# Volume Filling Effect in Modelling Chemotaxis

Mathematical Modelling of Natural Phenomena (2010)

- Volume: 5, Issue: 1, page 123-147
- ISSN: 0973-5348

## Access Full Article

top## Abstract

top## How to cite

topWrzosek, D.. "Volume Filling Effect in Modelling Chemotaxis." Mathematical Modelling of Natural Phenomena 5.1 (2010): 123-147. <http://eudml.org/doc/197715>.

@article{Wrzosek2010,

abstract = {The oriented movement of biological cells or organisms in response to a chemical gradient
is called chemotaxis. The most interesting situation related to self-organization
phenomenon takes place when the cells detect and response to a chemical which is secreted
by themselves. Since pioneering works of Patlak (1953) and Keller and Segel (1970) many
particularized models have been proposed to describe the aggregation phase of this
process. Most of efforts were concentrated, so far, on mathematical models in which the
formation of aggregate is interpreted as finite time blow-up of cell density. In recently
proposed models cells are no more treated as point masses and their finite volume is
accounted for. Thus, arbitrary high cell densities are precluded in such description and a
threshold value for cells density is a priori assumed. Different modeling
approaches based on this assumption lead to a class of quasilinear parabolic systems with
strong nonlinearities including degenerate or singular diffusion. We give a survey of
analytical results on the existence and uniqueness of global-in-time solutions, their
convergence to stationary states and on a possibility of reaching the density threshold by
a solution. Unsolved problems are pointed as well.},

author = {Wrzosek, D.},

journal = {Mathematical Modelling of Natural Phenomena},

keywords = {chemotaxis equations; quasilinear parabolic equations; Lyapunov functional; degenerate diffusion; no-flux boundary condition; semi-group theory; compactness method; attractor; convergence to steady states; nonlinear elliptic problem; existence; uniqueness},

language = {eng},

month = {2},

number = {1},

pages = {123-147},

publisher = {EDP Sciences},

title = {Volume Filling Effect in Modelling Chemotaxis},

url = {http://eudml.org/doc/197715},

volume = {5},

year = {2010},

}

TY - JOUR

AU - Wrzosek, D.

TI - Volume Filling Effect in Modelling Chemotaxis

JO - Mathematical Modelling of Natural Phenomena

DA - 2010/2//

PB - EDP Sciences

VL - 5

IS - 1

SP - 123

EP - 147

AB - The oriented movement of biological cells or organisms in response to a chemical gradient
is called chemotaxis. The most interesting situation related to self-organization
phenomenon takes place when the cells detect and response to a chemical which is secreted
by themselves. Since pioneering works of Patlak (1953) and Keller and Segel (1970) many
particularized models have been proposed to describe the aggregation phase of this
process. Most of efforts were concentrated, so far, on mathematical models in which the
formation of aggregate is interpreted as finite time blow-up of cell density. In recently
proposed models cells are no more treated as point masses and their finite volume is
accounted for. Thus, arbitrary high cell densities are precluded in such description and a
threshold value for cells density is a priori assumed. Different modeling
approaches based on this assumption lead to a class of quasilinear parabolic systems with
strong nonlinearities including degenerate or singular diffusion. We give a survey of
analytical results on the existence and uniqueness of global-in-time solutions, their
convergence to stationary states and on a possibility of reaching the density threshold by
a solution. Unsolved problems are pointed as well.

LA - eng

KW - chemotaxis equations; quasilinear parabolic equations; Lyapunov functional; degenerate diffusion; no-flux boundary condition; semi-group theory; compactness method; attractor; convergence to steady states; nonlinear elliptic problem; existence; uniqueness

