Waves for an hyperbolic Keller-Segel model and branching instabilities

Recent experiments for swarming of the bacteria Bacillus subtilis on nutrient rich media show that these cells are able to proliferate and spread out in colonies exhibiting complex patterns as dendritic ramifications. Is it possible to explain this process with a model that does not use local nutrient depletion? We present a new class of models which is compatible with the experimental observations and which predict branching instabilities and does not use nutrient limitation. These conclusions are based on numerical simulations. The most complex of these models is also the biologically most accurate but the essential effects can also be obtained in simplified versions which are amenable to analysis. An example of instability mechanism is the transition from a shock wave to a rarefaction wave in a reduced two by two hyperbolic system.