Reaction-diffusion Systems Research Articles

Breathers and mixed oscillatory states near a Turing–Hopf instability in a two–component reaction–diffusion system

Breathers and mixed oscillatory states near a Turing–Hopf instability in a two–component reaction–diffusion system

Global Hopf bifurcation and positive periodic solutions of a delayed diffusive predator–prey model with weak Allee effect for predator

In this paper, we investigate a delayed diffusive predator–prey model with weak Allee effect for predator. First, we discuss the existence and uniqueness of positive steady state and the local Hopf bifurcation. Next, we obtain the permanence and uniform boundedness of positive periodic solutions of the delayed reaction–diffusion system. The existence range of positive periodic solutions is further compressed by using iterative approach. For the system without delay, we prove the global attractiveness of unique positive steady state by using Lyapunov function, which ensures that the periods of positive periodic solutions are uniformly bounded. Then we obtain the global Hopf bifurcation and extended existence of positive periodic solutions by using the global Hopf bifurcation theorem of partial functional differential equation. Finally, the validity of the conclusion is verified through numerical simulation.

Front propagation near the onset of instability

We describe the resulting spatiotemporal dynamics when a homogeneous equilibrium loses stability in a spatially extended system. More precisely, we consider reaction-diffusion systems, assuming only that the reaction kinetics undergo a transcritical, saddle-node, or supercritical pitchfork bifurcation as a parameter passes through zero. We construct traveling front solutions which describe the invasion of the now-unstable state by a nearby stable state. We show that these fronts are marginally spectrally stable near the bifurcation point, which, together with recent advances in the theory of front propagation into unstable states, establishes that these fronts govern the dynamics of localized perturbation to the unstable state. Our proofs are based on functional analytic tools to study the existence and eigenvalue problems for fronts, which become singularly perturbed after a natural rescaling.

Reaction–diffusion systems associated with replicator dynamics for a class of population games and turing instability conditions

Reaction–diffusion systems associated with replicator dynamics for a class of population games and turing instability conditions

The Efficacy of Haar Wavelets in Addressing Discontinuities of McKean Equations with Heaviside Functions

This study explores the application of Haar wavelets to solve the McKean equation, a reaction-diffusion equation with discontinuous Heaviside step function. Haar wavelets, with their compact support and orthogonality, offer straightforward but yet powerful tool for addressing the equation’s nonlinear dynamics. We focus on the time-independent solution of the McKean equation, which is crucial for understanding the threshold phenomenon that determines the system’s behavior. Despite the existence of analytical time-independent solution to the McKean equation, achieving such solutions in closed form is uncommon for more complicated systems, highlighting the utility of the Haar wavelet approach. The proposed method integrates the Haar series expansion of the highest order derivative, enabling systematic solution derivation. Through a detailed comparison with analytical solution, we validate the Haar wavelet approach as a robust and computationally feasible tool for solving complex reaction-diffusion system. The results also demonstrate the method’s accuracy and efficiency, offering insights into its broader applicability to more complex reaction-diffusion system, especially those with discontinuity and sharp transitions.

From spots to stripes: Evolution of pigmentation patterns in monkeyflowers via modulation of a reaction-diffusion system and its prepatterns.

The reaction-diffusion (RD) system is widely assumed to account for many complex, self- organized pigmentation patterns in natural organisms. However, the specific configurations of such RD networks and how RD systems interact with positional information (i.e., prepatterns) that may specify the initiation conditions for the RD operation remain largely unknown. Here, we introduced a three-substance RD system underlying the formation of repetitive pigment spots and stripes in Mimulus flowers. It consists of an R2R3-MYB activator (NEGAN), an R3-MYB inhibitor (RTO), and a coactivator represented by two paralogous bHLH proteins. Through fine- scale genetic analyses, transgenic experiments, and computer simulations, we identified the causal loci contributing to the evolutionary transition from sparsely dispersed spots to longitudinal stripes. Genetic changes at these loci modulate the prepatterns of the activator and coactivator expression and the promoter activities of the inhibitor and one of the coactivator paralogs. Our findings highlight the importance of prepatterns towards a realistic description of RD systems in natural organisms, and reveal the genetic mechanism generating pattern variation through modulation of the kinetics of the RD system and its prepatterns.

Blow-up for a reaction–diffusion system with gradient terms under Robin boundary conditions

This paper is concerned with a reaction–diffusion system with gradient terms under Robin boundary conditions { ( h ( u ) ) t = ∇ ⋅ ( | ∇u | p − 2 ∇u ) + f ( u , v , | ∇u | 2 , t ) in Ω × ( 0 , t ∗ ) , ( r ( v ) ) t = ∇ ⋅ ( | ∇v | q − 2 ∇v ) + g ( u , v , | ∇v | 2 , t ) in Ω × ( 0 , t ∗ ) , ∂ u ∂ ν + γu = 0 , ∂ v ∂ ν + σv = 0 on ∂Ω × ( 0 , t ∗ ) , u ( x , 0 ) = u 0 ( x ) ≥ 0 , v ( x , 0 ) = v 0 ( x ) ≥ 0 in Ω ¯ , where p, q>2 and Ω ⊂ R N ( N ≥ 2 ) is a bounded domain with smooth boundary. Under some suitable assumptions, a sufficient condition that ensures the positive solution blows up in a finite time is obtained. Meanwhile, the upper bound for the blow-up time and the upper estimate for the blow-up rate are derived. The key to the research is to use the maximum principles and differential inequality technique. In addition, as an application of the abstract results obtained in this paper, an example is given.

