If \(X\) is a compact metric space and \(\varphi\) is a continuous selfmapping of \(X\), then the orbit of a point \(x \in X\) is \(\{\varphi^n\}_{n=0}^\infty\), where \(\varphi^n\) denotes the \(n\)th-iterate of \(\varphi\). The \(\omega\)-limit of \(x\) under \(\varphi\), \(\omega_\varphi (x)\), is the set of limit points of its orbit. A set \(Y \subset X\) is said to be \(\varphi\)-periodic whenever there exists \(n \geq 1\) such that \(\varphi^n (Y) = Y\). In this paper, the authors extend to the two-dimensional setting a one-dimensional result due to Sharkovsky about the structure of the \(\omega\)-limits of selfmappings of \(I := [0,1]\). Specifically, they prove that if \(F:I^2 \to I^2\) is continuous and antitriangular (that is, \(F\) has the form \(F(x,y) = (\alpha (y), \beta (x))\), \((x,y) \in I^2\)) and \(L = \omega_F (x,y)\), then \(L\) can be one of the following types: a finite set;an infinite nowhere dense set;a finite union of nondegenerate \(F\)-periodic rectangles whose interiors are pairwise disjoint. Antitriangular maps are closely related with the model of an economic process callet Cournot duopoly.