September  2017, 12(3): 381-401. doi: 10.3934/nhm.2017017

Nonlinear flux-limited models for chemotaxis on networks

Technische Universität Kaiserslautern, Department of Mathematics, Erwin-Schrödinger-Straße, 67663 Kaiserslautern, Germany

* Corresponding author: klar@mathematik.uni-kl.de

Received  September 2016 Revised  May 2017 Published  September 2017

Fund Project: The first author is supported by DFG grant BO 4768/1. The second author is supported by DFG grant 1105/27 and by DFG, RTG 1932

In this paper we consider macroscopic nonlinear moment models for the approximation of kinetic chemotaxis equations on a network. Coupling conditions at the nodes of the network for these models are derived from the coupling conditions of kinetic equations. The results of the different models are compared and relations to a Keller-Segel model on networks are discussed. For a numerical approximation of the governing equations an asymptotic well-balanced schemes is extended to directed graphs. Kinetic and macroscopic equations are investigated numerically and their solutions are compared for tripod and more general networks.

Citation: Raul Borsche, Axel Klar, T. N. Ha Pham. Nonlinear flux-limited models for chemotaxis on networks. Networks & Heterogeneous Media, 2017, 12 (3) : 381-401. doi: 10.3934/nhm.2017017
References:
[1]

N. BellomoA. BellouquidJ. Nieto and J. Soler, Multiscale biological tissue models and flux-limited chemotaxis for multicellular growing systems, Math. Models Methods Appl. Sci, 20 (2010), 1179-1207.  doi: 10.1142/S0218202510004568.  Google Scholar

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C. BertonP. Charrier and B. Dubroca, An asymptotic preserving relaxation scheme for a moment model of radiative transfer, C.R. Acad. Sci. Paris, Ser. I, 344 (2007), 467-472.  doi: 10.1016/j.crma.2007.02.004.  Google Scholar

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R. BorscheS. GöttlichA. Klar and P. Schillen, The scalar Keller-Segel model on networks, Math. Models Methods Appl. Sci., 24 (2014), 221-247.  doi: 10.1142/S0218202513400071.  Google Scholar

[6]

R. BorscheJ. KallA. Klar and T. N. H. Pham, Kinetic and related macroscopic models for chemotaxis on networks, Mathematical Models and Methods in Applied Sciences, 26 (2016), 1219-1242.  doi: 10.1142/S0218202516500299.  Google Scholar

[7]

N. Bournaveas and V. Calvez, The one-dimensional Keller-Segel model with fractional diffusion of cells, Nonlinearity, 23 (2010), 923-935.  doi: 10.1088/0951-7715/23/4/009.  Google Scholar

[8]

G. BrettiR. Natalini and M. Ribot, A hyperbolic model of chemotaxis on a network: A numerical study, ESAIM: M2AN, 48 (2014), 231-258.  doi: 10.1051/m2an/2013098.  Google Scholar

[9]

V. Calvez and J. A. Carrillo, Volume effects in the Keller-Segel model: Energy estimates preventing blow-up, Journal de Mathematiques Pures et Appliquees, 86 (2006), 155-175.  doi: 10.1016/j.matpur.2006.04.002.  Google Scholar

[10]

F. Camilli and L. Corrias, Parabolic models for chemotaxis on weighted networks, to appear in Journal de Mathematiques Pures et Appliquees, 2017. Google Scholar

[11]

F. ChalubP. MarkowichB. Perthame and C. Schmeiser, Kinetic models for chemotaxis and their drift-diffusion limits, Monatsh. Math., 142 (2004), 123-141.  doi: 10.1007/s00605-004-0234-7.  Google Scholar

[12]

P. H. Chavanis, Jeans type instability for a chemotactic model of cellular aggregation, Eur. Phys. J. B, 52 (2006), 433-443.  doi: 10.1140/epjb/e2006-00310-y.  Google Scholar

[13]

A. ChertockA. KurganovX. Wang and Y. Wu, On a chemotaxis model with saturated chemotactic flux, Kinet. Relat. Models, 5 (2012), 51-95.  doi: 10.3934/krm.2012.5.51.  Google Scholar

[14]

S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis, Math. Biosci., 56 (1981), 217-237.  doi: 10.1016/0025-5564(81)90055-9.  Google Scholar

[15]

S. Cordier and C. Buet, Asymptotic preserving scheme and numerical methods for radiative hydrodynamic models, C.R. Acad. Sci. Paris, Ser. I, 338 (2004), 951-956.  doi: 10.1016/j.crma.2004.04.006.  Google Scholar

[16]

R. E. Curto and L. A. Fialkow, Recursiveness, positivity, and truncated moment problems, Houston Journal of Mathematics, 17 (1991), 603-635.   Google Scholar

