In this paper, we rigorously
prove the existence and stability of K-peaked asymmetric
patterns for the Gierer-Meinhardt system in a two dimensional domain
which are far from
spatial homogeneity.
We show that given any positive integers k_1,\,k_2 \geq 1
with k_1+k_2=K,
there are asymmetric patterns with
k_1 large peaks and k_2 small peaks.
Most of these asymmetric patterns are shown
to be unstable. However,
in a narrow range of parameters,
asymmetric patterns may be stable
(in contrast to the one-dimensional case)
