Related papers: Parametric pattern selection in a reaction-diffusi…
The amplitude equation of Gierer-Mainhardt model has been actually derived near the boundary abuot which Turing and Hopf modes exist. In a parameter region where Hopf-Turing mixed mode solution is stable, a chaotic state that generally…
The theory of pattern formation in reaction-diffusion systems is extended to the case of a directed network. Due to the structure of the network Laplacian of the scrutinised system, the dispersion relation has both real and imaginary parts,…
The modelling of linear and nonlinear reaction-subdiffusion processes is more subtle than normal diffusion and causes different phenomena. The resulting equations feature a spatial Laplacian with a temporal memory term through a time…
The phenomenon of pattern formation in nonlinear optical resonators is commonly related to an off-resonance excitation mechanism, where patterns occur due to mismatch between the excitation and resonance frequency. In this paper we show…
We consider the classical Turing instability in a reaction-diffusion system as the secend part of our study on pattern formation. We prove that nonlinear dynamics of a general perturbation of the Turing instability is determined by the…
Cross-diffusion systems play a central role in mathematical modelling, in which density-dependent dispersal and multiscale mechanisms can lead to spatial segregation and diffusion-driven instabilities. In several relevant examples,…
Coupling diffusion process of signaling molecules with nonlinear interactions of intracellular processes and cellular growth/transformation leads to a system of reaction-diffusion equations coupled with ordinary differential equations…
Motivated by numerical simulations showing the emergence of either periodic or irregular patterns, we explore a mechanism of pattern formation arising in the processes described by a system of a single reaction-diffusion equation coupled…
Symmetry-breaking instabilities play an important role in understanding the mechanisms underlying the diversity of patterns observed in nature, such as in Turing's reaction--diffusion theory, which connects cellular signalling and transport…
Patterns of different symmetries may arise after solution to reaction-diffusion equations. Hexagonal arrays, layers and their perturbations are observed in different models after numerical solution to the corresponding initial-boundary…
Pattern formation, arising from systems of autonomous reaction-diffusion equations, on networks has become a common topic of study in the scientific literature. In this work we focus primarily on directed networks. Although some work prior…
The formation of localized structures in the chlorine dioxide-idodine-malonic acid (CDIMA) reaction-diffusion system is investigated numerically using a realistic model of this system. We analyze the one-dimensional patterns formed along…
This paper addresses the question of how population diffusion affects the formation of the spatial patterns in the spatial epidemic model by Turing mechanisms. In particular, we present theoretical analysis to results of the numerical…
Turing patterns on unbounded domains have been widely studied in systems of reaction-diffusion equations. However, up to now, they have not been studied for systems of conservation laws. Here, we (i) derive conditions for Turing instability…
We consider pattern formation using reaction-diffusion equation on various non-uniformly curved surfaces. We explore how, in general, curvature and, in particular the domain shape would affect the pattern formation in these geometries. As…
Drifting pattern domains (DPDs), moving localized patches of traveling waves embedded in a stationary (Turing) pattern background and vice versa, are observed in simulations of a reaction-diffusion model with nonlocal coupling. Within this…
Turing (or double-diffusive) instabilities describe pattern formation in reaction-diffusion systems, and were proposed in 1952 as a potential mechanism behind pattern formation in nature, such as leopard spots and zebra stripes. Because the…
In certain biological contexts, such as the plumage patterns of birds and stripes on certain species of fishes, pattern formation takes place behind a so-called "wave of competency". Currently, the effects of a wave of competency on the…
We study the mechanisms of pattern formation for vegetation dynamics in water-limited regions. Our analysis is based on a set of two partial differential equations (PDEs) of reaction-diffusion type for the biomass and water and one ordinary…
In this paper we study pattern formation arising in a system of a single reaction-diffusion equation coupled with subsystem of ordinary differential equations, describing spatially-distributed growth of clonal populations of precancerous…