  It is well-known that both branching random walk models and trap models can
exhibit intermittency and localisation phenomena; the prototypical examples
being the parabolic Anderson and Bouchaud trap models respectively. Our aim is
to investigate how these localisation phenomena interact. To do so, we study a
hybrid model combining the dynamics of both the parabolic Anderson and the
Bouchaud trap models; more precisely, we consider a variant of the parabolic
Anderson model in which the underlying random walk driving the model is
replaced with the Bouchaud trap model.
  In this initial study, we consider the model where the potential field
distribution has Weibull tail decay and the trap distribution is bounded away
from zero. In dimension one, we further assume that the trap distribution
decays sufficiently fast. Under these conditions, we show that the localisation
effects in the hybrid model are strongly dominated by the influence of the
parabolic Anderson model; the Bouchaud trap model plays at most a minor role.
Moreover, we distinguish regimes in which the Bouchaud trap model acts to
strengthen or weaken the localisation effects due to the parabolic Anderson
model, and also identify regimes in which the Bouchaud trap model has no
influence at all on localisation in the hybrid model.
