Kinetic Models for Chemotaxis and their Drift-Diffusion Limits

Chalub, Fabio A. C. C.; Markowich, Peter A.; Perthame, Benôıt; Schmeiser, Christian

doi:10.1007/s00605-004-0234-7

Cited by 211 publications

(294 citation statements)

References 28 publications

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“…The existence theory for the nonlinear system (12)- (13) was settled in [13] (thus extending a result in [33] in the linear case) and yields the Theorem 4.1 In dimension d = 3, assume that V is bounded, that (14) holds and that f 0 ∈ L ∞ (R d ×V ), then there is a unique solution to the system (12)- (13),…”

Section: Nonlinear Scattering Equationmentioning

confidence: 93%

“…Escherichia Coli bacterium for instance is known to move with a sequence of runs and tumbles, see [1,54,53,62,13] and the references therein. The advantage of the kinetic model is that, not only it provides a better detailed description of the movement, but also unifies the two macroscopic models by asymptotic derivations using either a diffusion or a hydrodynamic scale.…”

Section: Introductionmentioning

confidence: 99%

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PDE Models for Chemotactic Movements: Parabolic, Hyperbolic and Kinetic

Perthame

2004

Appl Math

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116

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Modeling the movement of cells (bacteria, amoeba) is a long standing subject and Partial Differential equations have been used several times. The most classical and successful system was proposed by Patlak [55] and Keller & Segel [39] and is formed of parabolic or elliptic equations coupled through a drift term. This model exhibits a very deep mathematical structure because smooth solutions exist for small initial norm (in the appropriate space) and blow-up for large norms. This reflects experiments on bacteria like Escherichia Coli or amoeba like Dictyostelium Discoïdeum exhibiting pointwise concentrations.For human endothelial cells, several experiments show the formation of networks that can be interpreted as the initiation of angiogenesis. To recover such patterns a hydrodynamical model seems better adapted.The two systems can be unified by a kinetic approach that was proposed for Escherichia Coli, based on more precise experiments showing a movement by 'jump and tumble'. This nonlinear kinetic model is interesting by itself and the existence theory is not complete. It is also interesting from a scaling point of view; in a diffusion limit one recovers the Keller-Segel model and in a hydrodynamical limit one recovers the model proposed for human endothelial cells.We also mention the mathematical interest of a analyzing another degenerate parabolic system (exhibiting different properties) proposed to describe the angiogenesis phenomena i.e. the formation of capillary blood vessels. 1

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Section: Nonlinear Scattering Equationmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

PDE Models for Chemotactic Movements: Parabolic, Hyperbolic and Kinetic

Perthame

2004

Appl Math

Self Cite

116

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“…The influence of S, the density of the chemical substance, is highlighted in the notation T [S]. One could follow this model in [3][4][5][8][9][10]13,[15][16][17]21].…”

Section: T [S](t X V → V)f (T X V ) Dv − λ[S](t V X)f (T V X)mentioning

confidence: 99%

“…By assuming a very fast diffusion of the chemical substance, one could use the following elliptic equation, (e.g. see [2,4,5,9,17])…”

Section: T [S](t X V → V) = χ[V · ∇S(t X)] + mentioning

confidence: 99%

One-dimensional chemotaxis kinetic model

tabar

2010

Nonlinear Differ. Equ. Appl.

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Abstract. In this paper, we study a variation of the equations of a chemotaxis kinetic model and investigate it in one dimension. In fact, we use fractional diffusion for the chemoattractant in the Othmar-Dunbar-Alt system (Othmer in J Math Biol 26(3): 1988). This version was exhibited in Calvez in Amer Math Soc, pp [45][46][47][48][49][50][51][52][53][54][55][56][57][58][59][60][61][62] 2007 for the macroscopic well-known Keller-Segel model in all space dimensions. These two macroscopic and kinetic models are related as mentioned in Bournaveas, Ann Inst H Poincaré

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“…Various contributions are given by papers [3], [4], [5], [17], [18], [20], [25], [27], while the book [11] and the survey [8] report about applications of the kinetic theory to model complex systems in biology. Moreover different models in life sciences are presented in the book [2] and the survey [7].…”

Section: Introductionmentioning

confidence: 99%

From kinetic models of multicellular growing systems to macroscopic biological tissue models

Bellouquid

Angelis

2011

Nonlinear Analysis: Real World Applications

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This paper deals with the derivation of macroscopic equations for a class of equations modelling complex multicellular systems. Cellular interactions generate both modification of biological functions and proliferative/destructive events. The asymptotic analysis refers to the derivation of hyperbolic models focused on the influence of existence of a global equilibrium solution. The asymptotic analysis shows how the macroscopic tissue behavior can be described from the underlying cellular description.

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Kinetic Models for Chemotaxis and their Drift-Diffusion Limits

Cited by 211 publications

References 28 publications

PDE Models for Chemotactic Movements: Parabolic, Hyperbolic and Kinetic

PDE Models for Chemotactic Movements: Parabolic, Hyperbolic and Kinetic

One-dimensional chemotaxis kinetic model

From kinetic models of multicellular growing systems to macroscopic biological tissue models

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