Pattern formation controlled by time-delayed feedback in bistable media

Influence of temperature on Turing pattern formation

The Turing instability is one of the most commonly studied mechanisms leading to pattern formation in reaction–diffusion systems, yet there are still many open questions on the applicability of the Turing mechanism. Although experiments on pattern formation using chemical systems have shown that temperature differences play a role in pattern formation, there is far less theoretical work concerning the interplay between temperature and spatial instabilities. We consider a thermodynamically extended reaction–diffusion system, consisting of a pair of reaction–diffusion equations coupled to an energy equation for temperature, and use this to obtain a natural extension of the Turing instability accounting for temperature. We show that thermal contributions can restrict or enlarge the set of unstable modes possible under the instability, and in some cases may be used to completely shift the set of unstable modes, strongly modifying emergent Turing patterns. Spatial heterogeneity plays a role under several experimentally feasible configurations, and we give particular consideration to scenarios involving thermal gradients, thermodynamics of chemicals transported within a flow, and thermodiffusion. Control of Turing patterns is also an area of active interest, and we also demonstrate how patterns can be modified using time-dependent control of the boundary temperature.

Pattern dynamics of a diffusive predator–prey model with delay effect

We study the spatiotemporal dynamics in a diffusive predator–prey system with time delay. By investigating the dynamical behavior of the system in the presence of Turing–Hopf bifurcations, we present a classification of the pattern dynamics based on the dispersion relation for the two unstable modes. More specifically, we researched the existence of the Turing pattern when control parameters lie in the Turing space. Particularly, when parameter values are taken in Turing–Hopf domain, we numerically investigate the formation of all the possible patterns, including time-dependent wave pattern, persistent short-term competing dynamics and stationary Turing pattern. Furthermore, the effect of time delay on the formation of spatial pattern has also been analyzed from the aspects of theory and numerical simulation. We speculate that the interaction of spatial and temporal instabilities in the reaction–diffusion system might bring some insight to the finding of patterns in spatial predator–prey models.

Delay-induced instability in a nutrient-phytoplankton system with flow

In this paper, a nutrient-phytoplankton system described by a couple of advection-diffusion-reaction equations with delay was studied analytically and numerically. The aim of this research was to provide an understanding of the impact of delay on instability. Significantly, delay cannot only induce instability, but can also promote the formation of spatial pattern via a Turing-like instability. In addition, the theoretical analysis indicates that the flow (advection term) may lead to instability when the delay term exists. By comparison, diffusion cannot result in Turing instability when flow does not exist. Results of numerical simulation were consistent with the analytical results.

Interaction of multiarmed spirals in bistable media

We study the interaction of both dense and sparse multiarmed spirals in bistable media modeled by equations of the FitzHugh-Nagumo type. A dense one-armed spiral is characterized by its fixed tip. For dense multiarmed spirals, when the initial distance between tips is less than a critical value, the arms collide, connect, and disconnect continuously as the spirals rotate. The continuous reconstruction between the front and the back drives the tips to corotate along a rough circle and to meander zigzaggedly. The rotation frequency of tip, the frequency of zigzagged displacement, the frequency of spiral, the oscillation frequency of media, and the number of arms satisfy certain relations as long as the control parameters of the model are fixed. When the initial distance between tips is larger than the critical value, the behaviors of individual arms within either dense or sparse multiarmed spirals are identical to that of corresponding one-armed spirals.