Emerging patterns in a hyperbolic model for locally interacting cell systems
Frithjof Lutscher and Angela Stevens
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Submission date: 11. Dec. 2001
published in: Journal of nonlinear science, 12 (2002) 6, p. 619-640
DOI number (of the published article): 10.1007/s00332-002-0510-4
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Morphogenetic processes like, for example, neurulation and gastrulation involve coordinated movements of cells. It is assumed that these processes happen due to long range signaling, although the detailed mechanisms are not completely understood. Therefore, one is interested in biological ``model-systems'' where self-organization of cells and in particular the mechanisms of signaling can be analyzed in greater detail. A major question is whether or not also short range signaling or local interaction of cells can be the cause of coordinated movement and morphogenetic processes. As a model problem we analyze ripple formation of myxobacteria due to purely local interaction, a hypothesis which is discussed in the biological literature. These ripples can be observed before the final aggregation of the bacteria and fruiting body formation takes place.
Our basic mathematical model is a one-dimensional hyperbolic system of Goldstein-Kac type with density dependent coefficients. Conditions for the existence of traveling waves are discussed by means of linear analysis and the construction of invariant domains.