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.

/pdf/a-user-s-guide-to-pde-models-for-chemotaxis-4zxd8lr4pw.pdf

A user’s guide to PDE models for chemotaxis

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

We introduce new Godunov-type semidiscrete central schemes for hyperbolic systems of conservation laws and Hamilton--Jacobi equations. The schemes are based on the use of more precise information about the local speeds of propagation and can be viewed as a generalization of the schemes from [A. Kurganov and E. Tadmor, J. Comput. Phys., 160 (2000), pp. 241--282; A. Kurganov and D. Levy, SIAM J. Sci. Comput., 22 (2000), pp. 1461--1488; A. Kurganov and G. Petrova, A third-order semidiscrete genuinely multidimensional central scheme for hyperbolic conservation laws and related problems, Numer. Math., to appear] and [A. Kurganov and E. Tadmor, J. Comput. Phys., 160 (2000), pp. 720--742].
The main advantages of the proposed central schemes are the high resolution, due to the smaller amount of the numerical dissipation, and the simplicity. There are no Riemann solvers and characteristic decomposition involved, and this makes them a universal tool for a wide variety of applications.
At the same time, the developed schemes have an upwind nature, since they respect the directions of wave propagation by measuring the one-sided local speeds. This is why we call them central-upwind schemes.
The constructed schemes are applied to various problems, such as the Euler equations of gas dynamics, the Hamilton--Jacobi equations with convex and nonconvex Hamiltonians, and the incompressible Euler and Navier--Stokes equations. The incompressibility condition in the latter equations allows us to treat them both in their conservative and transport form. We apply to these problems the central-upwind schemes, developed separately for each of them, and compute the corresponding numerical solutions.

Semidiscrete Central-Upwind Schemes for Hyperbolic Conservation Laws and Hamilton--Jacobi Equations

Finite-time blow-up in the higher-dimensional parabolic–parabolic Keller–Segel system

We consider two simple conservative systems of parabolic-elliptic and parabolic-degenerate type arising in modeling chemotaxis and angiogenesis. Both systems share the same property that when the

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 $$L^{\frac{d}
{2}} $$
 norm of initial data is small enough, where d ≥ 2 is the space dimension, then there is a global (in time) weak solution that stays in all the L
 p
 spaces with max

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 $$\left\{ {\left. {1;\frac{d}
{2} - 1} \right\} \leq p < \infty .} \right.$$
 This result is already known for the parabolic-elliptic system of chemotaxis, moreover blow-up can occur in finite time for large initial data and Dirac concentrations can occur. For the parabolic-degenerate system of angiogenesis in two dimensions, we also prove that weak solutions (which are equi-integrable in L1) exist even for large initial data. But break-down of regularity or propagation of smoothness is an open problem.

/pdf/global-solutions-of-some-chemotaxis-and-angiogenesis-systems-1oek7w6wb7.pdf

Global Solutions of Some Chemotaxis and Angiogenesis Systems in High Space Dimensions

This paper is devoted mainly to the global existence problem for the two-dimensional parabolic-parabolic Keller-Segel system in the full space. We derive a critical mass threshold below which global existence is ensured. Carefully using energy methods and ad hoc functional inequalities, we improve and extend previous results in this direction. The given threshold is thought to be the optimal criterion, but this question is still open. This global existence result is accompanied by a detailed discussion on the duality between the Onofri and the logarithmic Hardy-Littlewood-Sobolev inequalities that underlie the following approach.

https://projecteuclid.org/download/pdf_1/euclid.cms/1214949930

The parabolic-parabolic Keller-Segel model in R2

A chemotaxis model motivated by angiogenesis

Multivalued solutions with a limited number of branches of the inviscid Burgers equation can be obtained by solving closed systems of moment equations. For this purpose, a suitable concept of entropy multivalued solutions with K branches is introduced.

/pdf/a-kinetic-formulation-for-multi-branch-entropy-solutions-of-1e1ic3axhz.pdf

A kinetic formulation for multi-branch entropy solutions of scalar conservation laws

In two space dimensions, the parabolic–parabolic Keller–Segel system shares many properties with the parabolic–elliptic Keller–Segel system. In particular, solutions globally exist in both cases as long as their mass is less than a critical threshold M
 c
 . However, this threshold is not as clear in the parabolic–parabolic case as it is in the parabolic–elliptic case, in which solutions with mass above M
 c
 always blow up. Here we study forward self-similar solutions of the parabolic–parabolic Keller–Segel system and prove that, in some cases, such solutions globally exist even if their total mass is above M
 c
 , which is forbidden in the parabolic–elliptic case.

/pdf/large-mass-self-similar-solutions-of-the-parabolic-parabolic-45n2k5b116.pdf

Lucilla Corrias

Papers

Global Solutions of Some Chemotaxis and Angiogenesis Systems in High Space Dimensions

The parabolic-parabolic Keller-Segel model in R2

A chemotaxis model motivated by angiogenesis

A kinetic formulation for multi-branch entropy solutions of scalar conservation laws

Large mass self-similar solutions of the parabolic–parabolic Keller–Segel model of chemotaxis