Travelling Chemotactic Aggregates at Mesoscopic Scale and BiStability

A model consisting of a kinetic equation for “run-and-tumble” biased bacteria motion, coupled with two reaction-diffusion equations for chemical signals, is studied. It displays time-asymptotic propagation at constant velocity, i.e., aggregated travelling (exponential) layers. To capture them for various parameters, a well-balanced setup is based on both “Case's elementary solutions" and $\mathcal{L}$-spline reconstruction. Far from the diffusive regime, waves travelling at different velocities (bistability) are proved to coexist. Numerics suggest that they are locally asymptotically stable, so that the resulting bifurcation diagram shows counterintuitive features.