  In this paper we focus on three problems about the spreading speeds of
nonlocal dispersal Fisher-KPP equations. First, we study the signs of spreading
speeds and find that they are determined by the asymmetry level of the nonlocal
dispersal and $f'(0)$, where $f$ is the reaction function. This indicates that
asymmetric dispersal can influence the spatial dynamics in three aspects: it
can determine the spatial propagation directions of solutions, influence the
stability of equilibrium states, and affect the monotone property of solutions.
Second, we give an improved proof of the spreading speed result by constructing
new lower solutions and using the new "forward-backward spreading" method.
Third, we establish the relationship between spreading speed and exponentially
decaying initial data. Our result demonstrates that when dispersal is
symmetric, spreading speed decreases along with the increase of the
exponentially decaying rate. In addition, the results on the signs of spreading
speeds are applied to two special cases where we present more details of the
influence of asymmetric dispersal.
