user's guide to PDE models for chemotaxis

• Published in: Journal of mathematical biology Vol. 58; no. 1-2; pp. 183 - 217
• Main Authors: Hillen, T, Painter, K. J
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Berlin/Heidelberg : Springer-Verlag 2009; Springer-Verlag; Springer Nature B.V
• Subjects: 92C17; Applications of Mathematics; Chemotaxis - physiology; Computer Simulation; Mathematical and Computational Biology; Mathematics; Mathematics and Statistics; Models, Biological; 92C17
• ISSN: 0303-6812; 1432-1416
• DOI: 10.1007/s00285-008-0201-3

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.