Emerging patterns in a hyperbolic model for locally interacting cell systems

Morphogenetic processes such as 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 short-range signaling or local interaction of cells can also 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 take 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.