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How can repulsive and attractive forces, acting on a conservative system,create stable traveling patterns or branching instabilities? We have proposedto study this question in the framework of the hyperbolic Keller-Segel systemwith logistic sensitivity. This is a model system motivated by experiments oncell communities auto-organization, a field which is also called socio-biology.We continue earlier modeling work, where we have shown numerically thatbranching patterns arise for this system and we have analyzed this instabilityby formal asymptotics for small diffusivity of the chemo-repellent. Here we areinterested in the more general situation, where the diffusivities of both thechemo-attractant and the chemo-repellent are positive. To do so, we develop anappropriate functional analysis framework. We apply our method to two cases.Firstly we analyze steady states. Secondly we analyze traveling waves whenneglecting the degradation coefficient of the chemo-repellent; the unique wavespeed appears through a singularity cancelation which is the main theoreticaldifficulty. This shows that in different situations the cell density takes theshape of a plateau. The existence of steady states and traveling plateaus are asymptom of how rich the system is and why branching instabilities can occur.Numerical tests show that large plateaus may split into smaller ones, whichremain stable.

Traveling plateaus for a hyperbolic Keller-Segel system with attraction and repulsion: existence and branching instabilities