  The spreading of evolutionary novelties across populations is the central
element of adaptation. Unless population are well-mixed (like bacteria in a
shaken test tube), the spreading dynamics not only depends on fitness
differences but also on the dispersal behavior of the species. Spreading at a
constant speed is generally predicted when dispersal is sufficiently
short-ranged. However, the case of long-range dispersal is unresolved: While it
is clear that even rare long-range jumps can lead to a drastic speedup, it has
been difficult to quantify the ensuing stochastic growth process. Yet such
knowledge is indispensable to reveal general laws for the spread of modern
human epidemics, which is greatly accelerated by aviation. We present a simple
iterative scaling approximation supported by simulations and rigorous bounds
that accurately predicts evolutionary spread for broad distributions of long
distance dispersal. In contrast to the exponential laws predicted by
deterministic "mean-field" approximations, we show that the asymptotic growth
is either according to a power-law or a stretched exponential, depending on the
tails of the dispersal kernel. More importantly, we provide a full
time-dependent description of the convergence to the asymptotic behavior which
can be anomalously slow and is needed even for long times. Our results also
apply to spreading dynamics on networks with a spectrum of long-range links
under certain conditions on the probabilities of long distance travel and are
thus relevant for the spread of epidemics.
