Turing instability and 2-D pattern formation in reaction-diffusion systems derived from kinetic theory

We investigate Turing instability and pattern formation in two-dimensional domains for two reaction-diffusion models, obtained as diffusive limits of kinetic equations for mixtures of monatomic and polyatomic gases. The first model is of Brusselator type, which, compared with the classical formulation, presents an additional parameter whose role in stability and pattern formation is discussed. In the second framework, the system exhibits standard nonlinear diffusion terms typical of predator-prey models, but differs in reactive terms. In both cases, the kinetic-based approach proves effective in relating macroscopic parameters, often set empirically, to microscopic interaction mechanisms, thereby rigorously identifying admissible parameter ranges for the physical description. Furthermore, weakly nonlinear analysis and numerical simulations extend previously known one-dimensional results and reveal a wider scenario of spatial structures, including spots, stripes, and hexagonal arrays, that better reflect the richness observed in real-world systems.