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In this paper we study the existence and uniqueness of positive and periodic solutions of nonlinear delay integral systems of the type $$x\left(t\right)={\int }_{t-{\tau }_1}^{t}f(s,x\left(s\right),y\left(s\right))ds$$ $$y\left(t\right)={\int }_{t-{\tau }_2}^{t}g(s,x\left(s\right),y\left(s\right))ds$$ which model population growth in a periodic environment when there is an interaction between two species. For the proofs, we develop an adequate method of sub-supersolutions which provides, in some cases, an iterative scheme converging to the solution.