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

TL;DR

This work addresses Turing instability and 2D pattern formation in reaction-diffusion systems derived as diffusive limits of kinetic equations for gas mixtures. It introduces two models: a Brusselator-type system with an additional parameter $d$ and a nonlinear cross-diffusion system with algebraic coupling, with macroscopic coefficients expressed in terms of microscopic data. Using linear stability analysis, weakly nonlinear amplitude equations (with $r_0$, $\mu$, $s_1$, $s_2$, $s_3$) and 2D simulations, the study identifies parameter regions where stripes, hexagons, and mixed patterns emerge, including threshold conditions like $b_c=(1+ a \sqrt{d D_1/D_2})^2$ and $k_c^2 = a \sqrt{d}/\sqrt{D_2}$. The findings extend known 1D results to 2D, revealing a richer palette of spatial structures and clarifying how microscopic collision mechanisms govern macroscopic pattern formation.

Abstract

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.

Figures (10)

• Figure 1: Values for energy levels $E_2$ and $E_Z$ relevant to Turing instability for system \ref{['brusselator']}. In region I, condition \ref{['constraint']} is not satisfied, in regions II, III, and IV \ref{['constraint']} holds, and either both conditions \ref{['TurBrussMac']} and \ref{['TurBrussMac2']}, only \ref{['TurBrussMac2']}, or neither of the conditions in \ref{['TurBrussMac']} and \ref{['TurBrussMac2']} are satisfied. Parameters are chosen as in \ref{['ParsBrus']}.
• Figure 2: Values for energy levels $E_2$ and $E_Z$ relevant to the stability of the different patterns system \ref{['brusselator']}, as reported in Table \ref{['tab1']}. Black lines define the region where Turing instability can occur. Parameters are chosen as in \ref{['ParsBrus']}.
• Figure 3: Pattern formation for system \ref{['brusselator']} in a squared domain $\Omega$ assuming no-flux conditions at the boundary $\partial\Omega$, and taking coefficient as reported in the first row of Table \ref{['tab2']}. Panel (a): density of $n_1$. Panel (b): density of $n_2$.
• Figure 4: Pattern formation for system \ref{['brusselator']} in a squared domain $\Omega$ assuming no-flux conditions at the boundary $\partial\Omega$, and taking coefficient as reported in the second row of Table \ref{['tab2']}. Panel (a): density of $n_1$. Panel (b): density of $n_2$.
• Figure 5: Pattern formation for system \ref{['brusselator']} in a squared domain $\Omega$ assuming no-flux conditions at the boundary $\partial\Omega$, and taking coefficient as reported in the third row of Table \ref{['tab2']}. Panel (a): density of $n_1$. Panel (b): density of $n_2$.