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Volume-filling and quorum-sensing in models for chemosensitive movement. (English) Zbl 1057.92013

Summary: Chemotaxis is one of many mechanisms used by cells and organisms to navigate through the environment, and has been found on scales varying from the microscopic to the macroscopic. Chemotactic movement has also attracted a great deal of computational and modelling attention. Some of the continuum models are unstable in the sense that they can lead to finite time blow-up, or “overcrowding” scenarios. Cell overcrowding is unrealistic from a biological context, as it ignores the finite size of individual cells and the behaviour of cells at higher densities.
We have previously presented a mathematical model of chemotaxis incorporating density dependence that precludes blow-up from occurring [Adv. Appl. Math. 26, 280–301 (2001; Zbl 0998.92006)]. In this paper, we consider a number of approaches by which such equations can arise based on biologically realistic mechanisms, including the finite size of individual cells – “volume filling” and the employment of cell density sensing mechanisms – “quorum-sensing”. We show the existence of nontrivial steady states and we study the traveling wave problem for these models.
A comprehensive numerical exploration of the model reveals a wide variety of interesting pattern forming properties. Finally, we turn our attention to the robustness of patterning under domain growth, and discuss some potential applications of the model.

MSC:

92C17 Cell movement (chemotaxis, etc.)
92C15 Developmental biology, pattern formation
35Q92 PDEs in connection with biology, chemistry and other natural sciences
35K57 Reaction-diffusion equations

Citations:

Zbl 0998.92006
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