Asymptotic Behavior of a Two-Dimensional Keller–Segel Model with and without Density Control

Cálvez, Vincent; Dolak‐Struss, Yasmin

doi:10.1007/978-0-8176-4556-4_29

Cited by 7 publications

(6 citation statements)

References 22 publications

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“…The process of letting ε → 0 corresponds to the assumption that the dynamics of the chemorepellent is fast compared to the evolution of the cell density. In the case of chemoattractant a variant of such a model was considered by Perthame and Dalibard [33], Calvez and Dolak-Struß [7]. Instead, the boundary condition here is induced by ∇P · ν = 0.…”

Section: )mentioning

confidence: 99%

A cell–cell repulsion model on a hyperbolic Keller–Segel equation

Griette

Magal

2020

J. Math. Biol.

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In this work, we discuss a cell-cell repulsion population dynamic model based on a hyperbolic Keller-Segel equation with two populations. This model can well describe the cell growth and dispersion in the cell co-culture experiment in the work of Pasquier et al. [31]. With the notion of solutions integrated along the characteristics, we prove the existence and uniqueness of the solution and the segregation property of the two species. From a numerical perspective, we can also observe that the model admits a competitive exclusion (the results are different from the corresponding ODE model). More importantly, our model shows the complexity of the short term (6 days) co-cultured cell distribution depending on the initial distribution of each species. Through numerical simulations, the impact of the initial distribution on the population ratio lies in the initial total cell number and our study shows that the population ratio is not impacted by the law of initial distribution. We also find that a fast dispersion rate gives a short-term advantage while the vital dynamic contributes to a long-term population advantage.

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Section: )mentioning

confidence: 99%

A cell–cell repulsion model on a hyperbolic Keller–Segel equation

Griette

Magal

2020

J. Math. Biol.

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“…On the other hand, there are several threshold values in the case of Neumann boundary conditions: if Ω is regular then solutions are global if χ M < 4π, and may blow up above this threshold, either on the boundary or inside the domain. If Ω is piecewise C 2 and Θ denotes the smallest angle then solutions are global if χ M < 4Θ, and may blow up above this threshold [20,8]. On a disc, the situation is more clear and has been analyzed in [3] and it turns out that, again, boundary conditions on c play an important role.…”

Section: Remark 22mentioning

confidence: 99%

Modified Keller-Segel system and critical mass for the log interaction kernel

Cálvez¹,

Perthame²,

tabar³

2007

Stochastic Analysis and Partial Differential Equations

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The Keller-Segel system has been widely validated as a model for cell movement under the combined effect of stochastic diffusion and a directed drift due to a chemoattractant emitted by the cells themselves (chemotaxis). In two dimensions, and in its simpler mathematical setting, it has been proved that the system admits a critical mass (below this mass there are global solutions, above the system blows-up). In higher dimensions this is not true and L d/2 is the critical norm but only less precise statements are available. Here, we study a variant for the diffusion law of the chemical potential in chemotaxis, based on a convolution operator or in other words on a fractional diffusion law. The aim of this paper is to enumerate the corresponding qualitative properties-global existence, blow up, stationary states-for the model with critical logarithmic kernel. In particular we show that this new system leads to a unified theory with critical mass in any dimension.

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“…whose properties have been studied in several papers. [8][9][10][11][12] In particular, in Cherniha and Didovych 9 and Didovych, 10 Lie symmetries have been found and applied for construction of exact solutions.…”

Section: Introductionmentioning

confidence: 99%