  We consider a branching random walk on the lattice, where the branching rates
are given by an i.i.d. Pareto random potential. We describe the process,
including a detailed shape theorem, in terms of a system of growing lilypads.
As an application we show that the branching random walk is intermittent, in
the sense that most particles are concentrated on one very small island with
large potential. Moreover, we compare the branching random walk to the
parabolic Anderson model and observe that although the two systems show
similarities, the mechanisms that control the growth are fundamentally
different.
