The electromagnetic proton cyclotron instability and the mirror instability are driven by the proton temperature anisotropy T⊥p/T∥p>1, where ⊥ and ∥ denote directions relative to the background magnetic field. Linear theory and one-dimensional hybrid simulations imply that the former mode grows more rapidly over 0.05≤&bgr;∥p≤5 and that wave-particle scattering by its enhanced fluctuations imposes an upper bound on the temperature anisotropy of the form T⊥p/T∥p-1=Sp/&bgr;∥p&agr;p where &bgr;∥p≡8&pgr;npT∥p/Bo2 and Bo is the background magnetic field. Here Sp and &agr;p are fitting parameters, and 0.4≲&agr;p≲0.5. This paper describes results from more general two-dimensional hybrid simulations, which permit both instabilities to grow simultaneously. These simulations confirm the one-dimensional results on the initial domain 0.05≤&bgr;∥p≲5; enhanced fluctuations display the properties of the proton cyclotron instability and &agr;p≃0.4. On this domain the two-dimensional simulations also yield an upper bound for the fluctuating field energy density of the form |ΔB|2/Bo2≡∑k|ΔBk|2/Bo2=SB&bgr;∥p&agr;B with fitting parameter 0.5≲&agr;B≲1. The simulations on the initial domain 10≤&bgr;∥p≤100 show spectral characteristics of both instabilities and exhibit a more stringent bound on the proton anisotropy, in agreement with observations in the terrestrial magnetosheath. ¿ 1997 American Geophysical Union