PD control of turing instability in a fractional-order Sel'kov-Schnakenberg reaction-diffusion model with slow-fast effects

In light of the current lack of understanding of how Turing patterns are regulated in the fractional-diffusion Sel'kov-Schnakenberg system, this study proposes a dynamic control framework based on proportional-derivative (PD) feedback. By formulating a fractional-order diffusion model that incorporates PD control alongside fast-slow dynamics, we systematically reveal the mechanism of Turing-pattern formation under the combined action of cross-diffusion and fractional diffusion. First, the stability of equilibrium points under non-diffusive conditions was systematically analyzed, and the critical conditions for Hopf bifurcation induced by slow-fast parameters were derived. Next, treating the cross-diffusion coefficient as the bifurcation parameter, we perform a diffusion-driven instability analysis to establish a quantitative relationship between the Turing modulus threshold and the cross-diffusion coefficient. Utilizing multi-scale analysis, the amplitude equations near the Turing bifurcation threshold were derived. Numerical simulations show that the cross-diffusion coefficient controls the type of Turing pattern that emerges, while variations in the fast-slow time-scale parameter induce transitions between different pattern modes. The nonlocal nature of fractional diffusion overcomes the limitations of integer-order diffusion, significantly altering the range of unstable wavenumbers and thus allowing adjustment of the pattern's spatial density. The designed PD controller effectively suppresses Turing instability, thereby ensuring global asymptotic stability of the system in both temporal evolution and spatial configuration.