Stable localized moving patterns in the 2-D Gray-Scott model

Abstract

I show stable, localized, single and multi-spot patterns of three classes -stationary, moving, and rotatingthat exist within a limited range of parameter values in the two-dimensional Gray-Scott reaction-diffusion model with σ = Du/Dv = 2. These patterns exist in domains of any size, and appear to derive their stability from a constructive reinforcement effect of the standing waves that surround any feature. There are several common elements -including a spot that behaves as a quasiparticle, a U-shaped stripe, and a ring or annulus, or a portion thereof -which combine to form a great variety of stable structures. These patterns interact with each other in a variety of ways. There are similarities to other reaction-diffusion systems and to physical experiments; I offer several suggestions for further research.

Exploring complex pattern formation with convolutional neural networks

“…Our training data includes only 15 unique pairs of f and k parameter values given for each class. 4,16 We now want to use CNN to scan a much larger and denser parameter space of f ∈ [0.02, 0.08] and k ∈ [0, 0.12]. This part of the parameter space is selected based on the existence criterion for stable solutions in Eq.…”

“…We initially classify patterns whose existence and lack of sensitivity to the initial conditions are well described in the literature. 4,5,16,22 Additional pattern types have been identified in Ref. 16 for slightly more complicated initial conditions.…”

“…4,5,16,22 Additional pattern types have been identified in Ref. 16 for slightly more complicated initial conditions. The known classes have been mostly identified visually and their descriptions are often only semantic, making it difficult to determine where certain types of patterns are found in the parameter space, because a large number of values of the parameters need to be investigated.…”

“…4 Twelve heterogenous steady and time-dependent solution classes were identified initially and further investigations revealed more possible classes. 16 Figure We can solve Eqs. (1) and (2) numerically using the forward time centered space method.…”

“…An alternative way to classify images is to use a CNN to fit the entire dataset to prelabeled data. 11 Because we already have a classification of patterns with corresponding parameters, 4,16 we can use this knowledge to train a neural network to classify patterns automatically. In this section we will introduce the basic terminology.…”