[17]

B. Dubroca and A. Klar, Half-moment closure for radiative transfer equations, Journal of Computational Physics, 180 (2002), 584-596.  doi: 10.1006/jcph.2002.7106.  Google Scholar

[18]

L. Fermo and A. Tosin, A fully-discrete-state kinetic theory approach to traffic flow on road networks, Math. Models Methods Appl. Sci., 25 (2015), 423-461.  doi: 10.1142/S0218202515400023.  Google Scholar

[19]

F. FilbetP. Laurençot and B. Perthame, Derivation of hyperbolic models for chemosensitive movement, J Math Biol., 50 (2005), 189-207.  doi: 10.1007/s00285-004-0286-2.  Google Scholar

[20]

L. Gosse, Localization effects and measure source terms in numerical schemes for balance laws, Mathematics of Computation, 71 (2002), 553-582.  doi: 10.1090/S0025-5718-01-01354-0.  Google Scholar

[21]

L. Gosse, Asymptotic-preserving and well-balanced schemes for the 1d cattaneo model of chemotaxis movement in both hyperbolic and diffusive regimes, Journal of Mathematical Analysis and Applications, 388 (2012), 964-983.  doi: 10.1016/j.jmaa.2011.10.039.  Google Scholar

[22]

L. Gosse and G. Toscani, An asymptotic-preserving well-balanced scheme for the hyperbolic heat equations, Comptes Rendus Mathematique, 334 (2002), 337-342.  doi: 10.1016/S1631-073X(02)02257-4.  Google Scholar

[23]

T. Goudon and C. Lin, Analysis of the m1 model: Well-posedness and diffusion asymptotics, Journal of Mathematical Analysis and Applications, 402 (2013), 579-593.  doi: 10.1016/j.jmaa.2013.01.042.  Google Scholar

[24]

J. M. Greenberg and A. Leroux, A well-balanced scheme for the numerical processing of source terms in hyperbolic equations, SIAM Journal on Numerical Analysis, 33 (1996), 1-16.  doi: 10.1137/0733001.  Google Scholar

[25]

F. R. Guarguaglini and R. Natalini, Global smooth solutions for a hyperbolic chemotaxis model on a network, SIAM J. Math. Anal., 47 (2015), 4652-4671.  doi: 10.1137/140997099.  Google Scholar

[26]

M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Ⅳ. Ser., 24 (1997), 633-683.   Google Scholar

[27]

M. Herty and S. Moutari, A macro-kinetic hybrid model for traffic flow on road networks, Comput. Methods Appl. Math., 9 (2009), 238-252.  doi: 10.2478/cmam-2009-0015.  Google Scholar

[28]

T. Hillen, Hyperbolic models for chemosensitive movement, Math. Models Methods Appl. Sci., 12 (2002), 1007-1034.  doi: 10.1142/S0218202502002008.  Google Scholar

[29]

T. Hillen and K. Painter, A user's guide to pde models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.  Google Scholar

[30]

S. JinL. Pareschi and G. Toscani, Diffusive relaxation schemes for multiscale discrete-velocity kinetic equations, SIAM J. Numer. Anal., 35 (1998), 2405-2439 (electronic).  doi: 10.1137/S0036142997315962.  Google Scholar

[31]

S. Jin and Z. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Communications on Pure and Applied Mathematics, 48 (1995), 235-276.  doi: 10.1002/cpa.3160480303.  Google Scholar

[32]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[33]

E. F. Keller and L. A. Segel, Model for chemotaxis, Journal of Theoretical Biology, 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6.  Google Scholar

[34]

D. S. Kershaw, Flux limiting nature's own way: A new method for numerical solution of the transport equation, Lawrence Livermore Laboratory, UCRL 78378. Google Scholar

[35]

C. D. Levermore, Relating eddington factors to flux limiters, Journal of Quantitative Spectroscopy and Radiative Transfer, 31 (1984), 149-160.  doi: 10.1016/0022-4073(84)90112-2.  Google Scholar

[36]

M. Rascle and C. Ziti, Finite time blow-up in some models of chemotaxis, J. Math. Biol., 33 (1995), 388-414.  doi: 10.1007/BF00176379.  Google Scholar

[37]

F. SchneiderG. AlldredgeM. Frank and A. Klar, Higher order mixed-moment approximations for the Fokker-Planck equation in one space dimension, SIAM Journal on Applied Mathematics, 74 (2014), 1087-1114.  doi: 10.1137/130934210.  Google Scholar

show all references

References:
[1]

N. BellomoA. BellouquidJ. Nieto and J. Soler, Multiscale biological tissue models and flux-limited chemotaxis for multicellular growing systems, Math. Models Methods Appl. Sci, 20 (2010), 1179-1207.  doi: 10.1142/S0218202510004568.  Google Scholar

