Stochastic Reaction-diffusion Systems Research Articles

Null controllability for a strongly coupled stochastic degenerate reaction-diffusion system

This paper concerns the null controllability for a strongly coupled stochastic degenerate reaction-diffusion system in cardiac electrocardiology, which describes the electrical activity in the cardiac tissue with random effects. To deal with degeneracy of this system, we first consider an approximate problem of this coupled stochastic degenerate system. By a weighted identity method, we then prove a uniform Carleman estimate for the adjoint system of this approximate problem, which is a strongly coupled backward stochastic parabolic system with homogeneous Neumann boundary conditions. Based on this Carleman estimate and a limit process, we finally obtain the null controllability result.

Computing macroscopic reaction rates in reaction-diffusion systems using Monte Carlo simulations.

Stochastic reaction-diffusion models are employed to represent many complex physical, biological, societal, and ecological systems. The macroscopic reaction rates describing the large-scale, long-time kinetics in such systems are effective, scale-dependent renormalized parameters that need to be either measured experimentally or computed by means of a microscopic model. In a Monte Carlo simulation of stochastic reaction-diffusion systems, microscopic probabilities for specific events to happen serve as the input control parameters. To match the results of any computer simulation to observations or experiments carried out on the macroscale, a mapping is required between the microscopic probabilities that define the Monte Carlo algorithm and the macroscopic reaction rates that are experimentally measured. Finding the functional dependence of emergent macroscopic rates on the microscopic probabilities (subject to specific rules of interaction) is a very difficult problem, and there is currently no systematic, accurate analytical way to achieve this goal. Therefore, we introduce a straightforward numerical method of using lattice Monte Carlo simulations to evaluate the macroscopic reaction rates by directly obtaining the count statistics of how many events occur per simulation time step. Our technique is first tested on well-understood fundamental examples, namely, restricted birth processes, diffusion-limited two-particle coagulation, and two-species pair annihilation kinetics. Next we utilize the thus gained experience to investigate how the microscopic algorithmic probabilities become coarse-grained into effective macroscopic rates in more complex model systems such as the Lotka-Volterra model for predator-prey competition and coexistence, as well as the rock-paper-scissors or cyclic Lotka-Volterra model and its May-Leonard variant that capture population dynamics with cyclic dominance motifs. Thereby we achieve a more thorough and deeper understanding of coarse graining in spatially extended stochastic reaction-diffusion systems and the nontrivial relationships between the associated microscopic and macroscopic model parameters, with a focus on ecological systems. The proposed technique should generally provide a useful means to better fit Monte Carlo simulation results to experimental or observational data.

Finite time stability analysis of the coupled stochastic reaction–diffusion systems on networks

In this work, we consider a class of coupled stochastic reaction–diffusion systems (CSRDSs, in short) on networks, the criterion on the finite time stability of the trivial solution for CSRDSs on networks is derived by means of the composite Lyapunov function method. To carry out the required results, we establish some useful lemmas firstly for preliminaries. For illustration, an example is given to verify the obtained results.

Fixed/Prescribed stability criterions of stochastic system with time-delay

<abstract><p>In this paper, the fixed/prescribed-time stability issues were considered for stochastic systems with time delay. First, some new fixed-time stability and prescribed-time stability criteria for stochastic systems with delay and multi-delay were established. Second, based on the new fixed/prescribed stability criteria, the fixed-time stabilization of the stochastic system with time-delay and the prescribed-time stabilization of the stochastic reaction-diffusion system with multi-delay were investigated, respectively. Third, two new fixed/prescribed-time delay-independent control mechanisms were designed. The primary advantage of the innovative fixed/prescribed-time controller lies in its independence from delayed states. This makes the controller applicable to systems with unknown delays. Finally, three numerical examples were provided to illustrate the feasibility of the stated theoretical results.</p></abstract>

When patterns come to life: Comment on “Unified representation of Life's basic properties by a 3-species Stochastic Cubic Autocatalytic Reaction-Diffusion system of equations” by A.P. Muñuzuri and J. Pérez-Mercader

When patterns come to life: Comment on “Unified representation of Life's basic properties by a 3-species Stochastic Cubic Autocatalytic Reaction-Diffusion system of equations” by A.P. Muñuzuri and J. Pérez-Mercader

Exponential stability of large-scale stochastic reaction-diffusion equations

In this paper, we consider a class of large-scale stochastic reaction-diffusion systems. To prove the exponential stability of the system, we introduce the corresponding isolated subsystems. We show that the exponential stability of the isolated systems implies the exponential stability of the large-scale stochastic reaction-diffusion system under some conditions. Furthermore, we discuss a special case where the large-scale stochastic reaction-diffusion system is described in a hierarchical form. In this case, we prove that the original system is exponentially stable if and only if the corresponding subsystems are exponentially stable.

