Heriot-Watt Mathematics Report Series
HWM07-16, 22 Jun 2007
A users guide to PDE models for chemotaxis
T Hillen and KJ Painter
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, inspecting aspects including its mechanistic
basis, the modelling of specific systems and
the mathematical behaviour of the underlying equations. The Keller-Segel model of
chemotaxis \cite{KS,kellersegel} has provided a cornerstone for much of
this work, its success 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 expansion on some of the
outstanding issues revealed as a result of this work.
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