We recently setup a high energy, tunable, injection seeded Ti:Sapphire laser source whose implementation was reported in [Rev. Sci. Instrum. 78, 033102 (2007)]. The analysis of the spectral linewidth has demonstrated that the pulses were time Fourier-transform limited. The present paper reports on the modeling of this source with the aim of understanding the energy conversion in the injected as well as in the free run regimes. The experimental lasing threshold, output energy, slope efficiencies, buildup time, and pulse duration are compared to the analytical solution obtained by solving the laser rate equations. The developed model deals with the spatial dependence of the photon and population inversion densities, the physical properties of the Ti:Sa medium, and the unstable resonator (based on a super-Gaussian output coupler). The energy-related as well as the temporal features are discussed. The use of a squared asymmetrical hyperbolic secant function to describe the temporal pulse shapes is justified. The pulse durations exhibit a simple analytical shape when they are plotted versus the output pulse energy. The model is applied to experimental data obtained at different wavelengths by tuning the laser amplifier previously described. It can readily be applied to other pulsed Ti:Sapphire sources.