Chemotactic traveling waves by metric of food

The meaningful distance to biological organisms is not necessarily one measured by the Euclidean metric but possibly one by a metric that counts the amount of resources such as food. It is assumed in this paper that the distance for biological organisms is measured by the amount of food between two places. A new chemotaxis model is introduced as an application of this "metric of food." It is shown that, if the walk length of a random walk system is given by such a metric, the well-known chemotactic traveling wave phenomena can be obtained without the typical assumption that microscopic scale bacteria may sense the macroscopic scale gradient of a chemical concentration. The uniqueness and the existence of a traveling wave solution are obtained.