A Discrete Velocity Kinetic Model with Food Metric: Chemotaxis Traveling Waves

We introduce a mesoscopic scale chemotaxis model for traveling wave phenomena which is induced by food metric. The organisms of this simplified kinetic model have two discrete velocity modes, ± s and a constant tumbling rate. The main feature of the model is that the speed of organisms is constant s>0 with respect to the food metric, not the Euclidean metric. The uniqueness and the existence of the traveling wave solution of the model are obtained. Unlike the classical logarithmic model case there exist traveling waves under super-linear consumption rates and infinite population pulse-type traveling waves are obtained. Numerical simulations are also provided.