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January  2014, 19(1): 131-152. doi: 10.3934/dcdsb.2014.19.131

Numerical study of two-species chemotaxis models

Alexander Kurganov 1, and  Mária Lukáčová-Medvidová 2,

1. 

Mathematics Department, Tulane University, 6823 St. Charles Ave, New Orleans, LA 70118, United States
2. 

Institute of Mathematics, University of Mainz, Staudingerweg 9, 55099 Mainz, Germany

Received  December 2011 Revised  July 2013 Published  December 2013

We first conduct a comparative numerical study of two recently proposed two-species chemotaxis models. We show that different scenarios are possible: depending on the initial masses, either one or both cell densities may blow up, or a global solution may exist. In particular, our numerical results indicate answers on some open questions of possible blow up stated in [4,7]. We then introduce two regularizations of the studied models and demonstrate that their solutions are capable of developing spiky structure without blowing up.
Keywords: Two-species chemotaxis models, second-order positivity preserving finite-volume method, finite-time blow-up..
Mathematics Subject Classification: Primary: 92C17, 76M12; Secondary: 65M06, 35K55, 35B4.
Citation: Alexander Kurganov, Mária Lukáčová-Medvidová. Numerical study of two-species chemotaxis models. Discrete and Continuous Dynamical Systems - B, 2014, 19 (1) : 131-152. doi: 10.3934/dcdsb.2014.19.131
References:
[1]

A. Chertock, Y. Epshteyn and A. Kurganov, High-order finite-difference and finite-volume methods for chemotaxis models,, in preparation., (). 

[2]

A. Chertock, A. Kurganov, X. 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.

[3]

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

[4]

C. Conca, E. Espejo and K. Vilches, Remarks on the blowup and global existence for a two species chemotactic Keller-Segel system in $\mathbbR^2$, European J. Appl. Math., 22 (2011), 553-580. doi: 10.1017/S0956792511000258.

[5]

E. E. Espejo, A. Stevens and T. Suzuki, Simultaneous blowup and mass separation during collapse in an interacting system of chemotactic species, Differential Integral Equations, 25 (2012), 251-288.

[6]

E. E. Espejo, A. Stevens and J. J. L. Velázquez, A note on non-simultaneous blow-up for a drift-diffusion model, Differential Integral Equations, 23 (2010), 451-462.

[7]

E. E. Espejo, K. Vilches and C. Conca, Sharp condition for blow-up and global existence in a two species chemotactic Keller-Segel system in $\mathbbR^2$, European J. Appl. Math., 24 (2013), 297-313. doi: 10.1017/S0956792512000411.

[8]

E. E. Espejo Arenas, A. Stevens and J. J. L. Velázquez, Simultaneous finite time blow-up in a two-species model for chemotaxis, Analysis (Munich), 29 (2009), 317-338. doi: 10.1524/anly.2009.1029.

[9]

A. Fasano, A. Mancini and M. Primicerio, Equilibrium of two populations subject to chemotaxis, Math. Models Methods Appl. Sci., 14 (2004), 503-533. doi: 10.1142/S0218202504003337.

[10]

S. Gottlieb, C.-W. Shu and E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Rev., 43 (2001), 89-112. doi: 10.1137/S003614450036757X.

[11]

M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Normale Superiore, 24 (1997), 633-683.

[12]

I. Higueras, Characterizing strong stability preserving additive Runge-Kutta methods, J. Sci. Comput., 39 (2009), 115-128. doi: 10.1007/s10915-008-9252-2.

[13]

T. Hillen and K. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. in Appl. Math., 26 (2001), 280-301. doi: 10.1006/aama.2001.0721.

[14]

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

[15]

T. Hillen, K. Painter and C. Schmeiser, Global existence for chemotaxis with finite sampling radius, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 125-144. doi: 10.3934/dcdsb.2007.7.125.

[16]

D. Horstmann, From 1970 until now: The Keller-Segel model in chemotaxis and its consequences I, Jahresber. DMV, 105 (2003), 103-165.

[17]

D. Horstmann, From 1970 until now: The Keller-Segel model in chemotaxis and its consequences II, Jahresber. DMV, 106 (2004), 51-69.

