Dynamics of a microorganism moving by chemotaxis in its own secretion

The Brownian dynamics of a single microorganism coupled by chemotaxis to a diffusing concentration field which is secreted by the microorganism itself is studied by computer simulations in spatial dimensions $d=1,2,3$. Both cases of a chemoattractant and a chemorepellent are discussed. For a chemoattractant, we find a transient dynamical arrest until the microorganism diffuses for long times. For a chemorepellent, there is a transient ballistic motion in all dimensions and a long-time diffusion. These results are interpreted with the help of a theoretical analysis.