Consider the class of planar systems of first-order rational difference equations 1´ where , and the parameters are nonnegative and such that both terms in the right-hand side of (1′) are nonlinear. In this paper, we prove the following discretized Poincare-Bendixson theorem for the class of systems (1′). If the map associated to system (1′) is bounded, then the following statements are true: In particular, system (1′) cannot exhibit chaos when its associated map is bounded. Moreover, we show that if both equilibrium curves of (1′) are reducible conics and the map associated to system (1′) is unbounded, then every solution converges to one of up to infinitely many equilibria or to or . MSC:39A05, 39A11.