We present a nonlinear dynamical model for DNA denaturation which includes cooperativity effects through [ital anharmonic] nearest-neighbor stacking interactions. Transfer-integral calculations show that this one-dimensional model, without including long-range interactions, exhibits a sharp denaturation reminiscent of a first-order transition at a temperature lower than a similar model with harmonic coupling. Self-consistent phonon calculations point out the essential role of nonlinear effects in the mechanism.