Nonlocal Delay Research Articles

Pattern dynamics in a water–vegetation model with cross‐diffusion and nonlocal delay

Pattern dynamics in a water–vegetation model with cross‐diffusion and nonlocal delay

Asymptotic stability of the nonlocal diffusion equation with nonlocal delay

This work focuses on the asymptotic stability of nonlocal diffusion equations in ‐dimensional space with nonlocal time‐delayed response term. To begin with, we prove and ‐decay estimates for the fundamental solution of the linear time‐delayed equation by Fourier transform. For the considered nonlocal diffusion equation, we show that if , then the solution converges globally to the trivial equilibrium time‐exponentially. If , then the solution converges globally to the trivial equilibrium time‐algebraically. Furthermore, it can be proved that when , the solution converges globally to the positive equilibrium time‐exponentially, and when , the solution converges globally to the positive equilibrium time‐algebraically. Here, , and are the coefficients of each term contained in the linear part of the nonlinear term . All convergence rates above are and ‐decay estimates. The comparison principle and low‐frequency and high‐frequency analyses are significantly effective in proofs. Finally, our theoretical results are supported by numerical simulations in different situations.

Thermoelastic laminated beam with nonlocal delay

Thermoelastic laminated beam with nonlocal delay

Interactions of cross-diffusion and nonlocal delay induce spatial vegetation patterning in semi-arid environments

Interactions of cross-diffusion and nonlocal delay induce spatial vegetation patterning in semi-arid environments

Propagation dynamics of diffusive disease models with infectious stages

This paper studies the spatial propagation dynamics in diffusive disease transmission models with infectious stages, which may be nonmonotonic and partially degenerate. Using the solutions of linear heat equations, we transform the system into a new system that contains nonlocal delays and two unknown functions. From this new system, it is easy to obtain an important threshold and the numerical estimation for this threshold is easy. When the initial infection satisfies proper decaying behavior, we find that this threshold is the spreading speed for all infectious subgroups. Then the minimal wave speed of traveling wave solutions is proved to be the threshold, which connects the disease-free state to the endemic case. Moreover, some numerical examples are presented to explore the possible extension.

Nonlocal delay gives rise to vegetation patterns in a vegetation-sand model.

The vegetation pattern generated by aeolian sand movements is a typical type of vegetation patterns in arid and semi-arid areas. This paper presents a vegetation-sand model with nonlocal interaction characterized by an integral term with a kernel function. The instability of the Turing pattern was analyzed and the conditions of stable pattern occurrence were obtained. At the same time, the multiple scales method was applied to obtain the amplitude equations at the critical value of Turing bifurcation. The spatial distributions of vegetation under different delays were obtained by numerical simulation. The results revealed that the vegetation biomass increased as the interaction intensity decreased or as the nonlocal interaction distance increased. We demonstrated that the nonlocal interaction between vegetation and sand is a crucial mechanism for forming vegetation patterns, which provides a theoretical basis for preserving and restoring vegetation.

A geometric construction of travelling waves to Keller-Segel type diffusion system with local or nonlocal delay

A geometric construction of travelling waves to Keller-Segel type diffusion system with local or nonlocal delay

Existence and stability of traveling wavefronts for a discrete diffusion system with nonlocal delay effects

Existence and stability of traveling wavefronts for a discrete diffusion system with nonlocal delay effects

Traveling waves in nonlocal delayed reaction–diffusion bistable equations and applications

In this paper, we are concerned with traveling waves in a class of reaction–diffusion equations with nonlocal delays. By introducing new variables, we transform equations with nonlocal delays to non‐delayed system; the existence of traveling waves is obtained by means of Routh–Hurwitz criterion, and by considering the regularity of the Cauchy problem and using spectral theory, we investigate the global stability of traveling waves and then the uniqueness of wave speeds with the help of upper and lower solution method. Finally, our results are applied to two nonlocal delayed reaction–diffusion equations.

Global dynamics of a reaction-diffusion brucellosis model with spatiotemporal heterogeneity and nonlocal delay

The increase of animal transportation and livestock lead to repeated outbreaks of brucellosis, and its transmission process is extremely complex. According to the clinical symptoms and infectious differences of the diseased sheep, it is further divided into acute infection and chronic infection, and a reaction-diffusion SLICB (susceptible-latent-acute infected-chronic infected-brucella) model with seasonality, spatial heterogeneity and nonlocal delay is proposed to understand the transmission law of brucellosis and analyze its transmission risk. Based on the basic reproduction number of the model, the final development trend of brucellosis transmission among sheep is analyzed by persistence theory. The of the model is numerically calculated by the generalized power method, and simulation analysis shows that: (i) extending the latent period and increasing the random walk rate of infected sheep can effectively prevent brucellosis from developing into an endemic disease; (ii) the greater the density of acute infections, the higher the risk of brucellosis transmission, and the density of infected sheep and the time to reach the stable state will have a large deviation if only consider acute or chronic sheep; (iii) reaching the peak time will be delayed if the peak time of sheep birth is delayed. These results can provide some suggestions for the control of brucellosis.

Thermoelastic diffusion system of type II with nonlocal delay

This article deals with the well-posedness and exponential stability of a thermoelastic diffusion system with delay under Green and Naghdi’s theory that predicts a finite speed for heat propagation and involves no energy dissipation. Through semigroup theory, we prove the well-posedness by the Lumer–Phillips theorem and the exponential stability by exploring the dissipative properties of the linear operator associated with the model using the Gearhart–Huang–Pruss theorem.

