In this article, using Mawhin's continuation theorem of coincidence degree theory, we obtain sufficient conditions for the existence of positive almost periodic solutions for the system of equations $$ dot{u}_i(t)=u_i(t)Big[r_i(t)-a_{ii}(t)u_i(t) -sum_{j=1, jeq i}^na_{ij}(t)u_jig(t-au_j(t,u_1(t), dots,u_n(t))ig)Big], $$ where $r_i,a_{ii}>0$, $a_{ij}geq0(jeq i$, $i,j=1,2,dots,n)$ are almost periodic functions, $au_iin C(mathbb{R}^{n+1},mathbb{R})$, and $au_i(i=1,2,dots,n)$ are almost periodic in $t$ uniformly for $(u_1,dots,u_n)^Tinmathbb{R}^n$. An example and its simulation figure illustrate our results.