The author considers the two-dimensional dynamical system given by
generated by the map . Assuming that and are nondecreasing, is a competitive map which leaves invariant the diagonal . General definitions of order relations and competitive and cooperative maps are given and, by an example, it is shown that complicated dynamics can occur for such maps, but for exceptional initial data. This paper focusses on the special class of orientation reversing competitive and cooperative maps. Such maps have relatively simple dynamics. Finally, several applications to two models in population biology are discussed.