Nonlocal Delay Research Articles

Traveling wave solutions in advection hyperbolic-parabolic system with nonlocal delay

This paper is concerned with a class of advection hyperbolic-parabolic systems with nonlocal delay. We prove that the wave profile is described by a hybrid system that consists of an integral transformation and an ordinary differential equation. By considering the same problem for a properly parameterized system and the continuous dependence of the wave speed on the parameter involved, we obtain the existence and uniqueness of traveling wave solutions in advection hyperbolic-parabolic system with nonlocal delay under bistable assumption. The influence of advection on the propagation speed is also considered.

Stability and Hopf bifurcation of a diffusive Gompertz population model with nonlocal delay effect

Stability and Hopf bifurcation of a diffusive Gompertz population model with nonlocal delay effect

Existence and asymptotic behaviors of traveling waves of a modified vector-disease model

In this paper, we are concerned with the existence and asymptotic behavior of traveling wave fronts in a modified vector-disease model. We establish the existence of traveling wave solutions for the modified vector-disease model without delay, then explore the existence of traveling fronts for the model with a special local delay convolution kernel by employing the geometric singular perturbation theory and the linear chain trick. Finally, we deal with the local stability of the steady states, the existence and asymptotic behaviors of traveling wave solutions for the model with the convolution kernel of a special non-local delay.

Patterns in a nonlocal time-delayed reaction–diffusion equation

In this paper, the existence, stability, and multiplicity of nontrivial (spatially homogeneous or nonhomogeneous) steady-state solution and periodic solutions for a reaction–diffusion model with nonlocal delay effect and Dirichlet/Neumann boundary condition are investigated by using Lyapunov–Schmidt reduction. Moreover, we illustrate our general results by applications to population models with one-dimensional spatial domain.

Existence, uniqueness and asymptotic behavior of traveling wave fronts for a generalized Fisher equation with nonlocal delay

This paper is concerned with existence, uniqueness and asymptotic behavior of traveling wave fronts for a generalized Fisher equation with nonlocal delay. The existence of traveling wave fronts is established by linear chain trick and geometric singular perturbation theory. The strategy is to reformulate the problem as the existence of a heteroclinic connection in R4. The problem is then tackled by using Fenichel’s invariant manifold theory. The asymptotic behavior and uniqueness of traveling wave fronts are also obtained by using standard asymptotic theory and sliding method.

Spatio-temporal patterns in a diffusive model with non-local delay effect

Spatio-temporal patterns in a diffusive model with non-local delay effect

Travelling Wavefronts in Nonlocal Diffusion Equations with Nonlocal Delay Effects

This paper deals with the existence, monotonicity, uniqueness, and asymptotic behaviour of travelling wavefronts for a class of temporally delayed, spatially nonlocal diffusion equations.

Stability and traveling waves in a Beddington-DeAngelis type stage-structured predator-prey reaction-diffusion systems with nonlocal delays and harvesting

The goal of this paper is to study the stability and traveling waves of stage-structured predator-prey reaction-diffusion systems of Beddington-DeAngelis functional response with both nonlocal delays and harvesting. By analyzing the corresponding characteristic equations, the local stability of various equilibria is discussed. We reduce the existence of traveling waves to the existence of a pair of upper-lower solutions by using the cross iteration method and the Schauder’s fixed point theorem. The existence of traveling waves connecting the zero equilibrium and the positive equilibrium is then established by constructing a pair of upper-lower solutions.

Approximate controllability of fractional nonlinear neutral stochastic differential inclusion with nonlocal conditions and infinite delay

In this paper we consider a class of fractional nonlinear neutral stochastic evolution inclusions with nonlocal initial conditions in Hilbert space. Using fractional calculus, stochastic analysis theory, operator semigroups and Bohnenblust–Karlin’s fixed point theorem, a new set of sufficient conditions are formulated and proved for the existence of solutions and the approximate controllability of fractional nonlinear stochastic differential inclusions under the assumption that the associated linear part of the system is approximately controllable. An example is provided to illustrate the theory.

Hopf bifurcation in a reaction-diffusive two-species model with nonlocal delay effect and general functional response

A nonlocal delayed reaction-diffusive two-species model with Dirichlet boundary condition and general functional response is investigated in this paper. Based on the Lyapunov–Schmidt reduction, the existence, bifurcation direction and stability of Hopf bifurcating periodic orbits near the positive spatially nonhomogeneous steady-state solution are obtained, where the time delay is taken as the bifurcation parameter. Moreover, the general results are applied to a diffusive Lotka–Volterra type food-limited population model with nonlocal delay effect, and it is found that diffusion and nonlocal delay can also affect the other dynamic behavior of the system by numerical experiments.

Mild Solution and Constrained Local Controllability of Semilinear Boundary Control Systems

The existence of mild solution and the constrained local controllability of a retarded boundary control system with nonlocal delay condition have been established. The theory of extrapolation spaces is applied to derive the mild solution. Then, the constrained local controllability is established using the generalized open mapping theorem. In the last section, application of the result is shown through examples of control systems represented by hyperbolic partial differential equations.

Traveling wave solutions of a reaction-diffusion predator-prey model

This paper is concerned with the dynamics of traveling wave solutions for a reaction-diffusion predator-prey model with a nonlocal delay. By using Schauder's fixed point theorem, we establish the existence result of a traveling wave solution connecting two steady states by constructing a pair of upper-lower solutions which are easy to construct in practice. We also investigate the asymptotic behavior of traveling wave solutions by employing the standard asymptotic theory.