UR - http://eudml.org/doc/197715

ER -

## References

top- R. A. Adams. Sobolev spaces. Academic Press, New York, 1975.
- M. Alber, R. Gejji B. Kaźmierczak. Existence of global solutions of a macroscopic model of cellular motion in a chemotactic field. Applied Mathematics Letters., 22 (2009), No. 11, 1645–1648
- B. Ainsebaa, M. Bendahmaneb, A. Noussairc. A reaction–diffusion system modeling predator–prey with prey-taxis. Nonlinear Anal. R. World Appl., 9 (2008), No. 5, 2086–2105.
- H. Amann. Dynamic theory of quasilinear parabolic systems III. Global existence. Math. Z., 202 (1989), No. 2, 219–250.
- H. Amann. Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems.9–126, in: (H. Triebel, H.J. Schmeisser., eds.), Function Spaces, Differential Operators and Nonlinear Analysis. Teubner-Texte Math., 133, Teubner, Stuttgart, 1993.
- D. G. Aronson. The porous medium equation., in: (A.Fasano, M.Primicerio.,eds.) Some Problems in Nonlinear Diffusion. Lecture Notes in Mathematics., 1224, Springer, Berlin, 1986.
- M. Bendahmane, K. H. Karlsen, J. M. Urbano. On a two-sidedly degenerate chemotaxis model with volume-filling effect. Math. Models Methods Appl. Sci., 17 (2007), No. 2, 783–804.
- P. Biler. Local and global solvability of some parabolic systems modelling chemotaxis. Adv. Math. Sci. Appl. Nachr., 195 (1998), No. 8, 76–114
- M. P. Brenner, L. S. Levitov and E. O. Budrene. Physical mechanism for chemotactic pattern formation by bacteria. Biophys. J., 74 (1998), No. 4, 1677–1693.
- H. M. Byrne M. R. Owen. A new interpretation of the Keller-Segel model based on multiphase modelling. J. Math. Biol., 49 (2004), No. 6, 604–626
- F. A. C. C. Chalub J. F. Rodrigues. A class of kinetic models for chemotaxis with threshold to prevent overcrowding. Portugaliae Math., 26 (2006), No. 2, 227–250
- V. Calvez J. A. Carillo. Volume effects in the KellerSegel model: energy estimates preventing blow-up. J. Math. Pures Appl., 86 (2006), No. 2, 155–175
- T. Cieślak . The solutions of the quasilinear Keller-Segel system with the volume filling effect do not blow up whenever the Lyapunov functional is bounded from below. 127–132, in: Self-similar solutions of nonlinear PDE, Banach Center Publ., 74, Warsaw, 2006.
- T. Cieślak. Quasilinear nonuniformly parabolic system modelling chemotaxis. J. Math. Anal. Appl., 326 (2007), No. 2, 1410–1426
- T. Cieślak C. Morales-Rodrigo. Quasilinear non-uniformly parabolic-elliptic system modelling chemotaxis with volume filling effect. Existence and uniqueness of global-in-time solutions. Topol. Methods Nonlinear Anal., 29 (2007), No. 2, 361–381
- T. Cieślak M. Winkler. Finite-time blow-up in a quasilinear system of chemotaxis. Nonlinearity., 21 (2008), No. 5, 1057–1076
- Y. S. Choi Z. A. Wang. Prevention of blow up by fast diffusion in chemotaxis. J. Math. Anal. Appl., 362 (2010), No. 2, 553-564
- M. DiFrancesco J. Rosado. Fully parabolic Keller-Segel model for chemotaxis with prevention of overcrowding. Nonlinearity., 21 (2008), No. 11, 2715–2730
- Y. Dolak C. Schmeiser. The Keller-Segel model with logistic sensitivity function and small diffusivity. SIAM J. Appl. Math., 66 (2005), No. 1Cell migration, 286–308
- E. Feireisl, Ph. Laurençot H. Petzeltova. On convergence to equilibria for the Keller-Segel chemotaxis model. J.Diff.Equations., 236 (2007), No. 2, 551–569
- H. Gajewski K. Zacharias. Global behavior of a reaction-diffusion system modelling chemotaxis. Math. Nachr., 195 (1998), No. 1Cell migration, 77–114
- D. Henry. Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1981.
- M. A. Herrero J. J. L Velázquez. A blow-up mechanism for a chemotaxis model. Ann. Scuola Norm. Sup. Pisa., 24 (1997), No. 4, 633–683
- M. A. Herrero J. J. L Velázquez. Chemotactic collapse for the Keller-Segel model. J. Math. Biol., 35 (1996), No. 2, 583–623
- T. Hillen K. J. Painter. A user’s guide to PDE models for chemotaxis. J. Math. Biol., 58 (2009), No. 1–2, 183–217
- T. Hillen K. Painter. Global existence for a parabolic chemotaxis model with prevention of overcrowding. Adv. Appl. Math., 26 (2001), No. 4, 280–301