[2]

N. Bellomo, A. Bellouquid, J. Nieto and J. Soler, On the asymptotic theory from microscopic to macroscopic growing tissue models: An overview with perspectives Mathematical Models and Methods in Applied Sciences, 22 (2012), 1130001, 37pp. doi: 10.1142/S0218202512005885.  Google Scholar

[3]

N. BellomoA. BellouquidY. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X.  Google Scholar

[4]

C. BertonP. Charrier and B. Dubroca, An asymptotic preserving relaxation scheme for a moment model of radiative transfer, C.R. Acad. Sci. Paris, Ser. I, 344 (2007), 467-472.  doi: 10.1016/j.crma.2007.02.004.  Google Scholar

[5]

R. BorscheS. GöttlichA. Klar and P. Schillen, The scalar Keller-Segel model on networks, Math. Models Methods Appl. Sci., 24 (2014), 221-247.  doi: 10.1142/S0218202513400071.  Google Scholar

[6]

R. BorscheJ. KallA. Klar and T. N. H. Pham, Kinetic and related macroscopic models for chemotaxis on networks, Mathematical Models and Methods in Applied Sciences, 26 (2016), 1219-1242.  doi: 10.1142/S0218202516500299.  Google Scholar

[7]

N. Bournaveas and V. Calvez, The one-dimensional Keller-Segel model with fractional diffusion of cells, Nonlinearity, 23 (2010), 923-935.  doi: 10.1088/0951-7715/23/4/009.  Google Scholar

[8]

G. BrettiR. Natalini and M. Ribot, A hyperbolic model of chemotaxis on a network: A numerical study, ESAIM: M2AN, 48 (2014), 231-258.  doi: 10.1051/m2an/2013098.  Google Scholar

[9]

V. Calvez and J. A. Carrillo, Volume effects in the Keller-Segel model: Energy estimates preventing blow-up, Journal de Mathematiques Pures et Appliquees, 86 (2006), 155-175.  doi: 10.1016/j.matpur.2006.04.002.  Google Scholar

[10]

F. Camilli and L. Corrias, Parabolic models for chemotaxis on weighted networks, to appear in Journal de Mathematiques Pures et Appliquees, 2017. Google Scholar

[11]

F. ChalubP. MarkowichB. Perthame and C. Schmeiser, Kinetic models for chemotaxis and their drift-diffusion limits, Monatsh. Math., 142 (2004), 123-141.  doi: 10.1007/s00605-004-0234-7.  Google Scholar

[12]

P. H. Chavanis, Jeans type instability for a chemotactic model of cellular aggregation, Eur. Phys. J. B, 52 (2006), 433-443.  doi: 10.1140/epjb/e2006-00310-y.  Google Scholar

[13]

A. ChertockA. KurganovX. Wang and Y. Wu, On a chemotaxis model with saturated chemotactic flux, Kinet. Relat. Models, 5 (2012), 51-95.  doi: 10.3934/krm.2012.5.51.  Google Scholar

[14]

S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis, Math. Biosci., 56 (1981), 217-237.  doi: 10.1016/0025-5564(81)90055-9.  Google Scholar

[15]

S. Cordier and C. Buet, Asymptotic preserving scheme and numerical methods for radiative hydrodynamic models, C.R. Acad. Sci. Paris, Ser. I, 338 (2004), 951-956.  doi: 10.1016/j.crma.2004.04.006.  Google Scholar

[16]

R. E. Curto and L. A. Fialkow, Recursiveness, positivity, and truncated moment problems, Houston Journal of Mathematics, 17 (1991), 603-635.   Google Scholar

[17]

B. Dubroca and A. Klar, Half-moment closure for radiative transfer equations, Journal of Computational Physics, 180 (2002), 584-596.  doi: 10.1006/jcph.2002.7106.  Google Scholar

[18]

L. Fermo and A. Tosin, A fully-discrete-state kinetic theory approach to traffic flow on road networks, Math. Models Methods Appl. Sci., 25 (2015), 423-461.  doi: 10.1142/S0218202515400023.  Google Scholar

[19]

F. FilbetP. Laurençot and B. Perthame, Derivation of hyperbolic models for chemosensitive movement, J Math Biol., 50 (2005), 189-207.  doi: 10.1007/s00285-004-0286-2.  Google Scholar

[20]

L. Gosse, Localization effects and measure source terms in numerical schemes for balance laws, Mathematics of Computation, 71 (2002), 553-582.  doi: 10.1090/S0025-5718-01-01354-0.  Google Scholar

[21]

