  The dynamics of dispersal-structured populations, consisting of competing
individuals that are characterized by different diffusion coefficients but are
otherwise identical, is investigated. Competition is taken into account through
demographic processes. The problem addressed models natural selection. It is
observed that the mean value and the relative width of the initial distribution
of the diffusion coefficients characterizing the individuals together with the
temporal fluctuations determine the final distribution of the diffusivities
(diffusion coefficients leading to the competition success) as well as the
final diversity of the system at finite time (the number of different diffusion
coefficients present in the system). Large initial mean diffusivity of the
system leads to a rather fast disappearance of the diversity. Instead, small
initial mean diffusivity of the system leads to a diversity equal to the number
of niches forming in the system due to the competitive interactions. The
cluster formation is also associated to the competition success of the slower
diffusing individuals. The diversity is diminished by the increase of the
temporal fluctuations that give the competition advantage to the faster
diffusing individuals. Somewhat counterintuitively, under certain conditions
the competition success is given by intermediate values of the diffusion
coefficients.
