Hybrid control strategy for the Lévy superdiffusion Sel’kov-Schnakenberg model: formation, conversion, and annihilation of Turing patterns

As a typical reaction-diffusion model, the Sel’kov-Schnakenberg model has revealed its self-organization phenomena. However, the existing works have been limited to the consideration of normal self-diffusion, failing to account for interactions between different substances or more realistic forms of anomalous diffusion. This paper introduces a Lévy superdiffusion Sel’kov-Schnakenberg model with cross-diffusion. By analyzing the distribution of characteristic roots, we obtain the stability conditions for the non-spatial diffusion model, as well as the conditions for Turing pattern formation in a spatial diffusion model. The amplitude equation in the vicinity of the Turing bifurcation threshold is extracted to ascertain the precise structure of the Turing patterns. Additionally, the control of Turing patterns in two-dimensional space remains an open challenge. To address this, we present a hybrid control strategy utilizing two control parameters to precisely regulate the Turing patterns. The impact of the Lévy superdiffusion exponent on the form of patterns and the sensitivity analysis of model and control parameters are also discussed. The accuracy and efficiency of the proposed control strategy are confirmed through numerical simulations, which validates the theoretical findings. The results show that the coordinated control of the two parameters allows for the formation, transformation, and even annihilation of Turing patterns, offering valuable insights for biomedical applications, such as tissue patterning and morphogenesis.