- D. Horstmann. Lyapunov functions and Lp;-estimates for a class of reaction-diffusion systems. Colloq. Math., 87 (2001), No. 1Cell migration, 113–127
- D. Horstmann. From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. I. Jahresber. Deutsch. Math.-Verein., 105 (2003), No. 3, 103–165
- D. Horstmann. From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. I. Jahresber. Deutsch. Math.-Verein., 106 (2004), No. 2, 51–69
- J. Jiang Y. Zhang. On Convergence to equilibria for a Chemotaxis Model with Volume filling effect. Asymptotic Analysis., 65 (2009), No. 1–2, 79–102
- E. Keller and L. Segel. Initiation of slime mold aggregation viewed as an instability. J. Theor. Biology.26 (1970), No. 3, 399–415.
- R. Kowalczyk, A. Gamba and L. Preciosi. On the stability of homogeneous solutions to some aggregation models. Discrete Contin. Dynam. Systems-Series B.4 (2004), No. 1Cell migration, 204–220.
- Ph. Laurençot, D. Wrzosek. A chemotaxis model with threshold density and degenerate diffusion. 273-290 in: Progress in Nonlinear Differential Equations and Their Applications., 64, Birkhäuser, Basel, 2005.
- J.-L. Lions. Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Paris, 1969.
- P. M. Lushnikov, N. Chen and M. Alber. Macroscopic dynamics of biological cells interacting via chemotaxis and direct contact. Phys. Rev. E., 78 (2008), No. 6, 061904.
- T. Nagai. Blow-up of radially symetric solutions to a chemotaxis system. Adv. Math. Sci. Appl., 5 (1995), No. 2, 581–601
- T. Nagai, T. Senba T. Suzuki. Chemotaxis collapse in a parabolic system of mathematical biology. Hiroshima Math. J., 30 (2000), No. 3, 463–497
- K. Osaki A. Yagi. Finite dimensional attractors for one dimensional Keller-Segel equations. Funkcial. Ekvac., 44 (2001), No. 3, 441–469
- K. Osaki, A. Yagi. Global existence for a chemotaxis-growth system in ℝ2. Adv. Math. Sci. Appl., 12 (2002), No. 2, 587–606.
- K. Osaki, T. Tsujikawa, A. Yagi M. Mimura. Exponential attractor for a chemotaxis-growth system of equations. Nonlinear Anal., 51 (2002), No. 1Cell migration, 119–144
- K. Painter T. HillenVolume-filling and quorum-sensing in models for chemosensitive movement. Canadian Appl. Math. Q., 10 (2002), No. 4, 501–543
- C. S. Patlak. Random walk with persistence and external bias. Bull. Math. Biol. Biophys., 15 (1953), No. 3, 311–338
- B. Perthame A. -L. Dalibard. Existence of solutions of the hyperbolic Keller-Segel model. Trans. Amer. Math. Soc., 361 (2008), No. 5, 2319–2335
- A. B. Potapov T. Hillen. Metastability in Chemotaxis Models. J. Dyn. Diff. Eq., 17 (2005), No. 2, 293-330
- R. Schaaf. Stationary solutions of Chemotaxis systems. Trans. Am. Math. Soc., 292 (1985), No. 2, 531-556
- R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer- Verlag, New York, 1988.
- J. J. L Velázquez. Point dynamics in a singular limit of the Keller-Segel model 1: motion of the concentration regions. SIAM J. Appl. Math., 64 (2004), No. 4, 1198–1223
- M. Winkler. Does a volume filling effect always prevent chemotactic colapse. Math. Meth. Appl. Sci., 33 (2010), No. 1Cell migration, 12–24
- Z.A. Wang T. Hillen. Classical solutions and pattern formation for a volume filling chemotaxis model. Chaos., 17 (2007), No. 3, 037108–037121
- D. Wrzosek. Global attractor for a chemotaxis model with prevention of overcrowding. Nonlinear Anal. TMA., 59 (2004), No. 8, 1293–1310
- D. Wrzosek. Long time behaviour of solutions to a chemotaxis model with volume filling effect. Proc. Roy. Soc. Edinburgh., 136A (2006), No. 2, 431–444
- D. Wrzosek. Chemotaxis models with a threshold cell density. in: Parabolic and Navier-Stokes equations. Part 2, 553–566, Banach Center Publ., 81, Warsaw, 2008.
- D. Wrzosek. Model of chemotaxis with threshold density and singular diffusion. Nonlinear Anal. TMA.. to appear.
- Y. Zhang, S. Zheng. Asymptotic Behavior of Solutions to a Quasilinear Nonuniform Parabolic System Modelling Chemotaxis. J. Diff. Equations. in press.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.