The Scalar Keller–segel Model on Networks

Borsche, Raul; Göttlich, Simone; Klar, Axel; Schillen, Peter

doi:10.1142/s0218202513400071

Cited by 37 publications

(42 citation statements)

References 35 publications

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“…168 In particular, the specific features of the environment can oblige cells to move along networks rather than a less constrained movement in space homogeneity. 33,128 An additional difficulty is that cells may undergo mutations and selections through a proliferative or destructive dynamics. 63 Therefore, the properties of cells and their number evolve in time in a system, which is not mass conservative.…”

Section: A Micro-macro Approachmentioning

confidence: 99%

Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues

Bellomo

Bellouquid

Tao

et al. 2015

Math. Models Methods Appl. Sci.

866

810

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This paper proposes a survey and critical analysis focused on a variety of chemotaxis models in biology, namely the classical Keller-Segel model and its subsequent modifications, which, in several cases, have been developed to obtain models that prevent the non-physical blow up of solutions. The presentation is organized in three parts. The first part focuses on a survey of some sample models, namely the original model and some of its developments, such as flux limited models, or models derived according to similar concepts. The second part is devoted to the qualitative analysis of analytic problems, such as the existence of solutions, blow-up and asymptotic behavior. The third part deals with the derivation of macroscopic models from the underlying description, delivered by means of kinetic theory methods. This approach leads to the derivation of classical models as well as that of new models, which might deserve attention as far as 1663

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Section: A Micro-macro Approachmentioning

confidence: 99%

Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues

Bellomo

Bellouquid

Tao

et al. 2015

Math. Models Methods Appl. Sci.

866

810

View full text Add to dashboard Cite

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“…The FVM, which is referred to as the Dual-Finite Volume Method (DFVM) [1,13], is used for numerically solving (3). Detailed discretization of the DFVM for linear advection-diffusion equations is described in Yoshioka and Unami [1].…”

Section: Finite Volume Methodsmentioning

confidence: 99%

“…Diffusion-type partial differential equations (PDEs) on connected graphs arise as relevant mathematical models for transport phenomena in environmental [1,2], biological [3], and material problems [4,5]. Here a connected graph is defined as an union of 0-D vertices that links 1-D intervals [6].…”

Section: Introductionmentioning

confidence: 99%

Numerical modelling of nonlinear and degenerate diffusion equations on connected graphs: application to moisture dynamics in non-woven fibrous strip networks

2016

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A numerical model of a nonlinear and degenerate diffusion equation on connected graphs, arising in a diversity of industrial applications, is presented. This paper shows that concurrently using a staggered finite volume spatial discretization with an analytical solution-based flux evaluation method and an operator-splitting technique computes stable and physicallyconsistent numerical solutions to the equation. A demonstrative application example of the numerical model to moisture dynamics in a hypothetical non-woven fibrous strip network under an evaporative environment is presented in order to show its versatility. Mathematical issues to be addressed in a future for better comprehending the model are finally discussed.

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“…Results about hyperbolic models on networks can be found in [6,7,23,19,22], with different kinds of transmission conditions; moreover parabolic chemotaxis models on networks were studied in [1,5,17], with continuity conditions at the nodes.…”

Section: Introductionmentioning

confidence: 99%

Global solutions for a chemotaxis hyperbolic-parabolic system on networks with nonhomogeneous boundary conditions

Guarguaglini¹

2020

Communications on Pure &Amp; Applied Analysis

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In this paper we study a semilinear hyperbolic-parabolic system as a model for some chemotaxis phenomena evolving on networks; we consider transmission conditions at the inner nodes which preserve the fluxes and nonhomogeneous boundary conditions having in mind phenomena with inflow of cells and food providing at the network exits. We give some conditions on the boundary data which ensure the existence of stationary solutions and we prove that these ones are asymptotic profiles for a class of global solutions.

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The Scalar Keller–segel Model on Networks

Cited by 37 publications

References 35 publications

Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues

Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues

Numerical modelling of nonlinear and degenerate diffusion equations on connected graphs: application to moisture dynamics in non-woven fibrous strip networks

Global solutions for a chemotaxis hyperbolic-parabolic system on networks with nonhomogeneous boundary conditions

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