Individual dispersal propensity is a crucial feature in ecological systems for organisms while searching for a mate or a suitable landscape, foraging, or diminishing the competition. The different ways individuals move is primarily the result of evolution, being survival the ultimate reason for them to disperse. As discussed in [1], dispersal ability can vary within a species as among species. Indeed, especially in fragmented habitats, dispersal polymorphisms, namely the coexistence of individuals belonging to the same species with different dispersal rates, are commonplace in nature and a cause of diversity in populations: for instance, the coexistence of flight-capable and flightless individuals in insect populations competing for mate location [2]. The object of the present work is not to study any particular species, or the dynamics of any particular manner to move, but the emerging effect due to a heterogeneity in the individual dispersal ability within a population from a general ecological perspective. Doing this, we investigate the problem of the natural selection in dispersal-structured populations, namely a population of competing individuals that are characterized by different dispersal propensities but are otherwise identical. To this end, we develop an individual-based model consisting initially of an ensemble of N_0 organisms that move in real space undergoing Brownian motion with different diffusion constants κ_j with j=1,...,N_0 extracted randomly from a uniform distribution in the interval κ[(1-d),(1+d)], with average value κ, standard deviation κ d⁄√3 and relative width 0<d<1. Stochasticity of demographic events and competition for resources are considered in the model by introducing individual density-dependent birth and death rates as in Ref. [3]. The temporal dynamics is event-driven and advanced through time by using exponentially distributed inter-event times. At each time step τ: 1) the type of event (birth or death) is determined, 2) the bug performing the event is selected (the value of k_j is inherited by newborns), 3) all the organisms diffuse with normally distributed jump-length with standard deviation 2√(k_j τ). In the model, both the resulting diversity D (the number of different diffusion constant present in the system) and the distribution of diffusivities are determined by the level of temporal fluctuations, the values of κ and d, and the appearance of patch formation. As a result of the competition process, different ranges of diffusion constants can be selected based upon the values of the former parameters. We stress the fact, somewhat unexpected, that under certain initial conditions and small temporal fluctuations, the competition success is given to an optimal diffusivity resulting from the interplay between the inter- and intra-patch competition. Its value is obtained from typical first-passage time calculations. Regarding diversity, we observed that large values of κ lead to a rather faster disappearance of the initial diversity and result in a spatially homogeneous distribution of individuals. Yet, small values of κ, together with the spatial correlations, allow the individuals to form patches resulting in a diversity equal to the total number of them.
