A universal integral equation has been derived and solved for the nonlinear evolution of collective modes driven by kinetic wave-particle resonances near the instability threshold. The analysis always applies if there is dissipation present in the absence of the kinetic component. It is shown that the dominant nonlinearity stems from the dynamics of resonant particles and it can be taken into account perturbatively near the marginal state of the system. With a resonant particle source and classical relaxation processes included, the new equation allows one to determine the conditions for both the soft nonlinear regime, where the saturation level is proportional to the increment above threshold, and the hard nonlinear regime, where the saturation level is independent of the closeness to threshold. It has been found, both analytically and numerically, that in the hard regime the system exhibits explosive behavior with rapid oscillations of the mode amplitude. This suggests the application of this theoretical model to such phenomena as the onset of fishbone oscillations (the theory exhibits frequency chirping), evolution of interchange instabilities in the presence of hot electrons, collective mode excitation in storage rings etc. This theory has been used to interpret the experimental saturation levels of the Toroidal Alfven Eigenmodes in TFTR experiments. The theory provides a framework for fitting the existing experimental data and is capable of predicting fine features that are possible to measure with precision experiments. ^In collaboration with H.L.Berk, M.S.Pekker, N.V.Petviashvili, Institute for Fusion Studies, The University of Texas at Austin; F. Porcelli, Politechnic Institute of Turin, Italy; K.L.Wong, Princeton Plasma Physics Laboratory, Princeton University.