Model Hierarchies for Cell Aggregation by Chemotaxis

Chalub, Fabio A. C. C.; Dolak‐Struss, Yasmin; Markowich, Peter A.; Oelz, Dietmar; Schmeiser, Christian; Soreff, Alexander

doi:10.1142/s0218202506001509

Cited by 80 publications

(44 citation statements)

References 40 publications

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“…Other ways to model prevention of overcrowding can be found in [6,7,20]. Kinetic models for chemotaxis can be found in [12]. Numerics for the Keller-Segel model have been performed in [17,29].…”

Section: Motivationmentioning

confidence: 99%

A coupled Keller–Segel–Stokes model: global existence for small initial data and blow-up delay

Lorz

2012

Communications in Mathematical Sciences

110

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Abstract. We study a system consisting of the elliptic-parabolic Keller-Segel equations coupled to Stokes equations by transport and gravitational forcing. We show global-in-time existence of solutions for small initial mass in 2D. In 3D we establish global existence assuming that the initial L 3/2 -norm is small. Moreover, we give numerical evidence that for this extension of the Keller-Segel system in 2D, solutions exist with mass above 8π, which is the critical mass for the system without fluid.Key words. Keller-Segel, chemotaxis, Stokes, global existence, blow-up.AMS subject classifications. 35K55, 35Q92, 35Q35, 92C17. MotivationThe Keller-Segel system modelling chemotaxis is a very well studied model in mathematical biology. In the following, we investigate a system consisting of the elliptic-parabolic Keller-Segel equations coupled to Stokes equations:Here c denotes the concentration of a chemical, n a cell density, and u a fluid velocity field described by Stokes equations. The fluid couples to n and c through transport and gravitational forcing modelled by ∇φ. The pressure P can be seen as the Lagrange multiplier enforcing the incompressibility constraint. The chemical c diffuses, it is produced by the cells and it degrades. The cell density diffuses and it moves in the direction of the chemical gradient. The constant a 1 ≥ 0 measures self-degradation of the chemical and the constants a 2 ≥ 0, η > 0 determine the evolution undergone by u.The motivation for this model comes from experiments described in [21,30,33] for the case of bacteria consuming the chemical. There the authors observed large-scale convection patterns in a water drop sitting on a glass surface containing oxygensensitive bacteria, oxygen diffusing into the drop through the fluid-air interface and they proposed this model:The idea behind this paper is to take these equations and change consumption of the chemical to production i.e. make it the mathematically more interesting case of *

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Section: Motivationmentioning

confidence: 99%

A coupled Keller–Segel–Stokes model: global existence for small initial data and blow-up delay

Lorz

2012

Communications in Mathematical Sciences

110

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“…In Short et al [42] the "naïve" continuum limit (c.f. above remark) for the above system was derived using a common procedure in mathematical biology [3,4,15,16,20,21,24,32,35,36,45]:…”

Section: Formulation Of the Modelmentioning

confidence: 99%

Statistical Models of Criminal Behavior: The Effects of Law Enforcement Actions

Jones

2010

Math. Models Methods Appl. Sci.

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Abstract:We extend an agent-based model of crime-pattern formation initiated in Short et al. by incorporating the effects of law enforcement agents. We investigate the effect that these agents have on the spatial distribution and overall level of criminal activity in a simulated urban setting. Our focus is on a two-dimensional lattice model of residential burglaries, where each site (target) is characterized by a dynamic attractiveness to burglary and where criminal and law enforcement agents are represented by random walkers. The dynamics of the criminal agents and the target-attractiveness field are, with certain modifications, as described in Short et al. Here the dynamics of enforcement agents are affected by the attractiveness field via a biasing of the walk, the detailed rules of which define a deployment strategy. We observe that law enforcement agents, if properly deployed, will in fact reduce the total amount of crime, but their relative effectiveness depends on the number of agents deployed, the deployment strategy used, and spatial distribution of criminal activity. For certain policing strategies, continuum PDE models can be derived from the discrete systems. The continuum models are qualitatively similar to the discrete systems at large system sizes.

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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%

“…There are many choices for the evolution of the chemoattractant, one of them is to use a parabolic diffusion equation, (e.g. see [6,8,10,11,17,21])…”

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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Model Hierarchies for Cell Aggregation by Chemotaxis

Cited by 80 publications

References 40 publications

A coupled Keller–Segel–Stokes model: global existence for small initial data and blow-up delay

A coupled Keller–Segel–Stokes model: global existence for small initial data and blow-up delay

Statistical Models of Criminal Behavior: The Effects of Law Enforcement Actions

One-dimensional chemotaxis kinetic model

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