  Population dynamics of individuals undergoing birth and death and diffusing
by short or long ranged twodimensional spatial excursions (Gaussian jumps or
L\'{e}vy flights) is studied. Competitive interactions are considered in a
global case, in which birth and death rates are influenced by all individuals
in the system, and in a nonlocal but finite-range case in which interaction
affects individuals in a neighborhood (we also address the noninteracting
case). In the global case one single or few-cluster configurations are achieved
with the spatial distribution of the bugs tied to the type of diffusion. In the
L\'{e}vy case long tails appear for some properties characterizing the shape
and dynamics of clusters. Under non-local finite-range interactions periodic
patterns appear with periodicity set by the interaction range. This length acts
as a cut-off limiting the influence of the long L\'{e}vy jumps, so that spatial
configurations under the two types of diffusion become more similar. By
dividing initially everyone into different families and following their descent
it is possible to show that mixing of families and their competition is greatly
influenced by the spatial dynamics.
