Superlattice patterns

Coupling two systems that each generate spatial patterns can lead to what are referred to as “superlattice patterns.” Here we showcase a published example with nonidentical kinetics. This model combines a Brusselator and a Lengyll-Epstein reaction-diffusion model, with a nonlinear coupling between the two subsystems. The full system is given by:

\[\begin{aligned} \pd{u_{1}}{t} &= D_{u_{1}} \nabla^2 u_{1} + a-\left(b+1\right) u_{1}+{u_{1}}^{2} v_{1}+\alpha u_{1} u_{2} \left(u_{2}-u_{1}\right),\\ \pd{v_{1}}{t} &= D_{u_{2}} \nabla^2 v_{1} + b u_{1}-{u_{1}}^{2} v_{1},\\ \pd{u_{2}}{t} &= D_{u_{3}} \nabla^2 u_{2} + c - u_{2} -4 u_{2} v_{2}/\left(1+{u_{2}}^{2}\right)+\alpha u_{1} u_{2} \left(u_{1}-u_{2}\right),\\ \pd{v_{2}}{t} &= D_{u_{4}} \nabla^2 v_{2} + d \left[u_{2} - u_{2} v_{2}/\left(1+{u_{2}}^{2}\right)\right], \end{aligned}\]

with $\alpha$ denoting the coupling parameter. In the simulations below, we vary only $\alpha$ and the four diffusion coefficients to observe a range of different behaviours.

If you want to explore more, check out this full-page simulation.

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