Reaction-diffusion systems with stochastic time delay in kinetics

Role of Noise-Modulated Self-Propulsion in Driving Spatiotemporal Orders in Active Systems

Fluctuations play a pivotal role in driving spatiotemporal order in active matter systems. In this study, we employ a novel analytical framework to investigate the impact of dichotomous noise on the self-propelling velocity of active particle systems such as polymerizing actin filaments or reproducing elongated bacteria. By incorporating dichotomous fluctuations with Ornstein-Zernike correlations into a continuum-based model, we derive a bifurcation condition in the noise parameter space, revealing a noise-induced instability that drives the emergence of traveling waves. This approach demonstrates how specific noise strengths and correlation times expand the instability region by introducing effective new degrees of freedom that alter the system's stability matrix. Advance numerical simulations, meticulously designed to handle the properties of dichotomous noise, validate these theoretical predictions and reveal excellent agreement. A key finding is the observation of wave-reversal behavior, driven by the sign alternation of the noise-modulated advection term and modulated by the relaxation time. Remarkably, we identify a finite parameter range where this reversal is suppressed, offering new insights into noise-induced bifurcations and spatiotemporal dynamics. Our combined analytical and numerical approach provides a deeper understanding of the role of noise in shaping self-organization and pattern formation in biological and synthetic active systems.

Radially evolving spiral wave patterns in the Gierer-Meinhardt reaction-diffusion system

Spiral wave formation in spatially extended systems is a fascinating phenomenon that has garnered significant attention in reaction-diffusion systems. In this study, we explore the emergence of spiral wave-like patterns in the Gierer-Meinhardt reaction-diffusion model. By employing a multiple-time scale perturbation technique, we derive amplitude equations that reveal the conditions for spiral wave formation. Notably, our analysis shows that the amplitude of these spiral waves varies with the radial distance, introducing a distinctive feature to this pattern. Our theoretical predictions are further substantiated by numerical simulations, which confirm the emergence of spiral wave structures and validate the distinct radial dependence of their amplitude.

A heritable iron memory enables decision-making in Escherichia coli

The importance of memory in bacterial decision-making is relatively unexplored. We show here that a prior experience of swarming is remembered when Escherichia coli encounters a new surface, improving its future swarming efficiency. We conducted >10,000 single-cell swarm assays to discover that cells store memory in the form of cellular iron levels. This "iron" memory preexists in planktonic cells, but the act of swarming reinforces it. A cell with low iron initiates swarming early and is a better swarmer, while the opposite is true for a cell with high iron. The swarming potential of a mother cell, which tracks with its iron memory, is passed down to its fourth-generation daughter cells. This memory is naturally lost by the seventh generation, but artificially manipulating iron levels allows it to persist much longer. A mathematical model with a time-delay component faithfully recreates the observed dynamic interconversions between different swarming potentials. We demonstrate that cellular iron levels also track with biofilm formation and antibiotic tolerance, suggesting that iron memory may impact other physiologies.

Unravelling diverse spatiotemporal orders in chlorine dioxide-iodine-malonic acid reaction-diffusion system through circularly polarized electric field and photo-illumination

Designing and predicting self-organized pattern formation in out-of-equilibrium chemical and biochemical reactions holds fundamental significance. External perturbations like light and electric fields exert a crucial influence on reaction-diffusion systems involving ionic species. While the separate impacts of light and electric fields have been extensively studied, comprehending their combined effects on spatiotemporal dynamics is paramount for designing versatile spatial orders. Here, we theoretically investigate the spatiotemporal dynamics of chlorine dioxide-iodine-malonic acid reaction-diffusion system under photo-illumination and circularly polarized electric field (CPEF). By applying CPEF at varying intensities and frequencies, we observe the predominant emergence of oscillating hexagonal spot-like patterns from homogeneous stable steady states. Furthermore, our study unveils a spectrum of intriguing spatiotemporal instabilities, encompassing stripe-like patterns, oscillating dumbbell-shaped patterns, spot-like instabilities with square-based symmetry, and irregular chaotic patterns. However, when we introduce periodic photo-illumination to the hexagonal spot-like instabilities induced by CPEF in homogeneous steady states, we observe periodic size fluctuations. Additionally, the stripe-like instabilities undergo alternating transitions between hexagonal spots and stripes. Notably, within the Turing region, the interplay between these two external influences leads to the emergence of distinct superlattice patterns characterized by hexagonal-and square-based symmetry. These patterns include parallel lines of spots, target-like formations, black-eye patterns, and other captivating structures. Remarkably, the simple perturbation of the system through the application of these two external fields offers a versatile tool for generating a wide range of pattern-forming instabilities, thereby opening up exciting possibilities for future experimental validation.

Iron Memory in E. coli

The importance of memory in bacterial decision-making is relatively unexplored. We show here that a prior experience of swarming is remembered when E. coli encounters a new surface, improving its future swarming efficiency. We conducted >10,000 single-cell swarm assays to discover that cells store memory in the form of cellular iron levels. This memory pre-exists in planktonic cells, but the act of swarming reinforces it. A cell with low iron initiates swarming early and is a better swarmer, while the opposite is true for a cell with high iron. The swarming potential of a mother cell, whether low or high, is passed down to its fourth-generation daughter cells. This memory is naturally lost by the seventh generation, but artificially manipulating iron levels allows it to persist much longer. A mathematical model with a time-delay component faithfully recreates the observed dynamic interconversions between different swarming potentials. We also demonstrate that iron memory can integrate multiple stimuli, impacting other bacterial behaviors such as biofilm formation and antibiotic tolerance.

Deciphering electric field induced spatial pattern formation in the photosensitive chlorine-dioxide iodine malonic acid reaction and the Brusselator reaction-diffusion systems

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 sign and 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.

Delay-induced wave instabilities in single-species reaction-diffusion systems

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.

Dichotomous-noise-induced pattern formation in a reaction-diffusion system

We consider a generic reaction-diffusion system in which one of the parameters is subjected to dichotomous noise by controlling the flow of one of the reacting species in a continuous-flow-stirred-tank reactor (CSTR) -membrane reactor. The linear stability analysis in an extended phase space is carried out by invoking Furutzu-Novikov procedure for exponentially correlated multiplicative noise to derive the instability condition in the plane of the noise parameters (correlation time and strength of the noise). We demonstrate that depending on the correlation time an optimal strength of noise governs the self-organization. Our theoretical analysis is corroborated by numerical simulations on pattern formation in a chlorine-dioxide-iodine-malonic acid reaction-diffusion system.

Controlling birhythmicity in a self-sustained oscillator by time-delayed feedback

Time-delayed feedback is a practical method for controlling various nonlinear dynamical systems. We consider its influence on the dynamics of a multicycle van der Pol oscillator that is birhythmic in nature. It has been shown that depending on the strength of delay the bifurcation space can be divided into two subspaces for which the dynamical response of the system is generically distinct. We observe an interesting collapse and revival of birhythmicity with the variation of the delay time. Depending on the parameter space the system also exhibits a transition between birhythmicity and monorhythmic behavior. Our analysis of amplitude equation corroborates with the results obtained by numerical simulation of the dynamics.