Efficient uniformly convergent numerical methods for singularly perturbed parabolic reaction–diffusion systems with discontinuous source term

Efficient uniformly convergent numerical methods for singularly perturbed parabolic reaction–diffusion systems with discontinuous source term

A Dean–Kawasaki equation for reaction diffusion systems driven by Poisson noise

A Dean–Kawasaki equation for reaction diffusion systems driven by Poisson noise

Phase coexistence in a weakly stochastic reaction-diffusion system

Phase coexistence in a weakly stochastic reaction-diffusion system

Green behavior propagation analysis based on statistical theory and intelligent algorithm in data-driven environment.

The correlation between green behavior and energy efficiency is growing due to the heightened focus on energy efficiency among individuals. This paper introduces a three-layer network model to analyze the relationships among information diffusion, awareness and green behavior spreading. We have analyzed the probability tree of state transfer across 12 states by using Microscopic Markov Chain Approach (MMCA) and derived the state transfer equations for each state to compute the state transition threshold. In addition, we use the reaction-diffusion system to model the interaction between space and time changes for each state in the green behavior propagation layer. The equilibrium point of the system is defined, and the criteria for Turing bifurcation are identified. The optimal control approach achieves parameter identification, and this study validates the theory through several numerical simulations. Meanwhile, the effectiveness of parameter identification based on the convolutional neural network (CNN) and optimal control is compared. The data on China's electrical energy generation is predicted and compared by using three neural networks and an autoregressive integrated moving average (ARIMA) model. Further, considering clean energy generation as a green behavior, we fit the data on the percentage of clean energy generation by applying a Microscopic Markov Chain model and a reaction-diffusion system.

Stability analysis of a conservative reaction–diffusion system with rate controls

Stability analysis of a conservative reaction–diffusion system with rate controls

Chemical waves in reaction-diffusion networks of small organic molecules.

Chemical waves represent one of the fundamental behaviors that emerge in nonlinear, out-of-equilibrium chemical systems. They also play a central role in regulating behaviors and development of biological organisms. Nevertheless, understanding their properties and achieving their rational synthesis remains challenging. In this work, we obtained traveling chemical waves using synthetic organic molecules. To accomplish this, we ran a thiol-based reaction network in an unstirred flow reactor. Our observations revealed single or multiple waves moving in either the same or opposite directions, a behavior controlled by the geometry of our reactor. A numerical model can fully reproduce this behavior using the proposed reaction network. To better understand the formation of waves, we varied the diffusion coefficient of the fast inhibitor component of the reaction network by attaching polyethylene glycol tails with different lengths to maleimide and studied how these changes affect the properties of the waves and conditions for their sustained production. These studies point towards the importance of the molecular titration network motif in controlling the production of chemical waves in this system. Furthermore, we used machine learning (ML) tools to identify phase boundaries for classes of dynamic behaviors of this system, thus demonstrating the applicability of ML tools for the study of experimental nonlinear reaction-diffusion systems.

Chemical reactions with Liesegang rings: generation of non-permanent thermal patterns.

Patterns are encountered and employed in nature, such as in the communication or growth of organisms and sophisticated behaviors such as camouflage. Artificial patterns are not rare, either. They can also be used in sensing, recording information, and manipulating material properties. Natural or artificial, most patterns have the colors of visible light. Patterns of infrared radiation are rare, even though some organisms and artificial sensors can detect infrared. In this work, we display a formation of non-permanent infrared Liesegang patterns in different gel media. We used the exothermic neutralization of the solid Mg(OH)2 formed as Liesegang patterns (LPs) to obtain these patterns. The LPs of infrared radiation (IR patterns) appear due to a temperature increase of up to 4 °C on the gel surface. We also show that it is possible to 'read' the information with an IR camera when the patterns are concealed and invisible. The idea can be expanded to other chemical systems and used to communicate with IR. On the other hand, the chemical reactions on the patterned solids can also be used to pattern other artificial material patterns directly inaccessible with the existing reaction-diffusion systems.

