A set of easily verifiable sufficient
conditions are derived for the existence of positive
periodic solutions for delayed generalized predator-prey dispersion system
$$
\begin{array}{l}
x'_{1}(t)=x_1(t)g_1(t,x_1(t))-a_1(t)y(t)p_1(x_1(t))+ D_1(t)(x_2(t)-x_1(t)),\\
x'_{2}(t)=x_2(t)g_2(t,x_2(t))-a_2(t)y(t)p_2(x_2(t))+ D_2(t)(x_1(t)-x_2(t)),\\
y'(t)=y(t)\left[-h(t,y(t))+b_1(t)p_1(x_1(t-\tau_1))+ b_2(t)p_2(x_2(t-\tau_2))
\right],
\end{array}
$$
where $a_i(t),b_i(t)$ and $D_i(t)(i=1,2)$
are positive continuous $T$-periodic functions,
$g_i(t,x_i)$
$(i=1,2)$ and $h(t,y)$ are continuous and
$T$-periodic with respect to $t$ and $h(t,y)>0$
for $y>0,t,y\in R,p_i(x)(i=1,2)$ are continuous and
monotonously increasing functions, and
$p_i(x_i)>0$ for $x_i>0$.
