Instability and pattern formation in reaction-diffusion systems: A higher order analysis

The Turing (wave) instability is only possible in reaction-diffusion systems with more than one (two) components. Motivated by the fact that a time delay increases the dimension of a system, we investigate the presence of diffusion-driven instabilities in single-species reaction-diffusion systems with delay. The stability of arbitrary one-component systems with a single discrete delay, with distributed delay, or with a variable delay is systematically analyzed. We show that a wave instability can appear from an equilibrium of single-species reaction-diffusion systems with fluctuating or distributed delay, which is not possible in similar systems with constant discrete delay or without delay. More precisely, we show by basic analytic arguments and by numerical simulations that fast asymmetric delay fluctuations or asymmetrically distributed delays can lead to wave instabilities in these systems. Examples, for the resulting traveling waves are shown for a Fisher-KPP equation with distributed delay in the reaction term. In addition, we have studied diffusion-induced instabilities from homogeneous periodic orbits in the same systems with variable delay, where the homogeneous periodic orbits are attracting resonant periodic solutions of the system without diffusion, i.e., periodic orbits of the Hutchinson equation with time-varying delay. If diffusion is introduced, standing waves can emerge whose temporal period is equal to the period of the variable delay.

Reaction-diffusion systems involving ionic species are susceptible to an externally applied electric field. Depending on the charges on the ionic species and the intensity of the applied electric field, diverse spatiotemporal patterns can emerge. We here considered two prototypical reaction-diffusion systems that follow activator-inhibitor kinetics: the photosensitive chlorine dioxide-iodine-malonic acid (CDIMA) reaction and the Brusselator model. By theoretical investigation and numerical simulations, we unravel how and to what extent an externally applied electric field can induce and modify the dynamics of these two systems. Our results show that both the uni- and bi-directional electric fields may induce Turing-like stationary patterns from a homogeneous uniform state resulting in horizontal, vertical, or bent stripe-like inhomogeneity in the photosensitive CDIMA system. In contrast, in the Brusselator model, for the activator and the inhibitor species having the same positive or negative charges, the externally applied electric field cannot develop any spatiotemporal instability when the diffusion coefficients are identical. However, various spatiotemporal patterns emerge for the same opposite charges of the interacting species, including moving spots and stripe-like structures, and a phenomenon of wave-splitting is observed. Moreover, the same signand different magnitudes of the ionic charges can give rise to Turing-like stationary patterns from a homogeneous, stable, steady state depending upon the intensity of the applied electric field in the case of the Brusselator model. Our findings open the possibilities for future experiments to verify the predictions of electric field-induced various spatiotemporal instabilities in experimental reaction-diffusion systems.

Time delays, modelling the process of intracellular gene expression, have been shown to have important impacts on the dynamics of pattern formation in reaction–diffusion systems. In particular, past work has shown that such time delays can shrink the Turing space, thereby inhibiting patterns from forming across large ranges of parameters. Such delays can also increase the time taken for pattern formation even when Turing instabilities occur. Here, we consider reaction–diffusion models incorporating fixed or distributed time delays, modelling the underlying stochastic nature of gene expression dynamics, and analyse these through a systematic linear instability analysis and numerical simulations for several sets of different reaction kinetics. We find that even complicated distribution kernels (skewed Gaussian probability density functions) have little impact on the reaction–diffusion dynamics compared to fixed delays with the same mean delay. We show that the location of the delay terms in the model can lead to changes in the size of the Turing space (increasing or decreasing) as the mean time delay, tau , is increased. We show that the time to pattern formation from a perturbation of the homogeneous steady state scales linearly with tau , and conjecture that this is a general impact of time delay on reaction–diffusion dynamics, independent of the form of the kinetics or location of the delayed terms. Finally, we show that while initial and boundary conditions can influence these dynamics, particularly the time-to-pattern, the effects of delay appear robust under variations of initial and boundary data. Overall, our results help clarify the role of gene expression time delays in reaction–diffusion patterning, and suggest clear directions for further work in studying more realistic models of pattern formation.

