Travelling plateaus for a hyperbolic Keller–Segel system with attraction and repulsion: existence and branching instabilities
TL;DR: In this paper, the authors proposed a functional analysis framework for the hyperbolic Keller-Segel system with logistic sensitivity, where repulsive and attractive forces, acting on a conservative system, create stable travelling patterns or branching instabilities.
Abstract: How can repulsive and attractive forces, acting on a conservative system, create stable travelling patterns or branching instabilities? We have proposed to study this question in the framework of the hyperbolic Keller–Segel system with logistic sensitivity. This is a model system motivated by experiments on cell communities auto-organization, a field which is also called socio-biology. We continue earlier modelling work, where we have shown numerically that branching patterns arise for this system and we have analysed this instability by formal asymptotics for small diffusivity of the chemo-repellent.Here we are interested in the more general situation, where the diffusivities of both the chemo-attractant and the chemo-repellent are positive. To do so, we develop an appropriate functional analysis framework. We apply our method to two cases. Firstly we analyse steady states. Secondly we analyse travelling waves when neglecting the degradation coefficient of the chemo-repellent; the unique wave speed appears through a singularity cancellation which is the main theoretical difficulty. This shows that in different situations the cell density takes the shape of a plateau.The existence of steady states and travelling plateaus are a symptom of how rich the system is and why branching instabilities can occur. Numerical tests show that large plateaus may split into smaller ones, which remain stable.
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TL;DR: In this paper, the authors considered the attraction-repulsion chemotaxis system under homogeneous Neumann boundary conditions in a bounded domain with smooth boundary and proved that the system with τ = 0 is globally well-posed in high dimensions if repulsion prevails over attraction.
Abstract: We consider the attraction–repulsion chemotaxis system under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ ℝn with smooth boundary, where χ ≥ 0, ξ ≥ 0, α > 0, β > 0, γ > 0, δ > 0 and τ = 0, 1. We study the global solvability, boundedness, blow-up, existence of non-trivial stationary solutions and asymptotic behavior of the system for various ranges of parameter values. Particularly, we prove that the system with τ = 0 is globally well-posed in high dimensions if repulsion prevails over attraction in the sense that ξγ - χα > 0, and that the system with τ = 1 is globally well-posed in two dimensions if repulsion dominates over attraction in the sense that ξγ - χα > 0 and β = δ. Hence our results confirm that the attraction–repulsion is a plausible mechanism to regularize the classical Keller–Segel model whose solution may blow up in higher dimensions.
243 citations
TL;DR: A review and critical analysis of the asymptotic limit methods focused on the derivation of macroscopic equations for a class of equations modeling complex multicellular systems by methods of the kinetic theory for active particles is presented in this article.
Abstract: This paper proposes a review and critical analysis of the asymptotic limit methods focused on the derivation of macroscopic equations for a class of equations modeling complex multicellular systems by methods of the kinetic theory for active particles. Cellular interactions generate both modification of biological functions and proliferative/destructive events. The asymptotic analysis deals with suitable parabolic, hyperbolic, and mixed limits. The review includes the derivation of the classical Keller–Segel model and flux limited models that prevent non-physical blow up of solutions.
108 citations
TL;DR: In this paper, the authors considered the Keller-Segel system and established the global existence of uniformly in-time bounded classical solutions with large initial data if the repulsion dominates or cancels attraction (i.e., ξ γ ≥ α χ ).
103 citations
TL;DR: In this article, the authors studied the pattern formation of the attraction-repulsion Keller-Segel (ARKS) system and established the existence of time-periodic patterns and steady state patterns for the ARKS model in the full parameter regimes.
Abstract: In this paper, the pattern formation of the attraction-repulsion Keller-Segel (ARKS) system is studied analytically and numerically. By the Hopf bifurcation theorem as well as the local and global bifurcation theorem, we rigorously establish the existence of time-periodic patterns and steady state patterns for the ARKS model in the full parameter regimes, which are identified by a linear stability analysis. We also show that when the chemotactic attraction is strong, a spiky steady state pattern can develop. Explicit time-periodic rippling wave patterns and spiky steady state patterns are obtained numerically by carefully selecting parameter values based on our theoretical results. The study in the paper asserts that chemotactic competitive interaction between attraction and repulsion can produce periodic patterns which are impossible for the chemotaxis model with a single chemical (either chemo-attractant or chemo-repellent).
