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Pattern formation in a reaction‐advection model with delay: a continuum approach to myxobacterial rippling
Author(s) -
Börner U.,
Bär M.
Publication year - 2004
Publication title -
annalen der physik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.009
H-Index - 68
eISSN - 1521-3889
pISSN - 0003-3804
DOI - 10.1002/andp.200410086
Subject(s) - rippling , instability , chaotic , statistical physics , physics , nonlinear system , advection , homogeneous , pattern formation , term (time) , turbulence , classical mechanics , mechanics , computer science , quantum mechanics , artificial intelligence , biology , genetics , programming language
Standing wave oscillations of the cell density (rippling) are observed in premature aggregates of developing myxobacteria. Recently the underlying pattern formation mechanism was shown to be based on the interplay between active cell motion and local interactions triggering reversals in the cells' direction of motion. The propagation of information through the system is mediated by the internal state of moving cells rather than by diffusible chemical signals. Discrete cellular automata and coupled‐map lattices have been investigated earlier and indicate the importance of a minimum refractory period between subsequent reversals of a cell. In this paper we consider the continuum limit of the process, that yields a set of hyperbolic partial differential equations with a a single discrete time delay. The time delay corresponds to the duration of the mentioned refractory period of the cells. According to linear stability analysis a minimal time delay is required for a wave instability to occur. The results of the continuum model are in reasonable agreement with the findings in the discrete models adding credibility to the earlier studies.