Pattern formation in the nonlocal one-dimensional Fisher–Kolmogorov–Petrovskii–Piskunov model with fractional Laplacian

The dynamics of population density is studied within the framework of a generalized one-dimensional Fisher-Kolmogorov-Petrovskii-Piskunov model with nonlocal competitive losses and a fractional Laplace operator describing anomalous diffusion. Numerical solutions are obtained for selected parameter values considering two forms of the fractional Laplacian: the regional fractional Laplace operator and the Caputo-Fabrizio fractional Laplace operator. The simulations reveal that during the evolution of an initially localized population density distribution, spatially non-uniform patterns emerge in the form of local maxima. These maxima can be interpreted as dissipative structures. The impact of the fractional Laplacian parameter on the emergence and characteristics of these structures is also analyzed.

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