Information, computation, and causality in living systems: Comment on "Unified representation of life's basic properties by a 3-species stochastic cubic autocatalytic reaction-diffusion system of equations", by A.P. Muñuzuri and J. Pérez-Mercader.

Information, computation, and causality in living systems: Comment on "Unified representation of life's basic properties by a 3-species stochastic cubic autocatalytic reaction-diffusion system of equations", by A.P. Muñuzuri and J. Pérez-Mercader.

Conceptualizing life via non-equilibrium physics; is it enough?: Comment on "Unified representation of life's basic properties by a 3-species stochastic cubic autocatalytic reaction-diffusion system of equations" by A.P. Muñuzuri and J. Pérez-Mercader.

Conceptualizing life via non-equilibrium physics; is it enough?: Comment on "Unified representation of life's basic properties by a 3-species stochastic cubic autocatalytic reaction-diffusion system of equations" by A.P. Muñuzuri and J. Pérez-Mercader.

Waves in a Stochastic Cell Motility Model

In Bhattacharya et al. (Sci Adv 6(32):7682, 2020), a set of chemical reactions involved in the dynamics of actin waves in cells was studied at two levels. The microscopic level, where the individual chemical reactions are directly modelled using Gillespie-type algorithms, and on a macroscopic level where a deterministic reaction–diffusion equation arises as the large-scale limit of the underlying chemical reactions. In this work, we derive, and subsequently study, the related mesoscopic stochastic reaction–diffusion system, or chemical Langevin equation, that arises from the same set of chemical reactions. We explain how the stochastic patterns that arise from this equation can be used to understand the experimentally observed dynamics from Bhattacharya et al. In particular, we argue that the mesoscopic stochastic model better captures the microscopic behaviour than the deterministic reaction–diffusion equation, while being more amenable for mathematical analysis and numerical simulations than the microscopic model.

The Order of Convergence in the Averaging Principle for Slow-Fast Systems of Stochastic Evolution Equations in Hilbert Spaces

In this work we are concerned with the study of the strong order of convergence in the averaging principle for slow-fast systems of stochastic evolution equations in Hilbert spaces with additive noise. In particular the stochastic perturbations are general Wiener processes, i.e their covariance operators are allowed to be not trace class. We prove that the slow component converges strongly to the averaged one with order of convergence 1/2 which is known to be optimal. Moreover we apply this result to a slow-fast stochastic reaction diffusion system where the stochastic perturbation is given by a white noise both in time and space.

Mean square exponential input-to-state stability of stochastic Markovian reaction-diffusion systems with impulsive perturbations

Mean square exponential input-to-state stability (MSEISS) is considered for stochastic Markovian reaction-diffusion systems (SMRDSs) with impulsive perturbations. Both the boundary input and distributed input are considered in SMRDSs. With the Lyapunov–Krasovskii functional method, impulse theory and inequality techniques, a sufficient condition is established to achieve the MSEISS for SMRDSs with completely known transition rate matrix. Moreover, combined with the obtained sufficient conditions, the effects of the impulse and diffusion terms on MSEISS are demonstrated by examples. Then, the case is studied that the transition rate matrix is partially unknown and sufficient conditions are presented to ensure the MSEISS in light of the introduced free constants. Finally, two numerical examples are given to illustrate the validity of our theoretical results.

Numerical and renormalization group analysis of the phase diagram of a stochastic cubic autocatalytic reaction-diffusion system.

The renormalization group is a set of tools that can be used to incorporate the effect of fluctuations in a dynamical system as a rescaling of the system's parameters. Here, we apply the renormalization group to a pattern-forming stochastic cubic autocatalytic reaction-diffusion model and compare its predictions with numerical simulations. Our results demonstrate a good agreement within the range of validity of the theory and show that external noise can be used as a control parameter in such systems.

