Liouville's theorem and approximate but extremely accurate expressions which reflect the invariance of &mgr; and J can be used to determine analytically the evolution of an adiabatically convecting energetic particle distribution. Features of the convecting distribution which are reproduced by our model include positive pitch angle anisotropy, regions in velocity space in which ∂F?∂v⊥>0, and th energy dependence of the degree of particle injection. The energy dependence of the injection yeilds upper and lower cutoffs to the distribution within the plasmasphere, and only an upper cutoff outside. We have used this approach to study the evolution of ion cyclotron waves in a convecting particle distribution, and have found that the upper cutoff to the distribution restricts the growth of these waves, both on and off the geomagnetic equator, to a narrow range of frequencies which grow only inside the plasmasphere. In addition, we found that the upper cutoff limits wave growth to a feq hours on either side of dusk. We have examined the results for two source distributions, a power law and a Maxwellian,and we found that for distributions with equivalent mean energies the power law is unstable over a greater range of frequencies and radial distances than the Maxwellian.