  We study reaction-diffusion systems where diffusion is by jumps whose sizes
are distributed exponentially. We first study the Fisher-like problem of
propagation of a front into an unstable state, as typified by the A+B $\to$ 2A
reaction. We find that the effect of fluctuations is especially pronounced at
small hopping rates. Fluctuations are treated heuristically via a density
cutoff in the reaction rate. We then consider the case of propagating up a
reaction rate gradient. The effect of fluctuations here is pronounced, with the
front velocity increasing without limit with increasing bulk particle density.
The rate of increase is faster than in the case of a reaction-gradient with
nearest-neighbor hopping. We derive analytic expressions for the front velocity
dependence on bulk particle density. Compute simulations are performed to
confirm the analytical results.
