In this paper, an attempt to develop the original single mode theory describing the nonlinear dynamics of a linearly unstable plasma wave, excited by the resonant interaction with energetic ions near the stability threshold to the case of n interacting plasma modes has been made. The effects of an energetic ion source and classical collisional processes represented by the Krook, diffusion and dynamical friction (drag) collision operators are included in the model. For numerical purposes, the problem has been reduced to ten nonlinearly coupled integro-differential equations. In comparison to the previous papers, the system revealed similar (the steady-state, oscillation, and blow-up solutions), as well as quite new types of the amplitudes behaviour, i.e. different levels of competition between the modes.