The emergence of spatiotemporal structures is frequently observed in nature, especially in the case of collections of living beings. In order to describe the patterns that can emerge in populations of a single species, we construct a model that takes into account elementary processes such as growth, competition and dispersal. We also take into account the fact that the concentration of individuals can affect the rate of dispersion of the population (yielding sub-diffusive and super-diffusive processes) due to the reaction to overcrowding and sparseness. Concentration can also influence the growth rate, either spoiling or supporting individuals reproduction. Then, we formulate a generalization of the Fisher equation which includes nonlinearities in diffusion and reproduction rates to mimic the density-dependent feedbacks and including also the nonlocality of the competition for resources, to account for the spatial scales of interaction. In order to analyse the role of these nonlinearities in pattern formation, numerical simulations were carried out together with analytical studies of linear stability. We determine a phase diagram in the space of parameters that characterize the nonlinearities, exhibiting the domains where patterns emerge. We also identify and classify different shapes of patterns: continuous, fragmented, peaked, and discuss their implications for population stability and survival.