We study the stability of a four-fluid cosmic-ray-MHD system which comprises magnetized thermal plasma, cosmic rays, forward and backward propagating Alfvén waves. The coupling between the plasma, cosmic rays, and waves depends on the energy density of the waves. Local short-wavelength small perturbation analysis is performed on a background steady state. The magnetoacoustic modes of the plasma are modified and intertwined with cosmic ray and wave modes, while the plasma Alfvén mode is unaffected. The parameter space is large and the stable/unstable regions of the system are complicated. We discuss some special cases analytically and work out some general cases numerically. Roughly speaking, the system is more likely to be stable if the perturbations have very short wavelength, not propagating at large angle with background magnetic field, large cosmic ray energy density, not too small energy density for both waves and large thermal energy density (and no self-gravity).
- cosmic rays