  We introduce a new class of stochastic partial differential equations (SPDEs)
with seed bank modeling the spread of a beneficial allele in a spatial
population where individuals may switch between an active and a dormant state.
Incorporating dormancy and the resulting seed bank leads to a two-type coupled
system of equations with migration between both states. We first discuss
existence and uniqueness of seed bank SPDEs and provide an equivalent delay
representation that allows a clear interpretation of the age structure in the
seed bank component. The delay representation will also be crucial in the
proofs. Further, we show that the seed bank SPDEs give rise to an interesting
class of "on/off" moment duals. In particular, in the special case of the F-KPP
Equation with seed bank, the moment dual is given by an "on/off-branching
Brownian motion". This system differs from a classical branching Brownian
motion in the sense that independently for all individuals, motion and
branching may be "switched off" for an exponential amount of time after which
they get "switched on" again. Here, as an application of our duality, we show
that the spread of a beneficial allele, which in the classical F-KPP Equation,
started from a Heaviside intial condition, evolves as a pulled traveling wave
with speed $\sqrt{2}$, is slowed down significantly in the corresponding seed
bank F-KPP model. In fact, by computing bounds on the position of the rightmost
particle in the dual on/off-branching Brownian motion, we obtain an upper bound
for the speed of propagation of the beneficial allele given by
$\sqrt{\sqrt{5}-1}\approx 1.111$ under unit switching rates. This shows that
seed banks will indeed slow down fitness waves and preserve genetic
variability, in line with intuitive reasoning from population genetics and
ecology.
