Here we look at a model of vegetation patterning known as the Klausmeier model, written in terms of water $w$ and plant biomass $n$.

\[\begin{aligned}\pd{w}{t} &= a-w -wn^2+v\pd{w}{x} + \nabla^2w,\\ \pd{n}{t} &= wn^2 - mn + \nabla^2n\end{aligned}\]
  • Load the interactive simulation. Note that this model always has a plant-extinction state being stable.

  • The simulation starts in a parameter regime where pure stripes (regular patterns) of plants are favoured over time. If you instead set $a=0.4$, $m=0.4$, the environment will be much harsher, leading to less regular patterns. However, this also makes the extinction state more stable, so you may have to brush sufficient initial data to overcome this Allee-like effect.

Localised solutions

Similarly to the localised patterns in Swift–Hohenberg, Dan Hill has provided examples of dihedrally symmetric localised patterns in this model.

You can find initial conditions that simulate such patterns in this localised simulation, and more information about the rigorous theory underlying them in his 2024 paper.