Stability and traveling fronts for a food chain reaction-diffusion systems with nonlocal delays

This paper is purported to investigate a food chain reaction-diffusion predator-prey system with nonlocal delays in a bounded domain with no flux boundary condition. We investigate the global stability and find the sufficient conditions of global stability of the unique positive equilibrium for this system. The derived results show that delays often restrain stability. Using the method of linearizing this system, we see that the zero equilibrium is unstable. Moreover, by constructing upper-lower solutions, we find that there exist traveling wavefronts which connect the zero equilibrium and positive equilibrium when the wave speed is large enough and the prey intrinsic growth rate and the death rate of the predator are relatively big.

Bifurcation and stability of a Mimura–Tsujikawa model with nonlocal delay effect

In this paper, we investigate a Mimura–Tsujikawa model with nonlocal delay effect under the homogeneous Neumann boundary condition. By using Lyapunov–Schmidt reduction, we investigate the existence, multiplicity, stability, and Hopf bifurcation of nontrivial steady‐state solutions bifurcating from the nonzero steady‐state solution. Moreover, we illustrate our general results by applications to models with a one‐dimensional spatial domain. Copyright © 2016 John Wiley & Sons, Ltd.

Existence and uniqueness of periodic solutions for parabolic equation with nonlocal delay

This paper deals with the existence and uniqueness of time periodic solutions for the general periodic parabolic equation boundary problem with nonlocal delay. We apply operator semigroup theory and monotone iterative technique of lower and upper solutions to obtain the existence and uniqueness of ω-periodic mild solutions of some abstract evolution equation under some quasimonotone conditions. In the end, applying our abstract results to parabolic equation with nonlocal delay, we get the existence and uniqueness of ω-periodic solution, which generalize the recent conclusions on this issue.

Complex dynamic behavior of a rumor propagation model with spatial-temporal diffusion terms

Rumor propagation as a typical form of social communication in online social networks has had a significant negative impact on a harmonious and stable society. With the rapid development of mobile communication equipments, traditional rumor propagation models, which depend on ordinary differential equations (ODE), may not be suitable for describing rumor propagation in an online social network. In this paper, based on reaction-diffusion equations, we propose a novel epidemic-like model with both discrete and nonlocal delays for investigating the spatial-temporal dynamics of rumor propagation. By analyzing the corresponding characteristic equations of this model, the local stability conditions of a boundary equilibrium point and a positive equilibrium point are established. By applying the linear approximation method of nonlinear systems, sufficient conditions are derived for the existence of Hopf bifurcation at the above two kinds of equilibrium points. Moreover, a sensitivity analysis method based on the density of spreading users is proposed, and then in theoretical and experimental aspect we identify some sensitive parameters in the process of rumor propagation. Finally, numerical simulations are performed to illustrate the theoretical results.

Hopf bifurcation in a diffusive Lotka–Volterra type system with nonlocal delay effect

The dynamics of a diffusive Lotka–Volterra type model for two species with nonlocal delay effect and Dirichlet boundary conditions is investigated in this paper. The existence and multiplicity of spatially nonhomogeneous steady-state solutions are obtained by means of Lyapunov–Schmidt reduction. The stability of spatially nonhomogeneous steady-state solutions and the existence of Hopf bifurcation with the changes of the time delay are obtained by analyzing the distribution of eigenvalues of the infinitesimal generator associated with the linearized system. By the normal form theory and the center manifold reduction, the stability and bifurcation direction of Hopf bifurcating periodic orbits are derived. Finally, our theoretical results are illustrated by a model with homogeneous kernels and one-dimensional spatial domain.

Stability of traveling fronts in a population model with nonlocal delay and advection

In this paper, we are concerned with the stability of traveling fronts in a population model with nonlocal delay and advection under the large initial perturbation (i.e. the initial perturbation around the traveling wave decays exponentially as $x \rightarrow-\infty$, but it can be arbitrarily large in other locations). The globally exponential stability of traveling fronts is established by the weighted-energy method combining with comparison principle, including even the slower waves whose wave speed are close to the critical speed.

Stability and bifurcation of a reaction–diffusion predator–prey model with non-local delay and Michaelis–Menten-type prey-harvesting

We investigate a reaction–diffusion predator–prey system with homogeneous Neumann boundary conditions and non-local delay due to predator gestation. By analysing the corresponding characteristic equations, we establish the local stability of the positive steady state. We also discuss the existence of Hopf bifurcations at the positive steady state. We derive sufficient conditions for the global stability of the positive steady state of the proposed problem using the Lyapunov functional. Numerical simulations illustrate the results and reveal that as the discrete delay τ increases, the species may tend to extinction. Changes in the harvesting effort E and non-local delay β can transform an unstable system into a stable one.

Spatial dynamics of a diffusive predator-prey model with stage structure

In this paper, we propose a nonlocal and time-delayed reaction-diffusion predator-prey model with stage structure. It is assumed that prey individuals undergo two stages, immature and mature, and the conversion of consumed prey biomass into predator biomass has a retardation. In terms of the principal eigenvalue of nonlocal elliptic eigenvalue problems, we establish the uniform persistence and global extinction for the model. In particular, the uniform persistence implies the existence of positive steady states. Finally, we investigate a specially spatially homogeneous predator-prey system and show the complicated dynamics of the system due to the non-local delay in the prey equation.