Self-catalyzed GaAs nanowire (NW) growth via the vapor-liquid-solid mechanism is investigated by a theoretical model including the kinetics of material transport inside the catalyst droplet. The proposed model allows the description of nucleation and growth of 2D islands on the top facet of GaAs NWs. Analytical expressions for the growth rate of the disk-shaped GaAs island due to the volume diffusion of species in the droplet and for the attachment rate of GaAs pairs to the critical island are derived. As a result, the duration of the droplet refilling stage and the island growth stage at typical growth conditions of self-catalyzed GaAs NWs are obtained by a self-consistent calculation. Also, the time evolution of the droplet composition and the island radius are found. The derived equations for the island growth rate can be applied for modeling of catalyst-assisted growth of other III-V compounds. The results of the modeling are in good agreement with the experimental data on self-catalyzed GaAs NW growth via molecular beam epitaxy and can be used for the optimization of the NW growth conditions.