Delayed Reaction Diffusion Systems Research Articles

Spatial spread for a delayed and nonlocal foot-and-mouth disease model

Foot-and-mouth disease is a highly infectious zoonosis between human and cloven-hoofed animals and it has been endemic in almost every part of the world. In this paper, a novel mathematical model, in the form of a nonlocal delayed reaction–diffusion system on unbounded domain is studied. We concentrate on the spreading property in terms of traveling waves. By applying Schauder’s fixed point theorem on a large domain and constructing an invariant cone, we prove the existence of traveling weak-waves when R0>1 and c>c∗. Under an additional assumption that the infected individuals move more slowly than the susceptible individuals, we establish an uniform upper bound of the traveling weak-waves, which is independent of c. Based on this result, a new Lyapunov functional is then constructed to show that the traveling weak-waves converge to the endemic equilibrium at positive infinity and the existence of the critical traveling wave is proved by using a limiting argument. Finally, the nonexistence of the traveling wave is proved when R0≤1 or 0<c<c∗.

Finite-time synchronization of intermittently controlled reaction–diffusion systems with delays: A weighted LKF method

Considering the fact that existing methodologies for finite-time control are difficult to simultaneously overcome the difficulties induced by the effects of reaction–diffusion and time delay when intermittent control is confronted, this paper explores a novel Lyapunov–Krasovskii functional (LKF) method to investigate the finite-time synchronization of delayed reaction–diffusion systems. By designing a simple intermittent control and a weighted LKF, a general finite-time stability criterion is established first. Then, sufficient conditions for the finite-time synchronization of the interested system are given, where the weight factor of the LKF has a heavy influence on the settling time. Several important corollaries are also given to specify the usefulness and generality of the weighted LKF method and the finite-time stability criterion. Finally, a numerical example is provided to verify the new findings, and an image encryption algorithm is presented to validate the useful application of theoretical results.

Spreading dynamics of a diffusive epidemic model with free boundaries and two delays

AbstractA delayed reaction-diffusion system with free boundaries is investigated in this paper to understand how the bacteria spread spatially to larger area from the initial infected habitat. Under the assumptions that the nonlinearities are of monostable type and the initial values satisfy some compatible condition, we show that the free boundary problem is well-posed and discuss the long-time behaviour of solution (including spreading and vanishing) in terms of the spatial-temporal risk index. Furthermore, to determine the spreading speed of free boundaries when spreading occurs, we first study the distribution of roots of a transcendental equation containing a polynomial of degree four and then establish the existence and uniqueness of monotone solution to a delay-induced nonlocal semi-wave problem by employing the approximation method, lower-upper solutions technique and Schauder fixed point theorem. It is shown that time delays slow down the spreading of bacteria.

Asymptotic behavior of the principal eigenvalue and basic reproduction ratio for time-periodic reaction-diffusion systems with time delay

This paper is devoted to asymptotic behavior of the basic reproduction ratio for time-periodic and delay reaction-diffusion systems in the cases of small and large diffusion coefficients. First, we employ the generalized Kerin-Rutman theorem to establish the existence of the principal eigenvalue for periodic and cooperative reaction-diffusion systems with delay. Then, motivated by Kerin-Rutman theorem, we provide a method to estimate the upper and lower bounds of principal eigenvalues so as to obtain the asymptotic behavior of the principal eigenvalue. In view of the relationship between the basic reproduction ratio and the principal eigenvalue, we obtain the limiting profile of the basic reproduction ratio according to the asymptotic behavior of the principal eigenvalue.

Fixed-time boundary stabilisation for delay reaction–diffusion systems

This paper considers the problem of the boundary fixed-time stabilisation for delay reaction–diffusion systems (DRDSs). A new boundary controller is designed to achieve fixed time stability of DRDSs. Firstly, a boundary fixed-time controller is presented for DRDSs. Then based on the designed boundary controller, using Lyapunov functional method, fixed-time stability lemma and Poincaré's inequality, we obtain the criteria to guarantee the fixed-time stability of DRDSs. In addition, boundary fixed-time controller is proposed for uncertain DRDSs to reach robust fixed-time stability. Both the cases, the systems with or without uncertainties, are severally followed by a fixed time, which is an upper-bound of the settling time and irrelevant to the initial conditions. Finally, two simulation examples are given to illustrate the fixed-time stability of the controlled systems, which show the validity of our theoretical results.

Passivity-based boundary control for stochastic delay reaction–diffusion systems

This paper considers the passivity-based boundary control for stochastic delay reaction–diffusion systems (SDRDSs) with boundary input–output. Delay-dependent sufficient conditions are obtained in the sense of expectation for SDRDSs by the use of Lyapunov functional method, for the input strict passivity and the output strict passivity, respectively. Then delay-dependent sufficient conditions are presented to ensure the uncertainty SDRDSs to achieve the robust passivity. Moreover, by utilising the techniques of Green formula and Poincare's inequality, the difficulty is handled caused by the boundary input and boundary controller. Besides, the troubles brought by the existence of the stochastic term and uncertain parameters are resolved by stochastic inequality techniques. Finally, numerical simulations are provided to show the validity of our theoretical results.

