Gray Scott
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The Gray Scott model is a classic reaction-diffusion model exhibiting pattern formation, popularized in the articles `Complex Patterns in a Simple System <https://www.science.org/doi/10.1126/science.261.5118.189>`_ and `Pattern Formation by Interacting Chemical Fronts <https://www.science.org/doi/10.1126/science.261.5118.192>`_. The model consists of two species :math:`u` and :math:`v` that react and diffuse in a single compartment. :math:`k1`, :math:`k` and :math:`f` are reaction parameters. This system a good starting point to explore the effect of the parameterization on nonlinear reaction-diffusion systems.
This model comes in a 2D and a 3D version. For a related system with different reaction terms and membrane fluxes, see `Fitzhugh-Nagumo model <fitzhughnagumo.html>`_.

Formulation
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   .. math::

      &\frac{\partial u}{\partial t} = D_{u} \nabla^2 u - k_{1} u v^{2} + f \left(1-u \right)

      &\frac{\partial v}{\partial t} = D_{v} \nabla^2 v + k_{1} u v^{2} - \left( f + k \right) v

Example Snapshot
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.. figure:: img/grayscott.png
   :alt: screenshot of the final step of the Grey-Scott 2D example model

   Screenshot of the result of running the Gray-Scott example model in 2D. Observe the formation of patterned concentration structures.