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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.