko, Inkyung Ahn. Pattern formation of a diffusive eco-epidemiological model with predator-prey interaction. Communications on Pure & Applied Analysis, 2018, 17 (2) : 375-389. doi: 10.3934/cpaa.2018021

[8]

Xiaoying Wang, Xingfu Zou. Pattern formation of a predator-prey model with the cost of anti-predator behaviors. Mathematical Biosciences & Engineering, 2018, 15 (3) : 775-805. doi: 10.3934/mbe.2018035

[9]

Julien Cividini. Pattern formation in 2D traffic flows. Discrete & Continuous Dynamical Systems - S, 2014, 7 (3) : 395-409. doi: 10.3934/dcdss.2014.7.395

[10]

Yuan Lou, Wei-Ming Ni, Shoji Yotsutani. Pattern formation in a cross-diffusion system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1589-1607. doi: 10.3934/dcds.2015.35.1589

[11]

Tian Ma, Shouhong Wang. Dynamic transition and pattern formation for chemotactic systems. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2809-2835. doi: 10.3934/dcdsb.2014.19.2809

[12]

Taylan Sengul, Shouhong Wang. Pattern formation and dynamic transition for magnetohydrodynamic convection. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2609-2639. doi: 10.3934/cpaa.2014.13.2609

[13]

Martin Baurmann, Wolfgang Ebenhöh, Ulrike Feudel. Turing instabilities and pattern formation in a benthic nutrient-microorganism system. Mathematical Biosciences & Engineering, 2004, 1 (1) : 111-130. doi: 10.3934/mbe.2004.1.111

[14]

Ping Liu, Junping Shi, Zhi-An Wang. Pattern formation of the attraction-repulsion Keller-Segel system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2597-2625. doi: 10.3934/dcdsb.2013.18.2597

[15]

Fengqi Yi, Eamonn A. Gaffney, Sungrim Seirin-Lee. The bifurcation analysis of turing pattern formation induced by delay and diffusion in the Schnakenberg system. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 647-668. doi: 10.3934/dcdsb.2017031

[16]

Hyung Ju Hwang, Thomas P. Witelski. Short-time pattern formation in thin film equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 867-885. doi: 10.3934/dcds.2009.23.867

[17]

H. Malchow, F.M. Hilker, S.V. Petrovskii. Noise and productivity dependence of spatiotemporal pattern formation in a prey-predator system. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 705-711. doi: 10.3934/dcdsb.2004.4.705

[18]

Kolade M. Owolabi. Numerical analysis and pattern formation process for space-fractional superdiffusive systems. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 543-566. doi: 10.3934/dcdss.2019036

[19]

Christian Kuehn, Pasha Tkachov. Pattern formation in the doubly-nonlocal Fisher-KPP equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2077-2100. doi: 10.3934/dcds.2019087

[20]

Mahadevan Ganesh, Brandon C. Reyes, Avi Purkayastha. An FEM-MLMC algorithm for a moving shutter diffraction in time stochastic model. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 257-272. doi: 10.3934/dcdsb.2018107

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]