Partial differential equations (PDEs) involving fractional Laplace operators
have been increasingly used to model non-local diffusion processes and are
actively investigated using both analytical and numerical approaches. The
purpose of this work is to study the effects of the spectral fractional
Laplacian on the bifurcation structure of reaction-diffusion systems on bounded
domains. In order to do this we use advanced numerical continuation techniques
to compute the solution branches. Since current available continuation packages
only support systems involving the standard Laplacian, we first extend the
pde2path software to treat fractional PDEs. The new capabilities are then
applied to the study of the Allen-Cahn equation, the Swift-Hohenberg equation
and the Schnakenberg system (in which the standard Laplacian is each replaced
by the spectral fractional Laplacian). Our study reveals some common effects,
which contributes to a better understanding of fractional diffusion in generic
reaction-diffusion systems. In particular, we investigate the changes in
snaking bifurcation diagrams and also study the spatial structure of
non-trivial steady states upon variation of the order of the fractional
Laplacian. Our results show that the fractional order can induce very
significant qualitative and quantitative changes in global bifurcation
structures