The Brownian dynamics of a single microorganism coupled by chemotaxis to a diffusing concentration field that 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.
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