Modeling a self-propelled autochemotactic walker

We develop a minimal model for the stochastic dynamics of microorganisms where individuals communicate via autochemotaxis. This means that microorganisms, such as bacteria, amoebae, or cells, follow the gradient of a chemical that they produce themselves to attract or repel each other. A microorganism is represented as a self-propelled particle or walker with constant speed while its velocity direction diffuses on the unit circle. We study the autochemotactic response of a single self-propelled walker whose dynamics is non-Markovian. We show that its long-time dynamics is always diffusive by deriving analytic expressions for its diffusion coefficient in the weak- and strong-coupling case. We confirm our findings by numerical simulations.