Pattern formation in a two-component reaction–diffusion system with delayed processes on a network

Abstract

Reaction-diffusion systems with time-delay defined on complex networks have been studied in the framework of the emergence of Turing instabilities. The use of the Lambert W -function allowed us get explicit analytic conditions for the onset of patterns as a function of the main involved parameters, the time-delay, the network topology and the diffusion coefficients. Depending on these parameters, the analysis predicts whether the system will evolve towards a stationary Turing pattern or rather to a wave pattern associated to a Hopf bifurcation. The possible outcomes of the linear analysis overcome the respective limitations of the single-species case with delay, and that of the classical activator-inhibitor variant without delay. Numerical results gained from the Mimura-Murray model support the theoretical approach.

Robust stochastic Turing patterns in the development of a one-dimensional cyanobacterial organism

“…In [ 65 ], typical Turing models are reported to lose robustness with the inclusion of delays. On other occasions, however, time delay can facilitate the onset of the instability [ 76 , 77 ]. Gauging the impact of the delay for the problem at hand is left for future analysis.…”

Turing patterns in a network-reduced FitzHugh-Nagumo model

“…[12,13], enabling to better capture geometries present in many real cases. Since then, a number of generalisations to various types of networks and dynamics has been made, including asymmetric, multiplex, time-varying, time-delayed, and non-normal networks [14][15][16][17][18][19], and intriguing relations between self-organisation and network topology have been revealed.…”

Stability and Dissipation of Many Components’ LC Systems and Application to Metamaterials

“…It is necessary that obtained results may be used for timedelayed systems and experiments [15,16] and for analysis of stability of signal of composite many components' metamaterials.…”