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I'm using following code to generate a picture of the reaction-diffusion process:

Explicit Euler method too slow for reaction-diffusion problem

I would like to obtain an animation effect. When I draw the solution step by step, it does give good animation. The problem is that the solution quickly converges to uniform, solid color (i.e. uniform concentration of chemical species).

How to sustain the animation? The solution should reverse the Euler method to some degree? Adding random noise sustains the animation, but also adds visible noise

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If I remember correctly from the original question, the main purpose is to have an "interesting" visual output. While this somewhat lines up with your noise comment, likely the best way to maintain a transient solution is to have time dependent boundary conditions. As time goes on and your animation stabilizes, choose a random point on the edge of the image and impose a non-equilibrium concentration (how much effort you want to put into making it appear smooth the current solution is up to you). This will cause your solution to remain transient while avoiding the visible noise you mentioned.

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Turing instability only happens in specific parameter ranges. If the solution goes from something randomish to uniform, then that means that you're parameter choices give a stable uniform steady state. This means that to getting Turing patterning you need different parameters.

An example to look at can be found here where the authors determine parameter ranges for Turing instability and show that inside the range you getting patterning whereas outside you get uniform steady states.

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