Practical Problem 14: Reaction Diffusion Systems

The goal of this set of practice problems is solving reaction diffusion systems, so systems in which

• particles of different types have spatially variable concentrations
• particles of different types interact
• and they diffuse in space

Let's say we have two types of particles and two concentration $u=u(x,t)$ and $w=w(x,t)$ where $x$ is the position and $t$ denotes time, in this case the dynamical equations are defines as

[ \partial _t u = f (u,w) + D_u \partial_x^2 u ]

[ \partial _t w = g (u,w) + D_w \partial_x^2 w ]

The constants $D_u$ and $D_w$ are the diffusion constants and quantify how strong diffusion is for the different types of particles.

Preparations

Reaction diffusion systems are patch-only models. For all the systems you should define, for each particle concentration a patches-ownvariable. So say you have three particle concentrations $u$, $v$ and $w$ you should have patches-own [u v w du dv dw]. The reason for this is that you need to compute the differentials by looping of the patches and then use them later in a second loop to update the states of the patches.

Set up you world to be $64x64$ patches, so things look smooth.

Simulation 1 : Fisher equation

This is a simulation of a simple propagating wave:

[ \partial _t u = \lambda u (1-u) + D_u \partial_x^2 u ]

Set up the concentration $u$ such that it is zero everywhere, except for in the center of the world, where you can set it to some small positive value less than 1. The growth parameter $\lambda>0$ and the diffusion constant $0<D_u<1$ should be slider parameters. Likewise you need a $dt$ parameter which will the time increment. This should be have a small value like 0.01 or so.

For the simulation you need three loops, one that computes the differential of the reaction part

ask patches [
     set du  dt *  * lambda * u * (1 - u)
]

then you need to update the state

ask patches [
     set u u + du
]

and finally you need to execute the diffusion:

diffuse u D

The diffuse command has two arguments, one is the patch variable to diffuse, the other the diffusion constant.

Make sure you write a paint function that colors the patches according to the concentration of u. You can use the scale-color command.

Simulation 2: Competition

Let's now turn to a system in which three diffusing species compete, governed by the system

[ \partial _t u = f (u,v,w) + D_u \partial_x^2 u ]

[ \partial _t v = f (v,w,u) + D_v \partial_x^2 v ]

[ \partial _t w = f (w,u,v) + D_w \partial_x^2 w ]

where the function $f=f(u,v,w)=u (1-u - R v - R w)$ and the parameter $R$ quantifies the competition and should be a slider.

Set up the system such that each concentration is random for each patch somewhere between 0 and 1.

You can color the concentrations according to this paint function:

to paint
   ask patches [set pcolor rgb (255 * u) (255 * v) (255 * w)]
end

Write a go function that computes the differentials for u, v, and w. updates the states and diffuses each concentration.

Simulation 3: Spiral Waves

The deterministic equations for the rock paper scissors game have the shape [ \dot u = f(u,v,w)\quad \dot v = g(u,v,w)\quad \dot w = h(u,v,w) ] as in the problem in the test. Here we now consider a variant that looks like this [ \dot u = u (\sigma (v-w) + u - u^2 -\mu (v+w) ) ] [ \dot v = v (\sigma (w-u) + v - v^2 -\mu (w+u) ) ] [ \dot w = w (\sigma (u-v) + w - w^2 -\mu (u+v) ) ]

The first term is the RPC interaction (the impact governed by parameter $\sigma$), the second and third term is normal logistic growth and the last term is competition (quantified by parameter $\mu$).

Now implement this system and include the diffusion of the three types of reactants.