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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 diffusion-driven Turing instability is a potential mechanism for spatial pattern formation in numerous biological and chemical systems. However, engineering these patterns and demonstrating that they are produced by this mechanism is…
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…
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 population model of Busenberg and Travis is a paradigmatic model in ecology and tumour modelling due to its ability to capture interesting phenomena like the segregation of populations. Its singular mathematical structure enforces the…
We investigate dynamics near Turing patterns in reaction-diffusion systems posed on the real line. Linear analysis predicts diffusive decay of small perturbations. We construct a "normal form" coordinate system near such Turing patterns…
In this letter we propose a Turing model of the formation of patterns of visible light emission intensity in atmospheric pressure gas discharges. The electron density and the electron temperature take the roles of activator and inhibitor…
We develop a general classification of the nature of the instabilities yielding spatial organization in open nonideal reaction-diffusion systems, based on linear stability analysis. This encompasses dynamics where chemical species diffuse,…
Nonlinear instabilities are responsible for spontaneous pattern formation in a vast number of natural and engineered systems ranging from biology to galaxies build-up. We propose a new instability mechanism leading to pattern formation in…
In this work, the influence of geometry and domain size on spatiotemporal pattern formation is investigated to establish parameter spaces for a cross-diffusive reaction-diffusion model on an annulus. By applying linear stability theory, we…
One of the classical models in mathematical biology is the Lotka-Volterra competition model, describing the dynamics of two populations competing for resources. Two possible regimes in this system are given by their coexistence or…
Reaction-diffusion (Turing) systems are fundamental to the formation of spatial patterns in nature and engineering. These systems are governed by a set of non-linear partial differential equations containing parameters that determine the…
We derive a necessary and sufficient condition for Turing instabilities to occur in two-component systems of reaction-diffusion equations with Neumann boundary conditions. We apply this condition to reaction-diffusion systems built from…
A systematic introduction to nonequilibrium thermodynamics of dynamical instabilities is considered for an open nonlinear system beyond conventional Turing pattern in presence of cross diffusion. An altered condition of Turing instability…
The existence and stability of localized patterns of criminal activity are studied for the reaction-diffusion model of urban crime that was introduced by Short et. al. [Math. Models. Meth. Appl. Sci., 18, Suppl. (2008), pp. 1249--1267].…
A general system of n ordinary differential equations coupled with one reaction-diffusion equation, considered in a bounded N-dimensional domain, with no-flux boundary condition is studied in a context of pattern formation. Such initial…
As proposed by Alan Turing in 1952 as a ubiquitous mechanism for nonequilibrium pattern formation, diffusional effects may destabilize uniform distributions of reacting chemical species and lead to both spatially and temporally…
Systems consisting of a single ordinary differential equation coupled with one reaction-diffusion equation in a bounded domain and with the Neumann boundary conditions are studied in the case of particular nonlinearities from the…
We study the onset of spatial instabilities in reaction networks where the spatially homogeneous system admits a steady state parameterization. We formulate a sufficient condition -- based on the signs of the constant and leading…
Mass-conserving reaction-diffusion systems with bistable nonlinearity are useful models for studying cell polarity formation, which is a key process in cell division and differentiation. We rigorously show the existence and stability of…