Nonlinear diffusion effects on biological population spatial patterns

TL;DR

This paper explores how nonlinear, anomalous diffusion affects spatial pattern formation in biological populations, revealing that diffusion type critically influences pattern emergence and shape.

Contribution

It introduces a generalized nonlinear diffusion model with density-dependent diffusion, analyzing its impact on pattern formation in a nonlocal competition framework.

Findings

Nonlinear diffusion alters the phase diagram of pattern formation.

Subdiffusion induces fragmentation of spatial patterns.

Critical thresholds for pattern onset can be analytically predicted.

Abstract

Motivated by the observation that anomalous diffusion is a realistic feature in the dynamics of biological populations, we investigate its implications in a paradigmatic model for the evolution of a single species density $u(x,t)$. The standard model includes growth and competition in a logistic expression, and spreading is modeled through normal diffusion. Moreover, the competition term is nonlocal, which has been shown to give rise to spatial patterns. We generalize the diffusion term through the nonlinear form $\partial_t u(x,t) = D \partial_{xx} u(x,t)^\nu$ (with $D, \nu>0$), encompassing the cases where the state-dependent diffusion coefficient either increases ($\nu>1$) or decreases ($\nu<1$) with the density, yielding subdiffusion or superdiffusion, respectively. By means of numerical simulations and analytical considerations, we display how that nonlinearity alters the phase diagram. The type of diffusion imposes critical values of the model parameters for the onset of patterns and strongly influences their shape, inducing fragmentation in the subdiffusive case. The detection of the main persistent mode allows analytical prediction of the critical thresholds.