  Many biological microswimmers are capable of chemotaxis, i.e., they can sense
an ambient chemical gradient and adjust their mechanism of motility to move
towards or away from the source of the gradient. Synthetic active colloids
endowed with chemotactic behavior hold considerable promise for targeted drug
delivery and the realization of programmable and reconfigurable materials. Here
we study the chemotactic behavior of a Janus particle, which converts ``fuel''
molecules, released at an axisymmetric chemical patch located on a planar wall,
into ``product'' molecules at its catalytic cap and moves by self-phoresis
induced by the product. The chemotactic behavior is characterized as a function
of the interplay between the rates of release (at the patch) and the
consumption (at the particle) of fuel, as well as of details of the phoretic
response of the particle (i.e., its phoretic mobility). Among others, we find
that, under certain conditions, the particle is attracted to a stable
``hovering state'' in which it aligns its axis normal to the wall and rests
(positions itself) at an activity-dependent distance above the center of the
patch.
