EXPLORABLES

by Dirk Brockmann , Benjamin F. Maier

With this explorable you can discover a variety of spatio-temporal patterns that can be generated with a very famous and simple autocatalytic reaction diffusion system known as the Gray-Scott model. In the model two substances \(U\) and \(V\) interact and diffuse in a two-dimensional container. Although only two types of simple reactions occur, the system generates a wealth of different stable and dynamic spatio-temporal patterns depending on system parameters.

Press play and observe different patters by selecting preset parameter values for the supply rate of \(U\) (parameter \(F\)) and the decay rate of \(V\) (parameter \(k\)). The panel depicts the concentration \(u(\mathbf{x},t)\) of substance \(U\) as a function of position \(\mathbf{x}\) and time \(t\).

Keep on reading to learn a bit more about what's going on.

This is how it works

Two reactions take place in the system. First, when a single \(U\)-particle encounters two \(V\) particles it is converted into \(V\) itself:

\[ U+2V\rightarrow 3V \]

increasing the amount of \(V\) in the system (and decreasing the amount of \(U\)). Also, the amount of substance \(V\) is decreased at rate \(k\) by spontaneous decay into some inert substance \(P\):

\[ V\xrightarrow{k} P \]

In addition, substance \(U\) is uniformly supplied at a constant rate \(F\). \(U\)-, \(V\)-, and \(P\)-particles are removed at rate \(F\), proportional to their concentration, keeping the total density of particles fixed. If we denote the spatial concentrations of \(U\) and \(V\) by \(u(\mathbf{x},t)\) and \(v(\mathbf{x},t)\), the dynamics is governed by the equations:

\[ \partial_t{u}=-uv^2+F(1-u)+ D_u \nabla^2 u \]

\[ \partial_t{v}=uv^2-(F+k)v + D_v \nabla^2 v \]

The last terms account for the spatial diffusion of \(U\) and \(V\) particles, the constants \(D _{u,v}\) denote the diffusion constants of sustances \(U,V\) and \(\nabla^2=\partial_x^2+\partial_y^2\) is the Laplacian.

The solutions to these equations are shown in the display panel when you press play. Initially the system is set up with randomly placed reactangles of different values for \(u\) and \(v\). With the tangent and normal sliders you can navigate to different parameter values of \(F\) and \(k\) in the dynamically most interesting region (red sausage).

Sometimes the system will go into a uniform state, in this case you can press the reset button (triangle pointing to the left).

Stationary states, Instabilities & Bifurcations

When looking at the possible stationary states, we see that when \(v=0,u=1\) everywhere we have \(\dot{u}=\dot{v}=0\) and nothing changes. This uniform state is stationary. This makes sense, because if substance \(V\) is absent it cannot be made. This solution always exists and is always stable with respect to small perturbations.

We can also imagine a spatially uniform state in which the supply of \(U\), the autocatalysis of \(V\) and the decay of \(V\) balance, so both \(U\) and \(V\) balance at some nonzero concentration.

In fact, if one does the math, one can show that to the left of the solid black line in the parameter space shown in the control panel, three stationary states exist including the trivial one. Above the dashed line, one of the two is stable and the other unstable. When crossing the dashed line from above the non-trivial stable state also loses stability by something called a Hopf bifurcation.

When we look beyond spatially homogeneous solutions, additional interesting things happen. Even in regions that would exhibit stable stationary states in a well-mixed system, the diffusion in the system can destabilize the uniform state by the Turing mechanism and spatial structure spontaneously emerges.

The complexity of the patterns of the Gray-Scott model emerges because in a narrow range in parameter space a Hopf bifurcation, a saddle node bifurcation and Turing instabilities are entangled.

Try this

You can try to discover new patterns by starting at a preset pattern and gently move the sliders. Often, very different patterns can exist in close proximity in parameter space.

Further information & References


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