[18]

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.

[19]

E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234. doi: 10.1016/0022-5193(71)90050-6.

[20]

M. Kurokiba and T. Ogawa, Finite time blow-up of the solution for a nonlinear parabolic equation of drift-diffusion type, Diff. Integral Eqns., 4 (2003), 427-452.

[21]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and quasilinear equations of parabolic type, Translated from the Russian by S. Smith, Translations of Mathematical Monographs, 23, American Mathematical Society, Providence, R.I., (1968).

[22]

K.-A. Lie and S. Noelle, On the artificial compression method for second-order nonoscillatory central difference schemes for systems of conservation laws, SIAM J. Sci. Comput., 24 (2003), 1157-1174. doi: 10.1137/S1064827501392880.

[23]

C.-S. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27. doi: 10.1016/0022-0396(88)90147-7.

[24]

T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433.

[25]

H. Nessyahu and E. Tadmor, Nonoscillatory central differencing for hyperbolic conservation laws, J. Comput. Phys., 87 (1990), 408-463. doi: 10.1016/0021-9991(90)90260-8.

[26]

W.-M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc., 45 (1998), 9-18.

[27]

C. S. Patlak, Random walk with persistence and external bias, Bull. Math: Biophys., 15 (1953), 311-338. doi: 10.1007/BF02476407.

[28]

B. Perthame, Transport Equations in Biology, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2007.

[29]

B. D. Sleeman, M. J. Ward and J. C. Wei, The existence and stability of spike patterns in a chemotaxis model, SIAM J. Appl. Math., 65 (2005), 790-817. doi: 10.1137/S0036139902415117.

[30]

P. K. Sweby, High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM J. Numer. Anal., 21 (1984), 995-1011. doi: 10.1137/0721062.

[31]

J. J. L. Velázquez, Point dynamics in a singular limit of the Keller-Segel model. I. Motion of the concentration regions, SIAM J. Appl. Math., 64 (2004), 1198-1223. doi: 10.1137/S0036139903433888.

[32]

J. J. L. Velázquez, Point dynamics in a singular limit of the Keller-Segel model. II. Formation of the concentration regions, SIAM J. Appl. Math., 64 (2004), 1224-1248. doi: 10.1137/S003613990343389X.

[33]

X. Wang, Qualitative behavior of solutions of chemotactic diffusion systems: effects of motility and chemotaxis and dynamics, SIAM J. Math. Anal., 31 (2000), 535-560. doi: 10.1137/S0036141098339897.

[34]

G. Wolansky, Multi-components chemotactic system in the absence of conflicts, European J. Appl. Math., 13 (2002), 641-661. doi: 10.1017/S0956792501004843.

show all references

References:
[1]

A. Chertock, Y. Epshteyn and A. Kurganov, High-order finite-difference and finite-volume methods for chemotaxis models,, in preparation., (). 

[2]

A. Chertock, A. Kurganov, X. 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.

[3]

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

[4]

C. Conca, E. Espejo and K. Vilches, Remarks on the blowup and global existence for a two species chemotactic Keller-Segel system in $\mathbbR^2$, European J. Appl. Math., 22 (2011), 553-580. doi: 10.1017/S0956792511000258.

[5]

E. E. Espejo, A. Stevens and T. Suzuki, Simultaneous blowup and mass separation during collapse in an interacting system of chemotactic species, Differential Integral Equations, 25 (2012), 251-288.

[6]

E. E. Espejo, A. Stevens and J. J. L. Velázquez, A note on non-simultaneous blow-up for a drift-diffusion model, Differential Integral Equations, 23 (2010), 451-462.

[7]

E. E. Espejo, K. Vilches and C. Conca, Sharp condition for blow-up and global existence in a two species chemotactic Keller-Segel system in $\mathbbR^2$, European J. Appl. Math., 24 (2013), 297-313. doi: 10.1017/S0956792512000411.

[8]

E. E. Espejo Arenas, A. Stevens and J. J. L. Velázquez, Simultaneous finite time blow-up in a two-species model for chemotaxis, Analysis (Munich), 29 (2009), 317-338. doi: 10.1524/anly.2009.1029.

