  In this paper, we investigate the effect of dispersal and advection on the
dynamics of a predator-prey model. More precisely, we show that the linear
stability of the semi-trivial steady state is determined by the dispersal rate,
the mortality rate of the predator and the advection rate. We point out that
compared to homogeneous intrinsic growth rate and carrying capacity, the case
in this paper is more complicated. This work gives a investigation to an open
problem proposed by Nie et al. in \cite{NWW} by considering a more general
model, and then, can be seen as a further development of their work \cite{NWW}.
