Fixed and Distributed Gene Expression Time Delays in Reaction–Diffusion Systems

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.

The role of a reaction-diffusion system in the initiation of skin organ primordia. I. The first wave of initiation

Nucleation kinetics and global coupling in reaction-diffusion systems

Pattern Formation in Reaction–Diffusion Models with Nonuniform Domain Growth

Advances in nonequilibrium pattern formation in reaction–diffusion systems are reviewed. Special emphasis is placed on patterns found in the spatially extended Belousov–Zhabotinsky reaction dispersed in aerosol OT water-in-oilmicroemulsions (BZ–AOT system): Turing patterns, packet and standing waves, antispirals and segmented spirals, and accelerating waves and oscillons. All experimental results are explained theoretically and reproduced in computer simulations.

The stability of a one-spike solution to a general class of reaction-diffusion (RD) system with both regular and anomalous diffusion is analyzed. The method of matched asymptotic expansions is used to construct a one-spike equilibrium solution and to derive a nonlocal eigenvalue problem (NLEP) that determines the stability of this solution on an O (1) time-scale. For a particular sub-class of the reaction kinetics, it is shown that the discrete spectrum of this NLEP is determined in terms of the roots of certain simple transcendental equations that involve two key parameters related to the choice of the nonlinear kinetics. From a rigorous analysis of these transcendental equations by using a winding number approach and explicit calculations, sufficient conditions are given to predict the occurrence of Hopf bifurcations of the one-spike solution. Our analysis determines explicitly the number of possible Hopf bifurcation points as well as providing analytical formulae for them. The analysis is implemented for the shadow limit of the RD system defined on a finite domain and for a one-spike solution of the RD system on the infinite line. The theory is illustrated for two specific RD systems. Finally, in parameter ranges for which the Hopf bifurcation is unique, it is shown that the effect of sub-diffusion is to delay the onset of the Hopf bifurcation.

This paper presents a unified and robust numerical framework that combines the Discrete Singular Convolution (DSC) method for spatial discretization with the Exponential Time Differencing Runge–Kutta (ETDRK4) scheme for temporal integration to solve reaction–diffusion systems. Specifically, we investigate the formation of Turing patterns – such as spots, stripes, and mixed structures – in classical models including the Gray–Scott, Brusselator, and Barrio–Varea–Aragón–Maini (BVAM) systems. The DSC method, employing the regularized Shannon’s delta kernel, delivers spectral-like accuracy in computing spatial derivatives on both regular and curved geometries. Coupled with the fourth-order ETDRK method, this approach enables efficient and stable time integration over long simulations. Importantly, we rigorously establish the necessary theoretical results – including convergence, stability, and consistency theorems, along with their proofs – for the combined DSC-ETDRK4 method when applied to both planar and curved surfaces. We demonstrate the capability of the proposed method to accurately reproduce and analyze complex spatiotemporal patterns on a variety of surfaces, including the plane, sphere, torus, and bumpy geometries. Numerical experiments confirm the method’s versatility, high accuracy, and computational efficiency, making it a powerful tool for the study of pattern formation in reaction–diffusion systems on diverse geometries.

Global asymptotic stability of Lotka–Volterra 3-species reaction–diffusion systems with time delays

We consider the effect of time delay and cross diffusion on the dynamics of a modified Leslie-Gower predator-prey model incorporating a prey refuge. Based on the stability analysis, we demonstrate that delayed feedback may generate Hopf and Turing instability under some conditions, resulting in spatial patterns. One of the most interesting findings is that the model exhibits complex pattern replication: the model dynamics exhibits a delay and diffusion controlled formation growth not only to spots, stripes, and holes, but also to spiral pattern self-replication. The results indicate that time delay and cross diffusion play important roles in pattern formation.

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

Global attractor of a coupled finite difference reaction diffusion system with delays

We introduce a DNA-based reaction-diffusion (RD) system in which reaction and diffusion terms can be precisely and independently controlled. The effective diffusion coefficient of an individual reaction component, as we demonstrate on a traveling wave, can be reduced up to 2.7-fold using a self-assembled hydrodynamic drag. The intrinsic programmability of this RD system allows us to engineer, for the first time, orthogonal autocatalysts that counterpropagate with minimal interaction. Our results are in excellent quantitative agreement with predictions of the Fisher-Kolmogorov-Petrovskii-Piscunov model. These advances open the way for the rational engineering of pattern formation in pure chemical RD systems.

In order to reduce the delay times from onset of symptoms to arrival in hospital, and increase the use of ambulance in patients with suspected acute myocardial infarction (AMI), a media campaign was initiated in an urban area. An initial 3-week intense campaign was followed by a maintenance phase of 1 year. Delay times and ambulance use during the campaign were compared with the previous 21 months. Among patients admitted to a coronary care unit (CCU) due to suspected AMI, the median delay time was reduced from 3 h to 2 h 40 min and the mean delay time was reduced from 11 h 33 min to 7 h 42 min (P less than 0.001). Among patients with confirmed AMI the median delay time was reduced from 3 h to 2 h 20 min and the mean delay time from 10 h to 6 h 27 min (P less than 0.001). We conclude that a 1-year media campaign can reduce delay times in suspected AMI, and that this effect appears to continue at 1 year, but ambulance use seems to be more difficult to influence.

By discretising space into compartments and letting system dynamics be governed by the reaction–diffusion master equation, it is possible to derive and simulate a stochastic model of reaction and diffusion on an arbitrary domain. However, there are many implementation choices involved in this process, such as the choice of discretisation and method of derivation of the diffusive jump rates, and it is not clear a priori how these affect model predictions. To shed light on this issue, in this work we explore how a variety of discretisations and methods for derivation of the diffusive jump rates affect the outputs of stochastic simulations of reaction–diffusion models, in particular using Turing’s model of pattern formation as a key example. We consider both static and uniformly growing domains and demonstrate that, while only minor differences are observed for simple reaction–diffusion systems, there can be vast differences in model predictions for systems that include complicated reaction kinetics, such as Turing’s diffusion-driven instability model of pattern formation. Our work highlights that care must be taken in using the reaction–diffusion master equation framework to make predictions as to the dynamics of stochastic reaction–diffusion systems.

We study the stability of the steady state of coupled chaotic maps with randomly distributed time delays evolving on a random network. An analysis method is developed based on the peculiar mathematical structure of the Jacobian of the steady state due to time-delayed coupling, which enables us to relate the stability of the steady state to the locations of the roots of a set of lower-order bound equations. For delta -distributed time delays (or fixed time delay), we find that the stability of the steady state is determined by the maximum modulus of the roots of a set of algebraic equations, where the only nontrivial coefficient in each equation is one of the eigenvalues of the normalized adjacency matrix of the underlying network. For general distributed time delays, we find a necessary condition for the stable steady state based on the maximum modulus of the roots of a bound equation. When the number of links is large, the nontrivial coefficients of the bound equation are just the probabilities of different time delays. Our study thus establishes the relationship between the stability of the steady state and the probability distribution of time delays, and provides a better way to investigate the influence of the distributed time delays in coupling on the global behavior of the systems.