Geometric control of honeycomb superlattice plasma photonic crystals in dielectric barrier discharge

Abstract We propose a scheme for dynamically controlling the geometric configurations of honeycomb superlattice plasma photonic crystals (HsPPCs) in dielectric barrier discharge (DBD). A rapid transition from a simple honeycomb lattice to diverse HsPPCs has been achieved in ambient air. The HsPPCs exhibit substantial omnidirectional band gaps (OBGs), approximately three times larger than those of simple honeycomb lattices. An experimental verification of the OBGs as well as their frequency shifting with reconfiguration of different HsPPCs has been provided. A phenomenological reaction-diffusion system is developed to unravel the generation mechanism of HsPPCs. The experimental observations are consistent with the numerical simulations. Our approach provides a unique method for manufacturing tunable HsPPCs which significantly enhance the band gap size and improve the photonic properties.

Pattern Formation Mechanisms of Spatiotemporally Discrete Activator–Inhibitor Model with Self- and Cross-Diffusions

In this paper, we aim to solve the issue of pattern formation mechanisms in a spatiotemporally discrete activator–inhibitor model that incorporates self- and cross-diffusions. We seek to identify the conditions that lead to the emergence of complex patterns and to elucidate the principles governing the dynamic behaviors that result in these patterns. We first construct a corresponding coupled map lattice (CML) model based on the continuous activator–inhibitor reaction–diffusion system. In the reaction stage, we examine the existence, uniqueness, and stability of the homogeneous stationary state and specify the parametric conditions for realizing these properties. Furthermore, by applying the center manifold theorem, we perform a flip bifurcation analysis and confirm that the model is capable of undergoing flip bifurcation. In the diffusion stage, we focus on the analysis of Turing bifurcation and determine the parameter conditions for the emergence of Turing instability. Through numerical simulations, we validate and explain the results of our theoretical analysis. Our study reveals various Turing instability mechanisms by coupling Turing and flip bifurcations, which include pure-self-diffusion-Turing instability, pure-cross-diffusion-Turing instability, flip-self-diffusion-Turing instability, flip-cross-diffusion-Turing instability, and chaos-self-diffusion-Turing instability mechanisms. Under different mechanisms, we illustrate the corresponding Turing patterns and discover a rich variety of pattern types such as labyrinthine, mosaic, alternating mosaic, colorful mottled grid patterns with winding and twisted bands, and patterns with dense patches and twisted bands nested together. Our research provides a theoretical framework and numerical support for understanding the complex dynamical behaviors and pattern formations in activator–inhibitor models with self- and cross-diffusions.

Existence of similarity profiles for diffusion equations and systems

We study the existence of similarity profiles for diffusion equations and reaction diffusion systems on the real line, where the different nontrivial limits are imposed for x→-∞ and x→+∞. These similarity profiles solve a coupled system of nonlinear ODEs that can be treated by monotone operator theory.

Global solutions for semilinear parabolic evolution problems with Hölder continuous nonlinearities

AbstractIt is shown that semilinear parabolic evolution equations featuring Hölder continuous nonlinearities with at most linear growth possess global strong solutions for a general class of initial data. The abstract results are applied to a recent model describing front propagation in bushfires and in the context of a reaction–diffusion system.

Unexplored aspects of spiral wave dynamics in the Barkley model within an extended parameter range

The Barkley model has been widely used to simulate diverse reaction-diffusion systems exhibiting various self-organization processes, including creation of spiral waves. Depending on the given parameters, this model is capable of simulating monostable, bistable, or oscillatory dynamics that are often observed in nature. This study is concentrated on the general features of spiral wave dynamics in the extended parameter range of the Barkley model, including both monostable and bistable dynamics. The numerical data obtained and the results of analytical studies illustrate the existence and the interplay between spiral wave dynamics in these two different but closely related regimes. In particular, the symmetry between the spiral waves in monostable and bistable regions is demonstrated, analytical expressions for the boundaries of these regions are found, the existence of the hysteresis phenomenon in the dynamics of spiral waves is discovered, and an additional region of existence of spiral waves is found in the region of a low excitability. Published by the American Physical Society 2024

Quantum Entanglement in Classical Systems: so what is the Subquantum Medium Made of?

As astounding as it may still seem to many, Bell’s theorems do not prove nonlocality. Non separable multipartite objects exist classically, meaning with local physics, the statistical state measurement of which violates the famous inequalities. Alleviating the almost century old confusion, the correct laws of statistics and logic pinpoint the true oddity of quantum objects: duality. As it is shown in the first part of this short essay, duality plus conservation laws allow the violation of Bell’s inequalities for any spatio-temporal separation. To dig deeper into particle dualism, in the second part, a class of models is proposed as a working framework. It encompasses some chaotic excitable reaction-diffusion systems, whose generalized susceptibilities make them compatible with quantized fields and excitations, of any desired symmetry group including the renormalization semigroup. Particles are supposed topological in nature. Bohr-Sommerfeld quantization takes place thanks to topological invariants stemming from densely dispersed defects generated by a multifractal background. Entanglement phenomenology arises because latent variables exist that are carried away, along with the moving particles that have interacted, and by which correlations are preserved. Conservation is assumed to be born in the phase, just as momentum is for instance. In other words, all known phenomena in physics are deterministic, classical and real, in the sense that information does propagate locally and experiments conducted statistically do hide latent variables.