Based on the diffusive equation of incoherent-feedback random lasers，whose gain is modified by adding a saturation factor，the evolution of the emitted light intensity in two-photon pumping random lasers varying with time is studied by use of finite difference method. Numerical results show how the intensity of emitted light varies from the spontaneous radiation，which undergoes amplification at small-signal gain，to the steady output due to the saturation effect. It is noted that the nonlinear effect，e.g. saturation of gain，in such laser system can be achieved under low intensity conditions due to the disorder of media. Our results give a theoretical support to obtaining the steady output of the two-photon pumping random lasers.