101 citations
TL;DR: In this article, the authors considered the initial-boundary value problem of the attraction-repulsion Keller-Segel model describing aggregation of Microglia in the central nervous system in Alzheimer's disease due to the interaction of chemoattractant and chemorepellent.
94 citations
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TL;DR: This paper explores in detail a number of variations of the original Keller–Segel model of chemotaxis from a biological perspective, contrast their patterning properties, summarise key results on their analytical properties and classify their solution form.
Abstract: Mathematical modelling of chemotaxis (the movement of biological cells or organisms in response to chemical gradients) has developed into a large and diverse discipline, whose aspects include its mechanistic basis, the modelling of specific systems and the mathematical behaviour of the underlying equations. The Keller-Segel model of chemotaxis (Keller and Segel in J Theor Biol 26:399–415, 1970; 30:225–234, 1971) has provided a cornerstone for much of this work, its success being a consequence of its intuitive simplicity, analytical tractability and capacity to replicate key behaviour of chemotactic populations. One such property, the ability to display “auto-aggregation”, has led to its prominence as a mechanism for self-organisation of biological systems. This phenomenon has been shown to lead to finite-time blow-up under certain formulations of the model, and a large body of work has been devoted to determining when blow-up occurs or whether globally existing solutions exist. In this paper, we explore in detail a number of variations of the original Keller–Segel model. We review their formulation from a biological perspective, contrast their patterning properties, summarise key results on their analytical properties and classify their solution form. We conclude with a brief discussion and expand on some of the outstanding issues revealed as a result of this work.
1,532 citations
Book•
05 Sep 2008
TL;DR: In this paper, the renewal equation is used to describe the structure of a population from a point of view of population balance equations, which is an asymptotic view of the population dynamics.
Abstract: From differential equations to structured population dynamics.- Adaptive dynamics an asymptotic point of view.- Population balance equations: the renewal equation.- Population balance equations: size structure.- Cell motion and chemotaxis.- General mathematical tools.
932 citations
01 Jan 2004
616 citations
Book•
25 Jun 2004
TL;DR: In this article, Quasilinear systems and conservation laws are discussed, including conservative schemes and non-conservative schemes, and a numerical test with source is proposed. But the test is based on a finite volume.
Abstract: Introduction.- 1. Quasilinear systems and conservation laws.- 2. Conservative schemes.- 3. Source terms.- 4. Nonconservative schemes.- 5. Multidimensional finite volumes with sources.- 6. Numerical test with source.- Bibliography
561 citations
01 Jan 2002
TL;DR: A number of approaches by which equations can arise based on biologically realistic mechanisms, including the finite size of individual cells “volume filling” and the employment of cell density sensing mechanisms “quorum-sensing” are considered.
Abstract: Chemotaxis is one of many mechanisms used by cells and organisms to navigate through the environment, and has been found on scales varying from the microscopic to the macroscopic Chemotactic movement has also attracted a great deal of computational and modelling attention Some of the continuum models are unstable in the sense that they can lead to finite time blow-up, or “overcrowding” scenarios Cell overcrowding is unrealistic from a biological context, as it ignores the finite size of individual cells and the behaviour of cells at higher densities We have previously presented a mathematical model of chemotaxis incorporating density dependence that precludes blow-up from occurring, [19] In this paper, we consider a number of approaches by which such equations can arise based on biologically realistic mechanisms, including the finite size of individual cells “volume filling” and the employment of cell density sensing mechanisms “quorum-sensing” We show the existence of nontrivial steady states and we study the traveling wave problem for these models A comprehensive numerical exploration of the model reveals a wide variety of interesting pattern forming properties Finally we turn our attention to the robustness of patterning under domain growth, and discuss some potential applications of the model
516 citations