We obtain a set of sufficient conditions under which all positive solutions of the nonlinear delay difference equation
x
n
+
1
=
x
n
f
(
x
n
−
k
)
,
n
=
0
,
1
,
2
,
…
{x_{n + 1}} = {x_n}f({x_{n - k}}),n = 0,1,2, \ldots
, are attracted to the positive equilibrium of the equation. Our result applies, for example, to the delay logistic model
N
t
+
1
=
α
N
t
/
(
1
+
β
N
t
−
k
)
{N_{t + 1}} = \alpha {N_t}/(1 + \beta {N_{t - k}})
and to the delay difference equation
x
n
+
1
=
x
n
e
r
(
1
−
x
n
−
k
)
{x_{n + 1}} = {x_n}{e^{r(1 - {x_{n - k}})}}
.