The authors study the following delayed Lotka-Volterra competitive system with dispersal and periodic coefficients \[ \begin{aligned} \dot x_1(t)=&x_1(t)(a_1(t)-a_{11}(t)x_1(t)-\frac{a_{13}(t)y(t-\tau)}{x_1(t-\tau)+\alpha(t)y(t-\tau)})+D_1(t)(x_2(t)-x_1(t)),\\ \\ \dot x_2(t)=&x_2(t)(a_2(t)-a_{22}(t)x_2(t))+D_2(t)(x_1(t)-x_2(t)),\\ \dot y(t)=&y(t)(a_3(t)-a_{33}(t)y(t)-\frac{a_{31}(t)x_1(t-\tau)}{x_1(t-\tau)+\alpha(t)y(t-\tau)}), \end{aligned}\eqno{(1)} \] where \(y(t)\) and \(x_1(t)\) represent the densities of species \(y\) and species \(x\) in patch 1, \(x_2(t)\) is the density of species \(x\) in patch 2. \(\tau>0\) is a constant delay due to the gestation. \(a_i(t), a_{ii}(t) (i,j=1,2,3), a_{13}(t), a_{31}(t), D_1(t), D_2(t)\) and \(\alpha(t)\) are strictly positive, periodic, continuous functions with period \(\omega>0\). Sufficient conditions are derived for the existence of periodic solutions to system (1) by using the continuation theorem developed by [R. E. Gaines and J. L. Mawhin, Coincidence degree and nonlinear differential equations. Lecture Notes in Mathematics. 568. Berlin-Heidelberg-New York: Springer-Verlag (1977; Zbl 0339.47031)].