The anisotropic loss‐cone distribution of fusion‐born alpha particles in a mirror system is capable of destabilizing low‐frequency plasma waves. A one‐and‐one‐half‐dimensional quasilinear diffusion model is developed to study the nonlinear evolution of the alpha distribution function resulting from enhanced diffusion in velocity space and radial position driven by the unstable Alfve´n wave. It is found that pitch‐angle scattering of the resonant alphas into the loss cone causes significant losses while radial losses are negligible for a reactor‐size plasma. For a typical tandem‐mirror reactor, the alpha plasma heating is greatly reduced and ignition in the central cell cannot be achieved.