We recently setup a high energy, tunable, injection seeded Ti:Sapphirelaser source whose implementation was reported in [Rev. Sci. Instrum.78, 033102 (2007)]. The analysis of the spectral linewidth has demonstratedthat the pulses were time Fourier-transform limited. The present paperreports on the modeling of this source with the aim of understandingthe 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 solutionobtained by solving the laser rate equations. The developed modeldeals with the spatial dependence of the photon and population inversiondensities, the physical properties of the Ti:Sa medium, and the unstableresonator (based on a super-Gaussian output coupler). The energy-relatedas well as the temporal features are discussed. The use of a squaredasymmetrical hyperbolic secant function to describe the temporal pulseshapes is justified. The pulse durations exhibit a simple analyticalshape when they are plotted versus the output pulse energy. The modelis applied to experimental data obtained at different wavelengthsby tuning the laser amplifier previously described. It can readilybe applied to other pulsed Ti:Sapphire sources.