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Motivated by the observation that anomalous diffusion is a realistic featurein the dynamics of biological populations, we investigate its implications in aparadigmatic model for the evolution of a single species density $u(x,t)$. Thestandard model includes growth and competition in a logistic expression, andspreading is modeled through normal diffusion. Moreover, the competition termis nonlocal, which has been shown to give rise to spatial patterns. Wegeneralize 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 thestate-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 ofdiffusion imposes critical values of the model parameters for the onset ofpatterns and strongly influences their shape, inducing fragmentation in thesubdiffusive case. The detection of the main persistent mode allows analyticalprediction of the critical thresholds.

Nonlinear diffusion effects on biological population spatial patterns