Recent observations and theoretical studies have established the importance of the tenuous helium ion component in determining instability growth and low-frequency magnetic fluctuation properties in the terrestrial magnetosheath. Under low-&bgr;, high-ion anisotropy sheath conditions, enhanced fluctuations observed below the proton cyclotron frequency have been attributed to the proton and helium cyclotron anisotropy instabilities. This paper uses second-order theory and one-dimensional hybrid computer simulations to examine the nonlinear properties of these two instabilities at relatively weak fluctuation levels and at propagation parallel to the background magnetic field. The simulations confirm the second-order predictions that both instabilities yield efficient wave-particle interactions, the rate at which the driving species anisotropy is reduced is much greater than the rate at which the species loses kinetic energy. This result suggest that these instabilites should saturate at relatively low levels; we derive an approximate expression for the fluctuating field energy at saturation of the fastest growing modes and find that it is in fair agreement with three computer simulations. In addition, second-order theory and simulations concur that the helium component enhances the wave-particle exchange rate for proton anisotropy reduction by the proton cyclotron instability; theory predicts and simulations confirm that the saturation energy of this mode is substantially lower than it is in the absence of helium ¿American Geophysical Union 1993