Emergence of Turing Patterns in a Simple Cellular Automata-Like Model via Exchange of Integer Values between Adjacent Cells

The Turing pattern model is one of the theories used to describe organism formation patterns. Using this model, self-organized patterns emerge due to differences in the concentrations of activators and inhibitors. Here a cellular automata (CA)-like model was constructed wherein the Turing patterns emerged via the exchange of integer values between adjacent cells. In this simple hexagonal grid model, each cell state changed according to information exchanged from the six adjacent cells. The distinguishing characteristic of this model is that it presents a different pattern formation mechanism using only one kind of token, such as a chemical agent that ages via spatial diffusion. Using this CA-like model, various Turing-like patterns (spots or stripes) emerge when changing two of four parameters. This model has the ability to support Turing instability that propagates in the neighborhood space; global patterns are observed to spread from locally limited patterns. This model is not a substitute for a conventional Turing model but rather is a simplified Turing model. Using this model, it is possible to control the formation of multiple robots into such forms as circle groups or dividing a circle group into two groups, for example. In the field of information networks, the presented model could be applied to groups of Internet-of-Things devices to create macroscopic spatial structures to control data traffic.

Studies on the Existence of Unstable Oscillatory Patterns Bifurcating from Hopf Bifurcations in a Turing Model

We revisit a homogeneous reaction-diffusion Turing model subject to the Neumann boundary conditions in the one-dimensional spatial domain. With the help of the Hopf bifurcation theory applicable to the reaction-diffusion equations, we are capable of proving the existence of Hopf bifurcations, which suggests the existence of spatially homogeneous and nonhomogeneous periodic solutions of this particular system. In particular, we also prove that the spatial homogeneous periodic solutions bifurcating from the smallest Hopf bifurcation point of the system are always unstable. This together with the instability results of the spatially nonhomogeneous periodic solutions by Yi et al., 2009, indicates that, in this model, all the oscillatory patterns from Hopf bifurcations are unstable.