Spatial pattern regulation strategy of bimolecular model with anomalous diffusion and nonlocal effects

The mechanisms underlying the formation of spatial patterns in chemical reaction processes have been well established. However, there is still a lack of effective control methods for regulating the formation of spatial patterns and facilitating the transitions between different patterns. This article proposes a novel bimolecular model, incorporating nonlocal effects in reactions and molecular-level anomalous diffusion, within the two-dimensional reaction domain based on the previous framework. Linear stability analysis provides the necessary and sufficient conditions for inducing Turing instability. By utilizing the multiscale analysis, the amplitude equations pertinent to the new model are derived. After identifying the parameter ranges conducive to the formation of fundamental patterns, we introduce the proportional-derivative (PD) control strategy to manage spatial pattern formation and transitions. Simulation results validate the theoretical analysis and demonstrate the efficacy of the PD control strategy.