# Multiscale Simulation of random walker

I want to simulate a system of random walkers (called A) with diffusion coefficient equal to D and other systems of random walkers (called type B) with diffusion coefficient equal to 1000 D. Second type of walkers are smaller than the first ones and are eaten by the first type. As diffusion coefficient of B walkers are much bigger than the first type, I don't know how to choose the time step and space step of the simulation. Could anyone help? The small walkers perform a discrete random walk on a lattice and the big ones are described by a Wiener process. Also, big walkers move to the direction of the gradient of small walkers and they have a velocity which pushes them in a preferred direction of their body

• Yes, exactly. other useful information is that the rate of eating is k f , which k is a constant and f concentration of small walkers. Also, big walkers move to the direction of the gradient of small walkers and they have a velocity which pushes them in a preferred direction of their body@Wrzlprmft Is it clear now? Could you help please? Commented May 17, 2017 at 18:06

In general, the only harm a small step size causes is that it costs runtime. Therefore the step size imposed by your small random walkers is a reasonable choice. The only reason why it wouldn’t be is that you need an even smaller step size, but I can only see that happening if:

• The velocity of your big walkers is not much smaller than the velocity of your small walkers.
• The concentration of your small walkers is inhomogeneous on scales of the movement of a big walker during one step (which should not happen unless it is imposed as an initial condition).
• The eating rate of your big walkers is so large that they eat a relevant portion of small walkers during one time step.

So, with other words, you should ensure that no considerable changes happen during one time step.

• Why do you say "which should not happen unless it is imposed as an initial condition"? Big walkers eat small walkers in each time step, so the concentration of small walkers becomes inhomogeneous. Commented May 17, 2017 at 19:11
• By the way, is there any way to choose a larger time scale, so that the costs runtime becomes less? Commented May 17, 2017 at 19:12
• Why do you say "which should not happen unless it is imposed as an initial condition"? Big walkers eat small walkers in each time step, so the concentration of small walkers becomes inhomogeneous. – But your eating should not create an overly fine structure, in particular given the high diffusion of the small walkers. Anyway, the crucial thing is that your big walker does not step over a considerable change of small-walker concentration. Commented May 18, 2017 at 8:50
• is there any way to choose a larger time scale, so that the costs runtime becomes less? – From all I know, I would guess that you can model the concentration of small walkers directly (instead of via actual walkers). If that is the case, go for it. Commented May 18, 2017 at 8:52
• @user24280: Right now, your model variables are a bunch of positions of small walkers. Instead make your model variables the concentration of food at discrete points in your system and describe their diffusion by some partial differential equation. With other words: Do not model individual small walkers but their concentration. Commented May 18, 2017 at 11:01