Passivity-based boundary control for delay reaction-diffusion systems

Passivity-based boundary control is considered for time-varying delay reaction-diffusion systems (DRDSs) with boundary input-output. By virtue of Lyapunov functional method and inequality techniques, sufficient conditions are obtained for input strict passivity and output strict passivity of DRDSs, respectively. When the parameter uncertainties appear in DRDSs, sufficient conditions are presented to guarantee the robust passivity. Moreover, we apply our theoretical results to the synchronization problem of coupled delay reaction-diffusion systems and get the criterion to ensure the asymptotic synchronization. Finally, numerical simulations are provided to show the validity of our theoretical results.

A hybrid impulsive and sampled-data control for fractional-order delayed reaction–diffusion system of mRNA and protein in regulatory mechanisms

In genetic regulatory networks, the diffusion effects of mRNA and protein play a key role in regulatory mechanisms of gene expression, especially in translation and transcription. Hybrid impulsive and sampled-data controller are proposed in this paper. By constructing a suitable Lyapunov functional, and utilizing average impulsive interval approach, finite-time stabilization criteria are derived for fractional-order delayed reaction–diffusion genetic regulatory networks (FDRDGRNs). Meanwhile, impulsive control gains and sampled-data control gains are obtained by solving a set of linear matrix inequalities (LMIs). Finally, a numerical example is presented to show the applicability of the proposed scheme.

Spatio‐temporal sampled‐data control for delay reaction‐diffusion systems

AbstractThis article considers the exponential stabilization and performance for delay reaction‐diffusion systems (DRDSs), with spatial sampled‐data controller (SSDC) and spatio‐temporal sampled‐data controller (STSDC). Firstly, an SSDC is designed to stabilize the DRDSs. Using Lyapunov functional and Wirtinger's inequality technique, we obtain sufficient conditions to ensure the exponential stability of DRDSs. When there exist external disturbances, performance is considered and criteria are provided for the disturbed DRDSs, under the designed SSDC. Then, an STSDC is adopted for DRDSs. Time discrete item brings new difficulties for the analysis of the desired properties. To overcome these difficulties, a novel Lyapunov functional is constructed and Halaney's inequality is used. Moreover, the descriptor method is adopted. Under these techniques, delay‐dependent conditions are obtained to guarantee the exponential stability. The performance is also considered for the disturbed DRDSs under the designed STSDC. A Razumikhin‐type method is introduced together with the Lyapunov‐like functional. Sufficient conditions are presented to achieve performance for DRDSs with STSDC. Our theoretical results show that the spatial sampling interval does affect the desired properties, the shorter the spatial sampling interval, the easier to achieve the desired properties. Moreover, the time delay also influences the exponential stability of DRDSs, and smaller time delay is beneficial to the achievement of the stability. Finally, examples are given to illustrate the validity of results.

A Hybrid Impulsive and Sampled-Data Control for Fractional-Order Delayed Reaction-Diffusion System of mRNA a and Protein in Regulatory Mechanisms

A Hybrid Impulsive and Sampled-Data Control for Fractional-Order Delayed Reaction-Diffusion System of mRNA a and Protein in Regulatory Mechanisms

Hopf-Hopf bifurcation in the delayed nutrient-microorganism model

In view of time delay in the transport of nutrients, a delayed reaction-diffusion system with homogeneous Neumann boundary conditions is presented to understand the formation of the heterogeneous distribution of bacteria and nutrients in the sediment. With the effects of time delay and diffusion, the system will experience various dynamical behaviors, such as stability, the Turing instability, successive switches of stability of equilibria, the Hopf and the Hopf-Hopf bifurcations. To further understand the dynamics of the Hopf-Hopf bifurcation, the multiple time scale (MTS) technique is employed to derive the amplitude equations at this co-dimensional bifurcation point, and the dynamical classification near such bifurcation point is also identified by analyzing the obtained amplitude equations. Some numerical simulations are carried out to demonstrate the validity of the theoretical analysis.

Global stability of traveling waves for delay reaction-diffusion systems without quasi-monotonicity

In this article, we consider the existence of positive solutions for a nonlinear system of fourth-order ordinary differential equations. By constructing a single cone \(P\) in the product space \(C[0, 1] \times C[0, 1]\) and applying fixed point theorem in cones, we establish the existence of positive solutions for a system in which the nonlinear terms are both superlinear or sublinear. In addition, by the construction of the product cone \(K_1 \times K_2 \subset C[0, 1] \times C[0, 1]\) along with the product formula of fixed point theory on a product cone, we investigate the existence of positive solutions involving nonlinear terms, one uniformly superlinear or sublinear, and the other locally uniformly sublinear or superlinear.&#x0D; For more information see https://ejde.math.txstate.edu/Volumes/2020/45/abstr.html

Global stability of traveling waves for delay reaction-diffusion systems without quasi-monotonicity

