  This paper is concerned with the asymptotic stability of the initial-boundary
value problem of a singular PDE-ODE hybrid chemotaxis system in the half space
$\R_+=[0, \infty)$. We show that when the non-zero flux boundary condition at
$x=0$ is prescribed and the initial data are suitably chosen, the solution of
the initial-boundary value problem converges, as time tend to infinity, to a
shifted traveling wavefront restricted in the half space $[0,\infty)$ where the
wave profile and speed are uniquely selected by the boundary flux data. The
results are proved by a Cole-Hopf type transformation and weighted energy
estimates along with the technique of taking {\color{black} the}
anti-derivative.
