The bacterium E. coli maneuvers itself to regions with high chemoattractant concentrations by performing two stereotypical moves:
`runs', in which it moves in near straight lines, and `tumbles', in which it does not advance but changes direction randomly. The duration of each move
is stochastic and depends upon the chemoattractant concentration experienced in the recent past. We relate this stochastic behavior to the steady- state
density of a bacterium population, and we derive the latter as a function of chemoattractant concentration. In contrast to earlier treatments, here we
account for the effects of temporal correlations and variable tumbling durations. A range of behaviors obtains, that depends subtly upon several
aspects of the system---memory, correlation, and tumbling stochasticity in particular.