Analysis and numerical simulations of fractional order model of ring formation due to plant-soil negative feedback

• How to generalize the integer order model into a time-fractional model and what are the specific benefits associated with it? • What are the differences in stability analysis for the integer and fractional order cases? • How to solve 2D time-fractional PDEs and What are the computational challenges associated with solving them specifically, when it comes to the Neumann boundary conditions to address boundary and corner points? • How do the results for the integer and fractional order differ from each other? • What are the effects of different fractional orders on the dynamics of the fairy circles and patterns? Vegetation ring formation is a spatial pattern observed in ecosystems influenced by plant–soil negative feedback. Classical integer-order models capture only instantaneous interactions and cannot represent the history-dependent processes shaping these dynamics. To overcome this limitation, we propose a new time-fractional reaction-diffusion problem, that extends the model of Cartenì et al. of 2012 [1], by incorporating memory effects through Caputo derivatives. The system, consisting of coupled fractional partial differential equations (FPDEs) for biomass and toxicity, is analyzed for existence and uniqueness of solutions, equilibrium states, and stability in both homogeneous and heterogeneous cases. Since the analytical solution is not available, a numerical approach has been proposed. The numerical experiments illustrate variations in ring formation throughout time and space. The study covers multiple aspects, such as the influence of the fractional derivation index κ on pattern formation, showing that when κ decreases, the biomass spreads over a larger area with fewer oscillations and amplitudes. Moreover, reducing κ has the effect of slowing down the dynamics, requiring more time to reach the equilibrium points, and causes the ring’s width to expand, which shrinks the internal ring diameter until only disks are visible as κ tends to 0.6.