Simple PDE Model of Spot Replication in Any Dimension

We propose a simple PDE model which exhibits self-replication of spot solutions in any dimension. This model is analysed in one and higher dimensions. In the radially symmetric case, we rigorously demonstrate that a weakened version of the conditions proposed by Nishiura and Ueyama for self-replication are satisfied. In dimension three, two different types of replication mechanisms are analysed. The first type is due to radially symmetric instability, whereby a spot bifurcates into a ring. The second type is non-radial instability, which causes a spot to deform into a peanut-like shape, and eventually split into two spots. Both types of replication are observed in our model, depending on parameter choice. Numerical simulations are shown confirming our analytical results.