Early warning of tipping in a chemical model with cross-diffusion via spatiotemporal pattern formation and transition.

The spatiotemporal pattern formation and transition driven by cross-diffusion of the Gray-Scott model are investigated for the early warning of tipping in this paper. The mathematical analyses of the corresponding non-spatial model and spatial model are performed first, which enable us to have a comprehensive understanding. Then, the linear stability analysis and the multiple scale analysis method exhibit that cross-diffusion is the key mechanism for the evolution of spatiotemporal patterns. Through selecting a cross-diffusion coefficient as the bifurcation parameter, the amplitude equations that can describe structural transition and determine the stability of different types of Turing patterns are derived. Ultimately, numerical simulations verify the validity of the theoretical results. It is demonstrated that in the absence of cross-diffusion, the spatiotemporal distribution of substances is homogeneous. Nevertheless, when the cross-diffusion coefficient exceeds its threshold value, the spatiotemporal distribution of substances will become inhomogeneous in space. As the cross-diffusion coefficient increases, the Turing instability region will be extended, leading to various types of Turing patterns: spots, stripes, and a mixture of spots and stripes.