On a Two-Sidedly Degenerate Chemotaxis Model With Volume-Filling Effect

Bendahmane, Mostafa; Karlsen, Kenneth H.; Urbano, José Miguel

doi:10.1142/s0218202507002108

Cited by 37 publications

(32 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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.

View full text Add to dashboard Cite

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.

show abstract

“…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.

View full text Add to dashboard Cite

show abstract

“…Then it was extended in [52] for the case of a system including model (TP) and independently for chemotaxis model with two sidedly degenerate diffusion [7]. Recently the concept was used for the case of fast diffusion of cells [53] including (VF) model (1.12)-(1.13).…”

Section: Weak Solutionsmentioning

confidence: 99%

Volume Filling Effect in Modelling Chemotaxis

Wrzosek

2010

Math. Model. Nat. Phenom.

View full text Add to dashboard Cite

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.

show abstract

“…Specifically for a chemotaxis model, Bendahmane, Karlsen, and Urbano [4] proved the existence and Hölder regularity of weak solutions for a version of (1.1) for p = 2. For a detailed description of the intrinsic scaling method and some applications we refer to the books [5,6].…”

Section: Related Workmentioning

confidence: 99%

“…We first prove the existence of solutions to a non-degenerate, regularized version of problem (1.1), using the Schauder fixed-point theorem, and our approach closely follows that of [4]. We define the following closed subset of the Banach space L p (Q T ):…”

Section: Existence Of Solutionsmentioning

confidence: 99%

See 1 more Smart Citation

On a doubly nonlinear diffusion model of chemotaxis with prevention of overcrowding

Bendahmane

Bürger

Ruiz–Baier

et al. 2008

Math Methods in App Sciences

Self Cite

View full text Add to dashboard Cite

Abstract. This paper addresses the existence and regularity of weak solutions for a fully parabolic model of chemotaxis, with prevention of overcrowding, that degenerates in a two-sided fashion, including an extra nonlinearity represented by a p-Laplacian diffusion term. To prove the existence of weak solutions, a Schauder fixed-point argument is applied to a regularized problem and the compactness method is used to pass to the limit. The local Hölder regularity of weak solutions is established using the method of intrinsic scaling. The results are a contribution to showing, qualitatively, to what extent the properties of the classical Keller-Segel chemotaxis models are preserved in a more general setting. Some numerical examples illustrate the model.

show abstract

On a Two-Sidedly Degenerate Chemotaxis Model With Volume-Filling Effect

Cited by 37 publications

References 23 publications

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

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

Volume Filling Effect in Modelling Chemotaxis

On a doubly nonlinear diffusion model of chemotaxis with prevention of overcrowding

Contact Info

Product

Resources

About