Travelling plateaus for a hyperbolic Keller–Segel system with attraction and repulsion: existence and branching instabilities

How can repulsive and attractive forces, acting on a conservative system, create stable travelling patterns or branching instabilities? We have proposed to study this question in the framework of the hyperbolic Keller–Segel system with logistic sensitivity. This is a model system motivated by experiments on cell communities auto-organization, a field which is also called socio-biology. We continue earlier modelling work, where we have shown numerically that branching patterns arise for this system and we have analysed this instability by formal asymptotics for small diffusivity of the chemo-repellent. Here we are interested in the more general situation, where the diffusivities of both the chemo-attractant and the chemo-repellent are positive. To do so, we develop an appropriate functional analysis framework. We apply our method to two cases. Firstly we analyse steady states. Secondly we analyse travelling waves when neglecting the degradation coefficient of the chemo-repellent; the unique wave speed appears through a singularity cancellation which is the main theoretical difficulty. This shows that in different situations the cell density takes the shape of a plateau. The existence of steady states and travelling plateaus are a symptom of how rich the system is and why branching instabilities can occur. Numerical tests show that large plateaus may split into smaller ones, which remain stable