We develop a notion of monotonicity for maps defined on Euclidean spaces Rk+, of arbitrary dimension k. This is a geometric approach that extends the classical notion of planar monotone maps or planar competitive difference equations. For planar maps, we show that our notion and the classical notion of monotonicity are equivalent. In higher dimensions, we establish certain verifiable conditions under which Kolmogorov monotone maps on Rk+ have a globally asymptotically stable fixed point. We apply our results to two competition population models, the Leslie–Gower and the Ricker models of two- and three-species. It is shown that these two models have a unique interior fixed point that is globally asymptotically stable.