Internal feedbacks are commonly present in biological populations and can
play a crucial role in the emergence of collective behavior. We consider a
generalization of Fisher-KPP equation to describe the temporal evolution of the
distribution of a single-species population. This equation includes the
elementary processes of random motion, reproduction and, importantly, nonlocal
interspecific competition, which introduces a spatial scale of interaction.
Furthermore, we take into account feedback mechanisms in diffusion and growth
processes, mimicked through density-dependencies controlled by exponents ν
and μ, respectively. These feedbacks include, for instance, anomalous
diffusion, reaction to overcrowding or to rarefaction of the population, as
well as Allee-like effects. We report that, depending on the dynamics in place,
the population can self-organize splitting into disconnected sub-populations,
in the absence of environment constraints. Through extensive numerical
simulations, we investigate the temporal evolution and stationary features of
the population distribution in the one-dimensional case. We discuss the crucial
role that density-dependency has on pattern formation, particularly on
fragmentation, which can bring important consequences to processes such as
epidemic spread and speciation