Fractional Gray-Scott Model: Well-posedness, Discretization, and Simulations
For researchers studying pattern formation in reaction-diffusion systems, this work provides a mathematical and numerical framework to incorporate anomalous diffusion, revealing new pattern behaviors not captured by the integer-order model.
This paper extends the Gray-Scott model to include anomalous diffusion via the fractional Laplacian, proving well-posedness and developing a numerical scheme with second-order convergence. Simulations reveal distinct pattern formation depending on the fractional order, quantified by radial distribution functions, and a scaling law for steady patterns is discovered.
The Gray-Scott (GS) model represents the dynamics and steady state pattern formation in reaction-diffusion systems and has been extensively studied in the past. In this paper, we consider the effects of anomalous diffusion on pattern formation by introducing the fractional Laplacian into the GS model. First, we prove that the continuous solutions of the fractional GS model are unique. We then introduce the Crank-Nicolson (C-N) scheme for time discretization and weighted shifted Grünwald difference operator for spatial discretization. We perform stability analysis for the time semi-discrete numerical scheme, and furthermore, we analyze numerically the errors with benchmark solutions that show second-order convergence both in time and space. We also employ the spectral collocation method in space and C-N scheme in time to solve the GS model in order to verify the accuracy of our numerical solutions. We observe the formation of different patterns at different values of the fractional order, which are quite different than the patterns of the corresponding integer-order GS model, and quantify them by using the radial distribution function (RDF). Finally, we discover the scaling law for steady patterns of the RDFs in terms of the fractional order $1<α\leq 2 $.