Simple PDE Model of Spot Replication in Any Dimension

Abstract

Abstract. 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 one dimension, we rigorously demonstrate that 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.

Pulse replication and accumulation of eigenvalues

“…Early work [32] identified the role of (1) saddle-node bifurcations of standing single and multi-pulses and (2) solutions that connect single to double pulses at these saddle-node bifurcation points as the building blocks that enable pulse replication in, for instance, the Gray-Scott model. While, to our knowledge, the existence of the connecting solution has been established only numerically, several papers [12,15,16,26,32,36] focused on proving the existence of saddle-node bifurcations and instability mechanisms of standing or slowly moving pulses in a range of models.…”

Moving and jumping spot in a two-dimensional reaction–diffusion model

“…All other eigenvalues are referred to as large eigenvalues. The mode m = 0 corresponds to purely radial perturbations and its instability can induce spike oscillation or collapse, whereas the instability with respect to mode m = 2 eigenvalues triggers self-replication [10,13,16,25]. Here, we only concentrate on modes m = 0, 1 since τ does not appear to trigger instability of the higher modes.…”