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I have a two species reaction-diffusion system which is a Turing-type (activator-inhibitor) equation. I am trying to transform my reaction-diffusion system into a system of multiple walkers on a lattice with interactions or with differing jump probabilities.

These are the type of my equations:

An example of an activator-inhibitor system of equations

I did an extensive search and I found this paper really relevant: https://arxiv.org/pdf/1211.6494.pdf

I am wondering if anyone here knows any relevant paper or a better way to transform a reaction-diffusion problem into a system of random walkers.

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  • $\begingroup$ Could you please specify your initial and/or boundary conditions? Also, what guarantees the existence of a non-exploding solution to these PDEs? I am a bit concerned by the $1/h$ nonlinearity. $\endgroup$
    – Nawaf Bou-Rabee
    Commented Dec 2, 2016 at 13:21
  • $\begingroup$ It is not quite clear what kind of lattice you are thinking of. Do you want to discretize your space domain? Or are you thinking of a different approximation? There is a paper by Bobrowski (link.springer.com/article/10.1007/s00023-012-0158-z) showing that under suitable b.c. and if $p=p'=0$ (so the PDEs are linear and uncoupled - but they are allowed to be coupled at the boundary), the solutions converge to a space-discrete dynamical system as $D_a\to \infty$ and $D_h\to \infty$. It could be worth a try to extend this result to your setting. $\endgroup$
    – Delio Mugnolo
    Commented Dec 6, 2016 at 10:07

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