L. Gosse, Asymptotic-preserving and well-balanced schemes for the 1d cattaneo model of chemotaxis movement in both hyperbolic and diffusive regimes, Journal of Mathematical Analysis and Applications, 388 (2012), 964-983.  doi: 10.1016/j.jmaa.2011.10.039.  Google Scholar

[22]

L. Gosse and G. Toscani, An asymptotic-preserving well-balanced scheme for the hyperbolic heat equations, Comptes Rendus Mathematique, 334 (2002), 337-342.  doi: 10.1016/S1631-073X(02)02257-4.  Google Scholar

[23]

T. Goudon and C. Lin, Analysis of the m1 model: Well-posedness and diffusion asymptotics, Journal of Mathematical Analysis and Applications, 402 (2013), 579-593.  doi: 10.1016/j.jmaa.2013.01.042.  Google Scholar

[24]

J. M. Greenberg and A. Leroux, A well-balanced scheme for the numerical processing of source terms in hyperbolic equations, SIAM Journal on Numerical Analysis, 33 (1996), 1-16.  doi: 10.1137/0733001.  Google Scholar

[25]

F. R. Guarguaglini and R. Natalini, Global smooth solutions for a hyperbolic chemotaxis model on a network, SIAM J. Math. Anal., 47 (2015), 4652-4671.  doi: 10.1137/140997099.  Google Scholar

[26]

M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Ⅳ. Ser., 24 (1997), 633-683.   Google Scholar

[27]

M. Herty and S. Moutari, A macro-kinetic hybrid model for traffic flow on road networks, Comput. Methods Appl. Math., 9 (2009), 238-252.  doi: 10.2478/cmam-2009-0015.  Google Scholar

[28]

T. Hillen, Hyperbolic models for chemosensitive movement, Math. Models Methods Appl. Sci., 12 (2002), 1007-1034.  doi: 10.1142/S0218202502002008.  Google Scholar

[29]

T. Hillen and K. Painter, A user's guide to pde models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.  Google Scholar

[30]

S. JinL. Pareschi and G. Toscani, Diffusive relaxation schemes for multiscale discrete-velocity kinetic equations, SIAM J. Numer. Anal., 35 (1998), 2405-2439 (electronic).  doi: 10.1137/S0036142997315962.  Google Scholar

[31]

S. Jin and Z. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Communications on Pure and Applied Mathematics, 48 (1995), 235-276.  doi: 10.1002/cpa.3160480303.  Google Scholar

[32]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[33]

E. F. Keller and L. A. Segel, Model for chemotaxis, Journal of Theoretical Biology, 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6.  Google Scholar

[34]

D. S. Kershaw, Flux limiting nature's own way: A new method for numerical solution of the transport equation, Lawrence Livermore Laboratory, UCRL 78378. Google Scholar

[35]

C. D. Levermore, Relating eddington factors to flux limiters, Journal of Quantitative Spectroscopy and Radiative Transfer, 31 (1984), 149-160.  doi: 10.1016/0022-4073(84)90112-2.  Google Scholar

[36]

M. Rascle and C. Ziti, Finite time blow-up in some models of chemotaxis, J. Math. Biol., 33 (1995), 388-414.  doi: 10.1007/BF00176379.  Google Scholar

[37]

F. SchneiderG. AlldredgeM. Frank and A. Klar, Higher order mixed-moment approximations for the Fokker-Planck equation in one space dimension, SIAM Journal on Applied Mathematics, 74 (2014), 1087-1114.  doi: 10.1137/130934210.  Google Scholar

Figure 1.  Sketch of a tripod network
Figure 2.  Numerical solutions of the four models on an interval at time $t=0.2$ with $\epsilon = 1$ (top) and $\epsilon = 0.5$ (bottom)
Figure 3.  Numerical solutions of the four models on an interval at time $t=0.2$ with $\epsilon = 0.1$ (top) and $\epsilon = 10^{-6}$ (bottom)
Figure 4.  Linear models ($P_1$) with negative full or half range densities and positivity preserving nonlinear models ($M_1$)
Figure 5.  Numerical solutions on a tripod network at time $t=0.2$, $\Delta x=0.02$, $\epsilon = 1$
Figure 6.  Numerical solutions on a tripod network at time $t=0.2$, $\Delta x=0.02$, $\epsilon = 0.5$
Figure 7.  Numerical solutions on a tripod network at time $t=0.2$, $\Delta x=0.02$, $\epsilon = 0.1$
Figure 8.  Numerical solutions on a tripod network at time $t=0.2$, $\Delta x=0.02$, $\epsilon = 10^-6$
Figure 9.  Comparison of the numerical solutions on a larger network at $t=5$
Figure 10.  Total mass over time in the large network
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