Consider the discrete delay logistic model
\[
N
t
+
1
=
α
N
t
1
+
β
N
t
−
k
,
(
1
)
{N_{t + 1}} = \frac {{\alpha {N_t}}}{{1 + \beta {N_{t - k}}}}, \qquad \left ( 1 \right )
\]
where
α
∈
(
1
,
∞
)
,
β
∈
(
0
,
∞
)
\alpha \in \left ( {1, \infty } \right ), \beta \in \left ( {0, \infty } \right )
, and
k
∈
N
=
{
0
,
1
,
2
,
.
.
.
}
k \in \mathbb {N} = \left \{{0, 1, 2,...} \right \}
. We obtain conditions for the oscillation and asymptotic stability of all positive solutions of Eq. (1) about its positive equilibrium
(
α
−
1
)
/
β
\left ( {\alpha - 1} \right )/\beta
. We prove that all positive solutions of Eq. (1) are bounded and that for
k
=
0
k = 0
and
k
=
1
k = 1
the positive equilibrium
(
α
−
1
)
/
β
\left ( {\alpha - 1} \right )/\beta
is a global attractor.