# Single-species fragmentation: the role of density-dependent feedbacks

## TL;DR

This paper explores how internal feedback mechanisms in a generalized Fisher-KPP model influence population fragmentation and pattern formation, highlighting the role of density-dependent diffusion and growth in self-organizing populations.

## Contribution

It introduces a generalized Fisher-KPP equation with density-dependent feedbacks and demonstrates their impact on population fragmentation and pattern formation through numerical simulations.

## Key findings

- Population can self-organize into disconnected sub-populations without environmental constraints.
- Density-dependent feedbacks significantly influence pattern formation and fragmentation.
- The model's dynamics have implications for epidemic spread and speciation processes.

## Abstract

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 $\nu$ and $\mu$, 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.