The nonautonomous system of delay differential equations \[ u_i'(t)=u_i(t)\biggl[ a_i(t) - \sum_{j=1}^{n} b_{ij}(t) u_j(t)- \sum_{j=1}^{n} c_{ij}(t) u_j(t-\tau_{ij}(t)) \biggr] \] is considered. Here, \(a_i(t)\), \(b_{ij}(t)\), \(c_{ij}(t)\), \(\tau_{ij}(t)\), \(i,j=1,\dots,n\), are continuous \(\omega\)-periodic functions with \(\int_0^\omega a_i(t)\,dt>0\), \(b_{ij}(t)>0\), \(c_{ij}(t)\geq 0\) and \(\tau_{ij}(t)\geq 0\).
The authors provide conditions under which there exists at least one positive \(\omega\)-periodic solution of the problem. Sufficient conditions are obtained for this solution to be globally asymptotically stable.