  The nonlocal Fisher equation is a diffusion-reaction equation with a nonlocal
quadratic competition, which describes the reaction between distant
individuals. This equation arises in evolutionary biological systems, where the
arena for the dynamics is trait space, diffusion accounts for mutations and
individuals with similar traits compete, resulting in partial niche overlap. It
has been found that the (non-cutoff) deterministic system gives rise to a
spatially inhomogeneous state for a certain class of interaction kernels, while
the stochastic system produces an inhomogeneous state for small enough
population densities. Here we study the problem of front propagation in this
system, comparing the stochastic dynamics to the heuristic approximation of
this system by a deterministic system where the linear growth term is cut off
below some critical density. Of particular interest is the nontrivial pattern
left behind the front. For large population density, or small cutoff, there is
a constant velocity wave propagating from the populated region to the
unpopulated region. As in the local Fisher equation, the spreading velocity is
much lower than the Fisher velocity which is the spreading velocity for
infinite population size. The stochastic simulations give approximately the
same spreading velocity as the deterministic simulation with appropriate birth
cutoff. When the population density is small enough, there is a different
mechanism of population spreading. The population is concentrated on clusters
which divide and separate. This mode of spreading has small spreading velocity,
decaying exponentially with the range of the interaction kernel.
