We discuss and apply a recently proposed model (Inglebert et al 2011 Euro. Phys. Lett.95 45002) using a Hamiltonian formalism for the study of Weibel-type instabilities in the relativistic limit. Taking advantage of the invariance of the generalized canonical momentum, we represent the plasma as a sum of N particle bunches invariant under the dynamics. This approach allows for a drastic reduction in the computational time when compared with the full Vlasov–Maxwell system of equations. Analytically, the model is exact and we recover the standard fluid dispersion relations in the case of the Weibel and filamentation instabilities. By initially selecting a specific class of particle bunches, we are able to give a fine description of the phase space dynamics interactions even in the saturation regime.