This article concerns the global stability of traveling waves of a reaction-diffusion system with delay and without quasi-monotonicity. We prove that the traveling waves (monotone or non-monotone) are exponentially stable in \(L^\infty(\mathbb{R})\) with the exponential convergence rate \(t^{-1/2}e^{-\mu t}\) for some constant \(\mu&gt;0\). We use the Fourier transform and the weighted energy method with a suitably weight function.&#x0D; For more information see https://ejde.math.txstate.edu/Volumes/2020/46/abstr.html

Mean square finite-time synchronization of impulsive stochastic delay reaction-diffusion systems

This paper considers mean square finite-time synchronization for coupled impulsive stochastic delay reaction-diffusion systems (ISDRDSs). Using Lyapunov–Krasovskii functional method, impulsive comparison lemma and Gronwall’s inequality, we obtain sufficient conditions that ensure mean square finite-time synchronization of coupled ISDRDSs. These sufficient conditions reveal the relationship of impulsive coefficients (including impulsive strength and impulsive frequency) and system coefficients (without impulsive effects) for mean square finite-time synchronization. Numerical simulations are carried out to show the effectiveness of our theoretical results.

Hopf bifurcation analysis on a delayed reaction-diffusion system modelling the spatial spread of bacterial and viral diseases

Abstract A delayed reaction-diffusion system with Neumann boundary conditions modelling the spatial spread of bacterial and viral diseases is considered. Sufficient conditions independent of diffusion and delay are obtained for the asymptotical stability of the spatially homogeneous positive steady state. We also perform a detailed Hopf bifurcation analysis by analyzing the corresponding characteristic equation and derive some formulae determining the direction of bifurcation and the stability of the bifurcating periodic solution by calculating the normal form on the center manifold. The delay driven instability of the positive steady state and the diffusion-driven instability of the spatially homogeneous periodic solution are investigated. Our results complement the main results in Tan et al. (2018) [10]. Some examples and numerical simulations are presented to illustrate our theoretical results.

Delay driven vegetation patterns of a plankton system on a network

A two-component delayed reaction–diffusion system is introduced to a complex network to describe the vegetation patterns in plankton system. The positive equilibrium is shown by the linear stability analysis to be asymptotically stable in the absence of time delay, but when the time delay increases beyond a threshold it loses its stability via the Hopf bifurcation. The stability and direction of the Hopf bifurcation is investigated with the method of center manifold theory. Our result reveals that the stability of Hopf bifurcation leads to the emergence of vegetation patterns. Numerical calculations are performed to confirm our theoretical results.

Double Hopf Bifurcation in Delayed reaction–diffusion Systems

Double Hopf bifurcation analysis can be used to reveal some complicated dynamical behavior in a dynamical system, such as the existence or coexistence of periodic orbits, quasi-periodic orbits, or even chaos. In this paper, an algorithm for deriving the normal form near a codimension-two double Hopf bifurcation of a reaction-diffusion system with time delay and Neumann boundary condition is rigorously established, by employing the center manifold reduction technique and the normal form method. We find that the dynamical behavior near bifurcation points are proved to be governed by twelve distinct unfolding systems. Two examples are performed to illustrate our results: for a stage-structured epidemic model, we find that double Hopf bifurcation appears when varying the diffusion rate and time delay, and two stable spatially inhomogeneous periodic oscillations are proved to coexist near the bifurcation point; in a diffusive predator-prey system, we theoretically proved that quasi-periodic orbits exist on two- or three-torus near a double Hopf bifurcation point, which will break down after slight perturbation, leaving the system a strange attractor.

Finite-time boundary control for delay reaction–diffusion systems

This paper considers finite-time stabilization and H∞ performance for delay reaction–diffusion systems by boundary control. First, a full-domain controller is designed and sufficient conditions are obtained to achieve finite-time stability using finite-time stability lemma and Wirtinger’s inequality method. Then a boundary controller furnished with sufficient conditions to achieve finite-time stability is presented. When taking into consideration external noise on a delay reaction–diffusion system, finite horizon H∞ boundary control with a criterion that guarantees the H∞ performance of delay reaction–diffusion systems is proposed. How to handle Neumann boundary conditions and mixed boundary conditions are discussed. Numerical simulations are carried out to verify the effectiveness of our theoretical results.

Existence and exponential stability of traveling waves for delayed reaction-diffusion systems

The purpose of this work is to investigate the existence and exponential stability of traveling wave solutions for general delayed multi-component reaction-diffusion systems. Following the monotone iteration scheme via an explicit construction of a pair of upper and lower solutions, we first obtain the existence of monostable traveling wave solutions connecting two different equilibria. Then, applying the techniques of weighted energy method and comparison principle, we show that all solutions of the Cauchy problem for the considered systems converge exponentially to traveling wave solutions provided that the initial perturbations around the traveling wave fronts belong to a suitable weighted Sobolev space.

Spreading speed and traveling waves for a non-local delayed reaction-diffusion system without quasi-monotonicity

A non-local delayed reaction-diffusion model with a quiescent stage is investigated. It is shown that the spreading speed of this model without quasi-monotonicity is linearly determinate and coincides with the minimal wave speed of traveling waves.