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

/pdf/maximal-regularity-in-weighted-spaces-nonlinear-boundary-1sdi7i9e97.pdf

Maximal Regularity in Weighted Spaces, Nonlinear Boundary Conditions, and Global Attractors

This paper is concerned with the existence of large positive spiky
steady states for S-K-T competition systems with cross-diffusion.
Firstly by detailed integral and perturbation estimates, the
existence and detailed fast-slow structure of a class of spiky
steady states are obtained for the corresponding shadow system,
which also verify and extend some existence results on spiky steady
states obtained in [10] by different method of proof. Further
by applying special perturbation method, we prove the existence of
large positive spiky steady states for the original competition
systems with large cross-diffusion rate.

The existence and structure of large spiky steady states for S-K-T competition systems with cross-diffusion

This paper extends the volume filling chemotaxis model [18, 26] by taking into account the cell population interactions. The extended chemotaxis models have nonlinear diffusion and chemotactic sensitivity depending on cell population density, which is a modification of the classical Keller-Segel model in which the diffusion and chemotactic sensitivity are constants (linear). The existence and boundedness of global solutions of these models are discussed and the numerical pattern formations are shown. The further improvement is proposed in the end.

/pdf/on-chemotaxis-models-with-cell-population-interactions-58efekh4bp.pdf

On Chemotaxis Models with Cell Population Interactions

Global attractors for cross diffusion systems on domains of arbitrary dimension

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Global attractors for cross diffusion systems on domains of arbitrary dimension