Reaction-Diffusion (RD) systems provide a computational framework that governs many pattern formation processes in nature. Current RD system design practices boil down to trial-and-error parameter search. We propose a differentiable optimization method for learning the RD system parameters to perform example-based texture synthesis on a 2D plane. We do this by representing the RD system as a variant of Neural Cellular Automata and using task-specific differentiable loss functions. RD systems generated by our method exhibit robust, non-trivial 'life-like' behavior.

Mathematical modeling of different types of instabilities in time fractional reaction-diffusion systems

Reaction–diffusion processes across layered media arise in several scientific domains such as pattern-forming E. coli on agar substrates, epidermal–mesenchymal coupling in development, and symmetry-breaking in cell polarization. We develop a modeling framework for bilayer reaction–diffusion systems and relate it to a range of existing models. We derive conditions for diffusion-driven instability of a spatially homogeneous equilibrium analogous to the classical conditions for a Turing instability in the simplest nontrivial setting where one domain has a standard reaction–diffusion system, and the other permits only diffusion. Due to the transverse coupling between these two regions, standard techniques for computing eigenfunctions of the Laplacian cannot be applied, and so we propose an alternative method to compute the dispersion relation directly. We compare instability conditions with full numerical simulations to demonstrate impacts of the geometry and coupling parameters on patterning, and explore various experimentally relevant asymptotic regimes. In the regime where the first domain is suitably thin, we recover a simple modulation of the standard Turing conditions, and find that often the broad impact of the diffusion-only domain is to reduce the ability of the system to form patterns. We also demonstrate complex impacts of this coupling on pattern formation. For instance, we exhibit non-monotonicity of pattern-forming instabilities with respect to geometric and coupling parameters, and highlight an instability from a nontrivial interaction between kinetics in one domain and diffusion in the other. These results are valuable for informing design choices in applications such as synthetic engineering of Turing patterns, but also for understanding the role of stratified media in modulating pattern-forming processes in developmental biology and beyond.

It is well known that noise and detection error can affect the performances of an adaptive optics (AO) system. Effects of noise and detection error on the phase compensation effectiveness in a dynamic AO system are investigated by means of a pure numerical simulation in this paper. A theoretical model for numerically simulating effects of noise and detection error in a static AO system and a corresponding computer program were presented in a previous article. A numerical simulation of effects of noise and detection error is combined with our previous numeral simulation of a dynamic AO system in this paper and a corresponding computer program has been compiled. Effects of detection error, readout noise and photon noise are included and investigated by a numerical simulation for finding the preferred working conditions and the best performances in a practical dynamic AO system. An approximate model is presented as well. Under many practical conditions such approximate model is a good alternative to the more accurate one. A simple algorithm which can be used for reducing the effect of noise is presented as well. When signal to noise ratio is very low, such method can be used to improve the performances of a dynamic AO system.

A self-organized mesh generator using pattern formation in a reaction–diffusion system

Nucleation kinetics and global coupling in reaction-diffusion systems

Noise is ubiquitous in a system and can induce some spontaneous pattern formations on a spatially homogeneous domain. In comparison to the Reaction-Diffusion System (RDS), Stochastic Reaction-Diffusion System (SRDS) is more complex and it is very difficult to deal with the noise function. In this paper, we have presented a method to solve it and obtained the conditions of how the Turing bifurcation and Hopf bifurcation arise through linear stability analysis of local equilibrium. In addition, we have developed the amplitude equation with a pair of wave vector by using Taylor series expansion, multiscaling, and further expansion in powers of small parameter. Our analysis facilitates finding regions of bifurcations and understanding the pattern formation mechanism of SRDS. Finally, the simulation shows that the analytical results agree with numerical simulation.

