  This work is concerned with the large time behavior of the solutions of a
parabolic-ODE hybrid system, modeling the competition of two populations which
are identical except their movement behaviors: one species moves by random
dispersal while the other does not diffuse. We show that the non-diffusing
population will always drive the diffusing one to extinction in environments
with sinks. In contract, the non-diffusing and diffusing populations can
coexist in environments without sinks.
