We consider the dynamics of a quantum degenerate trapped gas of 7Li atoms. Because the atoms have a negative s-wave scattering length, a Bose condensate of 7Li becomes mechanically unstable when the number of condensate atoms approaches a maximum value. We calculate the dynamics of the collapse that occurs when the unstable point is reached. In addition, we use the quantum Boltzmann equation to investigate the nonequilibrium kinetics of the atomic distribution during and after evaporative cooling. The condensate is found to undergo many cycles of growth and collapse before a stationary state is reached.