We consider, for
N
≥
2
N \geq 2
, the system of
N
N
competing species which are ecologically identical and having distinct diffusion rates
{
D
i
}
i
=
1
N
\{D_i\}_{i=1}^N
, in an environment with the carrying capacity
m
(
x
,
t
)
m(x,t)
. For a generic class of
m
(
x
,
t
)
m(x,t)
that varies with space and time, we show that there is a positive number
D
∗
D_*
independent of
N
N
so that if
D
i
≥
D
∗
D_i \geq D_*
for all
1
≤
i
≤
N
1\le i\le N
, then the slowest diffusing species is able to competitively exclude all other species. In the case when the environment is temporally constant or temporally periodic, our result provides some further evidence in the affirmative direction regarding the conjecture by Dockery et al. [J. Math. Biol. 37 (1998), pp. 61–83]. The main tool is the theory of the principal Floquet bundle for linear parabolic equations.