This work is inspired by recent experiments on the formation of vortices in exciton-polariton condensates placed in rotating optical traps. We study theoretically the dynamics of the formation of such vortices and elucidate the fundamental role of the mode competition effect in determining the properties of stationary polariton states triggered by stimulated scattering of exciton-polaritons. The interplay between linear and nonlinear effects is shown to result in peculiar polariton dynamics. However, near the lasing threshold, the predominant contribution of the nonlinear effects is the saturation of the linear gain.