  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 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 non-linear effects is
shown to result in a peculiar polariton dynamics. However, near the lasing
threshold, the predominant contribution of the nonlinear effects is the
saturation of the linear gain.