[9]

A. Fasano, A. Mancini and M. Primicerio, Equilibrium of two populations subject to chemotaxis, Math. Models Methods Appl. Sci., 14 (2004), 503-533. doi: 10.1142/S0218202504003337.

[10]

S. Gottlieb, C.-W. Shu and E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Rev., 43 (2001), 89-112. doi: 10.1137/S003614450036757X.

[11]

M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Normale Superiore, 24 (1997), 633-683.

[12]

I. Higueras, Characterizing strong stability preserving additive Runge-Kutta methods, J. Sci. Comput., 39 (2009), 115-128. doi: 10.1007/s10915-008-9252-2.

[13]

T. Hillen and K. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. in Appl. Math., 26 (2001), 280-301. doi: 10.1006/aama.2001.0721.

[14]

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

[15]

T. Hillen, K. Painter and C. Schmeiser, Global existence for chemotaxis with finite sampling radius, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 125-144. doi: 10.3934/dcdsb.2007.7.125.

[16]

D. Horstmann, From 1970 until now: The Keller-Segel model in chemotaxis and its consequences I, Jahresber. DMV, 105 (2003), 103-165.

[17]

D. Horstmann, From 1970 until now: The Keller-Segel model in chemotaxis and its consequences II, Jahresber. DMV, 106 (2004), 51-69.

[18]

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.

[19]

E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234. doi: 10.1016/0022-5193(71)90050-6.

[20]

M. Kurokiba and T. Ogawa, Finite time blow-up of the solution for a nonlinear parabolic equation of drift-diffusion type, Diff. Integral Eqns., 4 (2003), 427-452.

[21]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and quasilinear equations of parabolic type, Translated from the Russian by S. Smith, Translations of Mathematical Monographs, 23, American Mathematical Society, Providence, R.I., (1968).

[22]

K.-A. Lie and S. Noelle, On the artificial compression method for second-order nonoscillatory central difference schemes for systems of conservation laws, SIAM J. Sci. Comput., 24 (2003), 1157-1174. doi: 10.1137/S1064827501392880.

[23]

C.-S. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27. doi: 10.1016/0022-0396(88)90147-7.

[24]

T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433.

[25]

H. Nessyahu and E. Tadmor, Nonoscillatory central differencing for hyperbolic conservation laws, J. Comput. Phys., 87 (1990), 408-463. doi: 10.1016/0021-9991(90)90260-8.

[26]

W.-M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc., 45 (1998), 9-18.

[27]

C. S. Patlak, Random walk with persistence and external bias, Bull. Math: Biophys., 15 (1953), 311-338. doi: 10.1007/BF02476407.

[28]

B. Perthame, Transport Equations in Biology, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2007.

[29]

B. D. Sleeman, M. J. Ward and J. C. Wei, The existence and stability of spike patterns in a chemotaxis model, SIAM J. Appl. Math., 65 (2005), 790-817. doi: 10.1137/S0036139902415117.

[30]

P. K. Sweby, High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM J. Numer. Anal., 21 (1984), 995-1011. doi: 10.1137/0721062.

[31]

J. J. L. Velázquez, Point dynamics in a singular limit of the Keller-Segel model. I. Motion of the concentration regions, SIAM J. Appl. Math., 64 (2004), 1198-1223. doi: 10.1137/S0036139903433888.

[32]

J. J. L. Velázquez, Point dynamics in a singular limit of the Keller-Segel model. II. Formation of the concentration regions, SIAM J. Appl. Math., 64 (2004), 1224-1248. doi: 10.1137/S003613990343389X.

[33]

X. Wang, Qualitative behavior of solutions of chemotactic diffusion systems: effects of motility and chemotaxis and dynamics, SIAM J. Math. Anal., 31 (2000), 535-560. doi: 10.1137/S0036141098339897.

[34]

G. Wolansky, Multi-components chemotactic system in the absence of conflicts, European J. Appl. Math., 13 (2002), 641-661. doi: 10.1017/S0956792501004843.

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