Cooperative behaviors arising from bacterial cell-to-cell communication can\nbe modeled by reaction-diffusion equations having only a single diffusible\ncomponent. This paper presents the following three contributions for the\nsystematic analysis of Turing instability in such reaction-diffusion systems.\n(i) We first introduce a unified framework to formulate the reaction-diffusion\nsystem as an interconnected multi-agent dynamical system. (ii) Then, we\nmathematically classify biologically plausible and implausible Turing\ninstabilities and characterize them by the root locus of each agent's dynamics,\nor the local reaction dynamics. (iii) Using this characterization, we derive\nanalytic conditions for biologically plausible Turing instability, which\nprovide useful guidance for the design and the analysis of biological networks.\nThese results are demonstrated on an extended Gray-Scott model with a single\ndiffuser.\n

Systems of reaction-diffusion equations are widely used to model morphogenesis (organisation of forms and patterns in living organisms) in different biological contexts. A particular application describes the formation of animal coat patterns and distribution of structures on the epidermis. The patterns of interest are either an arrangement of local maxima or a stripe-like distribution of chemical concentrations over the domain of interest that persist over time. In mathematical terms, a pattern is a spatially inhomogeneous solution of the reaction-diffusion system which is asymptotically stable. The common feature of systems displaying pattern formation is the presence of a steady state that is asymptotically stable to spatially-homogeneous perturbations but asymptotically unstable to spatially-inhomogeneous perturbations.We study whether it is possible to relax this assumption with an analysis of a reaction-diffusion system with a singularity that is a slight modification of a model for hair follicle spacing . We prove existence of global solutions for the reaction-diffusion system and examine stability of spatially inhomogeneous stationary solutions.

The striking similarity between the nonlinear dynamical phenomena in chemical systems and those in biological systems, has enhanced the interest in the search of new reaction-diffusion systems. Here we report an oscillatory chemical system, which exhibits different types of chemical waves and patterns when kept unstirred. The reaction constituents are 1,2,3-trihydroxybenzene (pyrogallol), ferroin, sulphuric acid and potassium bromate. The system has a significant advantage of being gas-free.

The Brusselator model is widely used to illustrate and study basic features of models of chemical reactions involving trimolecular steps. We provide the necessary mathematical theory related to Reaction-Diffusion systems in general and to the Brusselator model in particular. Specifically, the issues of local and global existence of solutions, their uniqueness and regularity are discussed in detail. The theoretical and numerical investigation presented futher in the thesis provides an insight into the asymptotic behavior of the solutions of this model, characterizing the parameter region for each of the three qualitatively different cases: homogeneous steady state, Turing pattern and bulk oscillations. Particular attention is given to the supercritical Hopf bifurcation parameter domain where no substantial theory is available. This study was largely motivated by the observations of Young, Zhabotinsky and Epstein that Turing patterns eventually (for sufficiently small ratio of the diffusion coefficients) dominate the Hopf bifurcation induced bulk oscillations. In this work we confirm this observation and further establish more precisely the shape of the boundary separating the Turing pattern domain and the bulk oscillations domain in the parameter space. The obtained results are used in revealing an essential mechanism generating oscillating patterns in the coupled Brusselator model. It can be considered as a model of the reaction sequences in two thin layers of gel that meet at an interface. Each layer contains the same reactants with the same kinetics but with different diffusion coefficients. The occurrence of oscillating patterns is due to the fact that for the same values of the parameters of the model but with different diffusion coefficients the one system can be in the Turing pattern domain while the other is in the bulk oscillations domain. Hence, roughly speaking, one layer provides a pattern while the other layer drives the oscillations.

This paper is concerned with stochastic reaction-diffusion kinetics governed by the reaction-diffusion master equation. Specifically, the primary goal of this paper is to provide a mechanistic basis of Turing pattern formation that is induced by intrinsic noise. To this end, we first derive an approximate reaction-diffusion system by using linear noise approximation. We show that the approximated system has a certain structure that is associated with a coupled dynamic multi-agent system. This observation then helps us derive an efficient computation tool to examine the spatial power spectrum of the intrinsic noise. We numerically demonstrate that the result is quite effective to analyze noise-induced Turing pattern. Finally, we illustrate the theoretical mechanism behind the noise-induced pattern formation with a H2 norm interpretation of the multi-agent system.