Turing pattern of a diffusive predator-prey model with nonlocal delay and herd behavior

In this paper, we introduce nonlocal time delay into a diffusive predator-prey system with herd behavior. Special attention is paid to the modeling of the time delays to incorporate associated nonlocal spatial terms which account for the drift of individuals to their present position from their possible positions at previous times. We first discuss the stability of the positive equilibrium of system without diffusion and give the necessary and sufficient conditions for the stability of positive equilibrium of diffusion system. Then we investigate Turing instability and deduce the amplitude equation of Turing patterns by using multiple scale method. The numerical simulations show the spots pattern and stripes pattern and verify the results of theoretical analysis. Our research reveals that nonlocal time delay plays an important role in the formation of pattern.

Hopf bifurcation in a two-species reaction-diffusion-advection competitive model with nonlocal delay

In this paper, we are concerned with a two-species reaction-diffusion-advection competitive model with nonlocal delay subject to the homogeneous Dirichlet boundary conditions. The existence of the spatially inhomogeneous steady state is studied by using the implicit function theorem. The stability of the inhomogeneous steady state and the associated Hopf bifurcation are investigated by analyzing a non-self-adjoint linear operator. Taking the time delay as the bifurcation parameter, we obtain the critical value of time delay for the Hopf bifurcation and the stability switch phenomenon.

Regular traveling waves for a reaction-diffusion equation with two nonlocal delays

This article concerns regular traveling waves of a reaction-diffusionequation with two nonlocal delays arising from the study of a singlespecies with immature and mature stages and different ages at reproduction. Establishing a necessary condition on the regular traveling waves, we prove the uniqueness of noncritical regular traveling waves,regardless of being monotone or not. Under a quasi-monotone assumption and among other things, we further show that all noncritical monotone traveling waves are exponentially stable, by establishing two comparison theorems and constructing an auxiliary lower equation.

A two-dimensional stochastic fractional non-local diffusion lattice model with delays

The well-posedness, regularity and general stability of solutions to a two-dimensional stochastic non-local delay diffusion lattice system with a time Caputo fractional operator of order [Formula: see text] are investigated in [Formula: see text] spaces for [Formula: see text]. First, the global existence and uniqueness of solutions are established by using a temporally weighted norm, the Burkholder–Davis–Gundy inequality and the Banach fixed point theorem. Then the continuous dependence of solutions on initial values is established in the sense of [Formula: see text]th moment. In particular, the [Formula: see text]th moment Hölder regularities in time and [Formula: see text]th moment general stability, including polynomial and logarithmic stability of solutions, are obtained.

Solitary Waves of the Perturbed KdV Equation with Nonlocal Effects

In this paper, the Korteweg–de Vries (KdV) equation is considered, which is a shallow water wave model in fluid mechanic fields. First the existence of solitary wave solutions for the original KdV equation and geometric singular perturbation theory are recalled. Then the existence of solitary wave solutions is established for the equation with two types of delay convolution kernels by using the method of dynamical system, especially the geometric singular perturbation theory, invariant manifold theory and Melnikov method. Finally, the asymptotic behaviors of solitary wave solution are discussed by applying the asymptotic theory. Moreover, an interesting result is found for the equation without backward diffusion effect, there is no solitary wave solution in the case of local delay, but there is a solitary wave solution in the case of nonlocal delay.

Existence and stability of traveling wavefronts for a nonlocal delay Belousov–Zhabotinskii system

This paper is concerned with the existence and stability of traveling wavefronts for a nonlocal delay Belousov–Zhabotinskii system. We first introduce the new variable and change the two-dimensional system with nonlocal delay into a three-dimensional system without delay. Then the existence of traveling wavefronts is obtained by constructing a pair of suitable super- and sub-solutions and applying Schauder's fixed point theorem. Finally, we establish the exponential stability of traveling wavefronts with large speed by the weighted energy method together with the comparison principle, when the initial perturbation around the traveling wavefront decays exponentially as , but can be arbitrarily large in other locations.

Hopf bifurcation in a reaction-diffusion-advection model with nonlocal delay effect and Dirichlet boundary condition

In this paper, we investigate the dynamics of a reaction-diffusion-advection equation with nonlocal delay effect and Dirichlet boundary condition. The existence of spatially nonhomogeneous steady states and the associated Hopf bifurcation are obtained by using the Lyapunov-Schmidt reduction. We also give applications of the theoretical results to models with a logistic growth rate and a weak Allee growth rate.

Nonlocal delay driven spatiotemporal patterns in a single-species reaction–diffusion model

This paper presents Turing–Hopf bifurcation analysis and the resulting spatiotemporal dynamics in a single-species reaction–diffusion model with nonlocal delay. A linear stability analysis is performed to find the conditions for the occurrence of Turing–Hopf bifurcation. The weakly nonlinear analysis is then employed to derive the amplitude equations which describe the slow-time evolution of the critical Turing and Hopf modes. By the utilization of amplitude equations, stability conditions for different pattern formations can be obtained, helping to reveal the affluent spatiotemporal patterns near the Turing–Hopf bifurcation and predict when and where these patterns can emerge. Also, our analysis shows the role of the choice of initial conditions in pattern formations. Numerical simulations are given to verify the theoretical results.

Topological properties of solution sets for Riemann-Liouville fractional nonlocal delay control systems with noncompact semigroups and applications to approximate controllability

This paper investigates the topological properties of the mild solution set for a control system monitored by Riemann-Liouville fractional nonlocal delay evolution equations with noncompact semigroups. We first obtain the Rδ-property of the mild solution set by employing the method of the Hausdorff measure of noncompactness. Then, we apply the obtained result to show that the presented control problem admits a reachable invariant set under nonlinear perturbations. Furthermore, we apply the obtained result to characterize the approximate controllability of the presented control system. Finally, we provide an example to illustrate the application of the abstract results.