We set up a mapping procedure able to translate the evolution of the radial profile of fast ions, interacting with toroidal Alfvén Eigenmodes, into the dynamics of an equivalent one-dimensional bump-on-tail system. We apply this mapping technique to reproduce the relevant ITER simulations, which clearly outline deviations from the diffusive quasi-linear (QL) model. Our analysis demonstrates the capability of the one-dimensional beam-plasma dynamics to predict the relevant features of the non-linear hybrid LIGKA/HAGIS simulations. In particular, we clearly identify how the deviation from the QL evolutive profiles is due to the presence of avalanche processes. A detailed analysis regarding the reduced dimensionality is also addressed, by means of phase-space slicing based on constants of motion. In the conclusions, we outline the main criticalities and outcomes of the procedure, which must be satisfactorily addressed to make quantitative predictions on the observed outgoing fluxes in a Tokamak device.