On the robustness of the Grey-Scott system as a minimal model of life: Comments on "Unified representation of Life's basic properties by a 3-species Stochastic Cubic Autocatalytic Reaction-Diffusion system of equations" by A. P. Muñuzuri and J. Pérez-Mercader.

On the robustness of the Grey-Scott system as a minimal model of life: Comments on "Unified representation of Life's basic properties by a 3-species Stochastic Cubic Autocatalytic Reaction-Diffusion system of equations" by A. P. Muñuzuri and J. Pérez-Mercader.

A novel recipe for prebiotic systems chemistry arising from autocatalytic relationships: Comment on "Unified representation of life's basic properties by a 3-species stochastic cubic autocatalytic reaction-diffusion system of equations", by A.P. Muñuzuri and J. Pérez-Mercader.

A novel recipe for prebiotic systems chemistry arising from autocatalytic relationships: Comment on "Unified representation of life's basic properties by a 3-species stochastic cubic autocatalytic reaction-diffusion system of equations", by A.P. Muñuzuri and J. Pérez-Mercader.

Singular limit of deterministic and stochastic reaction-diffusion systems

Singular limit of deterministic and stochastic reaction-diffusion systems

Theoretical physics unified model building meets biology: Comment on "Unified representation of life's basic properties by a 3-species stochastic cubic autocatalytic reaction-diffusion system of equations", by A.P. Muñuzuri and J. Pérez-Mercader.

Theoretical physics unified model building meets biology: Comment on "Unified representation of life's basic properties by a 3-species stochastic cubic autocatalytic reaction-diffusion system of equations", by A.P. Muñuzuri and J. Pérez-Mercader.

Impulsive effects on stabilization of stochastic nonlinear reaction-diffusion systems with time delays and boundary feedback control

In this paper, we investigate the stabilization of stochastic nonlinear impulsive reaction-diffusion systems (SNIRDSs) with time delays and boundary feedback control via average impulsive interval approach. Boundary feedback control strategy are designed to stabilization of SNIRDSs. By constructing a Lyapunov-Krasovskii functional (LKF), and using Wirtinger's inequality, Gronwall inequality, average impulsive interval approach, sufficient conditions are derived to guarantee the finite-time stability (FTS) of proposed systems. We investigate the stabilization results by designing the control gain matrices for boundary feedback controller. The criterions are expressed in terms of linear matrix inequalities (LMIs) that can be verified by Matlab LMI toolbox. Finally, numerical example are given to verify the efficiency and superiority of proposed stabilization criterions.

Stability of switched stochastic reaction–diffusion systems

As one of important hybrid dynamical systems, switched system has practical applications in some practical systems. Generally, switched system is made up of a family of continuous-time subsystems and discrete switching signals. Affected by environmental disturbance, stochastic switched system should be considered. This paper deals with a class of switched stochastic reaction-diffusion systems (SSRDSs, in short), where the discrete switching signal is a right continuous and piecewise function. The switching instants are stopping times or deterministic times. Criterions on stability in probability and exponential stability in mean square of the trivial solution for SSRDSs are derived by means of Lyapunov functional methods. We propose two examples to verify the obtained theoretical results.

Finite-time stability analysis of switched stochastic reaction-diffusion systems

In this work, we consider a class of switched stochastic reaction-diffusion systems (SSRDSs, in short), where the discrete switching signal is a right continuous, piecewise function and the switching instants are stopping times or deterministic times. Criteria on finite-time stability in the probability of the trivial solution for SSRDSs are derived by means of Lyapunov functional methods. We propose an example to verify the obtained theoretical results.

Analysis of a stochastic reaction–diffusion Alzheimer’s disease system driven by space–time white noise

Alzheimer’s disease (AD) is a neurodegenerative disorder with multifactorial etiology that poses a serious threat to human health. This work is devoted to a stochastic reaction–diffusion system for AD with space–time white noise accounting, which describes the positive feedback loop among the Aβ and the Ca2＋. The existence and uniqueness of mild solutions for the system are obtained by using the truncation method. Then, we study the long-time behaviors of the system. Further, an illustrative example is shown to support the obtained analytic results.