By using Krasnoselskii’s fixed point theorem, we prove that the following periodic
n
−
n-
species Lotka-Volterra competition system with multiple deviating arguments
(
∗
)
x
˙
i
(
t
)
=
x
i
(
t
)
[
r
i
(
t
)
−
∑
j
=
1
n
a
i
j
(
t
)
x
j
(
t
−
τ
i
j
(
t
)
)
]
,
i
=
1
,
2
,
…
,
n
,
\begin{equation*} (\ast )\quad \quad \dot {x}_i(t)=x_i(t)\left [r_i(t)-\sum _{j=1}^{n}a_{ij}(t)x_j(t-\tau _{ij}(t)) \right ],\quad i=1, 2, \ldots , n,\qquad \quad \end{equation*}
has at least one positive
ω
−
\omega -
periodic solution provided that the corresponding system of linear equations
(
∗
∗
)
∑
j
=
1
n
a
¯
i
j
x
j
=
r
¯
i
,
i
=
1
,
2
,
…
,
n
,
\begin{equation*} (\ast \ast )\qquad \qquad \qquad \qquad \quad \sum _{j=1}^{n}\bar {a}_{ij}\ x_j= \bar {r}_i, \quad i=1, 2, \ldots , n,\qquad \qquad \qquad \qquad \quad \end{equation*}
has a positive solution, where
r
i
,
a
i
j
∈
C
(
R
,
[
0
,
∞
)
)
r_i, a_{ij}\in C({\mathbf {R}}, [0, \infty ))
and
τ
i
j
∈
C
(
R
,
R
)
\tau _{ij}\in C({\mathbf {R}}, {\mathbf {R}})
are
ω
−
\omega -
periodic functions with
\[
r
¯
i
=
1
ω
∫
0
ω
r
i
(
s
)
d
s
>
0
;
a
¯
i
j
=
1
ω
∫
0
ω
a
i
j
(
s
)
d
s
≥
0
,
i
,
j
=
1
,
2
,
…
,
n
.
\bar {r}_i=\frac {1}{\omega }\int _{0}^{\omega }r_i(s)ds >0;\ \ \ \bar {a}_{ij}=\frac {1}{\omega }\int _{0}^{\omega }a_{ij}(s)ds \ge 0, \quad i, j=1, 2, \ldots , n.
\]
Furthermore, when
a
i
j
(
t
)
≡
a
i
j
a_{ij}(t)\equiv a_{ij}
and
τ
i
j
(
t
)
≡
τ
i
j
\tau _{ij}(t)\equiv \tau _{ij}
,
i
,
j
=
1
,
…
,
n
i,j =1,\ldots ,n
, are constants but
r
i
(
t
)
,
i
=
1
,
…
,
n
r_i(t),\ i=1, \ldots ,n
, remain
ω
\omega
-periodic, we show that the condition on
(
∗
∗
)
(\ast \ast )
is also necessary for
(
∗
)
(\ast )
to have at least one positive
ω
−
\omega -
periodic solution.