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.

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A user’s guide to PDE models for chemotaxis

This article summarizes various aspects and results for some general formulations of the classical chemotaxis models also known as Keller-Segel models. It is intended as a survey of results for the most common formulation of this classical model for positive chemotactical movement and offers possible generalizations of these results to more universal models. Furthermore it collects open questions and outlines mathematical progress in the study of the Keller-Segel model since the first presentation of the equations in 1970.

/pdf/from-1970-until-present-the-keller-segel-model-in-chemotaxis-23f2x935xe.pdf

From 1970 until present: the Keller-Segel model in chemotaxis and its consequences

Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model

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 the related analytic problems are concerned. Finally, an overview of the entire contents leads to suggestions for future research activities.

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

http://www.mi.uni-koeln.de/~dhorst/Winkler-JDE.pdf

Boundedness vs. blow-up in a chemotaxis system

Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen

Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen.

Using Lyapunov functionals the global behaviour of the solutions of a reaction-diffusion system modelling chemotaxis is studied for bounded piecewise smooth domains in the plane. Geometric criteria can be given so that this dynamical system tends to a (not necessarily trivial) stationary state.

Global Behaviour of a Reaction‐Diffusion System Modelling Chemotaxis

Asymptotic Behavior of Solutions to a Class of Diffusion-Reaction Equations

This paper concerns boundary value problems for quasilinear second order elliptic systems which are, for example, of the type 

$$ \begin{aligned} \partial _{j} {\left( {a^{{ij}}_{{\alpha \beta }} {\left( {u,\lambda } \right)}\partial _{i} u^{\alpha } + b^{j}_{\beta } {\left( {u,\lambda } \right)}} \right)} + c^{i}_{{\alpha \beta }} {\left( {u,\lambda } \right)}\partial _{i} u^{\alpha } & = d_{\beta } {\left( {u,\lambda } \right)}{\text{ in }}\Omega {\text{,}} \\ {\left( {a^{{ij}}_{{\alpha \beta }} {\left( {u,\lambda } \right)}\partial _{i} u^{\alpha } + b^{j}_{\beta } {\left( {u,\lambda } \right)}} \right)}\nu _{j} & = e_{\beta } {\left( {u,\lambda } \right)}{\text{ on }}\Gamma _{\beta } , \\ u^{\beta } & = \varphi ^{\beta } {\text{ on }}\partial \Omega \backslash \Gamma _{\beta } . \\ \end{aligned} $$

Here Ω is a Lipschitz domain in $\mathbb{R}^{N},$ ν j are the components of the unit outward normal vector field on ∂Ω, the sets Γβ are open in ∂Ω and their relative boundaries are Lipschitz hypersurfaces in ∂Ω. The coefficient functions are supposed to be bounded and measurable with respect to the space variable and smooth with respect to the unknown vector function u and to the control parameter λ. It is shown that, under natural conditions, such boundary value problems generate smooth Fredholm maps between appropriate Sobolev-Campanato spaces, that the weak solutions are Holder continuous up to the boundary and that the Implicit Function Theorem and the Newton Iteration Procedure are applicable.

Konrad Gröger

Papers

Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen

Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen.

Global Behaviour of a Reaction‐Diffusion System Modelling Chemotaxis

Asymptotic Behavior of Solutions to a Class of Diffusion-Reaction Equations

Applications of differential calculus to quasilinear elliptic boundary value problems with non-smooth data