Spatiotemporal dynamics for a Belousov–Zhabotinsky reaction–diffusion system with nonlocal effects

In this paper, a generalized toxic-phytoplankton–zooplankton system with nonlinear cross-diffusion and non-local competition effect subject to the Neumann boundary conditions is investigated. We mainly show the results on the bifurcation analysis and spatiotemporal patterns by bifurcation theory and numerical simulations. First, we discuss the stability of non-negative constant equilibria and Hopf bifurcation of the corresponding non-spatial system. Then the linear stability analysis presents that the nonlinear cross-diffusion is the crucial mechanism for the occurrence of Turing instability without non-local effect and non-local interaction effect has a significant impact on the stability of positive constant steady state solution. By choosing proper bifurcation parameter, the existence of the steady state bifurcation and non-homogeneous Hopf bifurcation at the positive constant steady state is derived by Lyapunov–Schmidt reduction. Finally, some numerical simulations are executed to demonstrate our theoretical results. Spatial patterns and spatiotemporal patterns appear near steady state-Hopf bifurcation point.

The global dynamics of a nonlocal predator–prey model with mutation are investigated in this paper. First, by building a new comparison principle and constructing monotone iterative sequences, we give the existence of the solution for the corresponding initial boundary value problem. Then the uniqueness and uniformly boundedness of the solution are established by using the fundamental solution and quasi-fundamental solution of the heat equation, Grönwall’s inequality and auxiliary functions. Finally, under the truncation function method, we discuss the asymptotic behavior of the solution. It is worth mentioning that the asymptotic behavior here is different from the conclusion of the classical model and we use more refined analysis and estimation in the research process.

This paper presents a new analytical bar-substrate model for the analysis of an isotropic and homogeneous nanowire embedded in an elastic substrate. A fourth-order strain gradient model based on a thermodynamic approach is employed to represent the small-scale effect (nonlocal effect) while the Gurtin-Murdoch continuum model based on the surface elastic theory is used to account for the size-dependent effect (surface energy effect). The proposed model is derived from the virtual displacement principle, leading to the governing differential equations and its associated natural boundary conditions. The analytical solutions of the sixth-order governing differential equation for the nanowire-substrate element are provided, and were employed in numerical simulations. Two numerical simulations are used to demonstrate the performance and to investigate the characteristics of the fourth-order strain gradient model on nanowire responses, when compared to the classical model and the second-order strain gradient model. The first simulation investigates the influences of nonlocal and surface effects on the responses of a nanowire embedded in an elastic substrate, while the second simulation study assessed the sensitivity of system stiffness on parameters in the nanowire-substrate model.

This paper aims to inspect the nonlocal Erigen’s elasticity model for free vibrations of thermoelastic diffusive sphere in the reference of dual-phase-lag (DPL) model of generalized thermoelasticity. The effect of non-locality and different models of generalized thermoelasticity, i.e. dual-phase-lag (DPL), Lord–Shulman (LS) models and coupled thermoelasticity (CTE) in the presence and absence of diffusion, has been taken under investigation. Due to time harmonic vibrations, the constitutive relations and governing equations are transformed into ordinary differential equations. By applying the stress free thermal boundary conditions, the frequency equation has been taken under investigation for the possible modes of vibrations in compact form. To discover free vibration analysis from frequency equations, the numerical functional iteration technique has been applied for generating numerical data with the assistance of MATLAB software tools. The obtained analytical results have been presented graphically from numerical computations and simulations. The effect of elastic nonlocality on the frequency shift and thermoelastic damping is represented graphically in some cases of interest and analyzed the obtained numerical results.

Surface effect and nonlocal effect are incorporated into classical Euler beam equation to study the static bending behavior of nanobridges. The generalized Young-Laplace equation and core-shell model are used to model the surface effect. The nonlocal effect is introduced in bending moment equation. Results show that a positive surface tension causes the nanobridge more difficult to bend while the nonlocal effect causes the nanobridge easier to bend. Moreover, the influences of the surface effect and the nonlocal effect on the nanobridge bending behavior are found to be dependent of the applied force position. This study indicates the importance of the excitation force position in the characterizations and applications of nanobridges.

PurposeGoal for the present research is investigating the axisymmetric wave propagation behaviors of fluid-filled carbon nanotubes (CNTs) with low slenderness ratios when the nanoscale effects contributed by CNT and fluid flow are considered together.MethodAn elastic shell model for fluid-conveying CNTs is established based on theory of nonlocal elasticity and nonlocal fluid dynamics. The effects of stress non-locality and strain gradient at nanoscale are simulated by applying nonlocal stress and strain gradient theories to CNTs and nonlocal fluid dynamics to fluid flow inside the CNTs, respectively. The equilibrium equations of axisymmetric wave motion in fluid-conveying CNTs are derived. By solving the governing equations, the relationships between wave frequency and all small-scale parameters, as well as the effects caused by fluid flow on different wave modes, are analyzed.ResultsThe numerical simulation indicates that nonlocal stress effects damp first-mode waves but promote propagation of second-mode waves. The strain gradient effect promotes propagation of first-mode waves but has no influence on second-mode waves. The nonlocal fluid effect only causes damping of second-mode waves and has no influence on first-mode waves. Damping caused by nonlocal effects are most affect on waves with short wavelength, and the effect induced by strain gradient almost promotes the propagation of wave with all wavelengths.

A new Reynolds stress anisotropy closure that includes nonlocal and nonequilibrium effects in turbulent flows has been obtained from a recently proposed nonlocal anisotropy formulation. This formulation is based on a new nonlocal derivation of the rapid pressurestrain correlation, which rigorously accounts for nonlocal effects on the anisotropy due to spatial variations in the mean velocity gradient tensor. The present nonlocal and nonequilibrium anisotropy model is obtained as a quasi-linear solution to the anisotropy transport equation, and directly replaces the classical local equilibrium Boussinesq closure in standard two-equation turbulence models. This allows straightforward implementation of the present approach in existing computational frameworks for solving the Reynolds averaged Navier-Stokes equations. Here we present the first assessment of the model in inhomogeneous flows – where nonlocal effects are expected to be important – by comparing with results from direct numerical simulations of turbulent channel flow.

Spatiotemporal dynamics of a diffusive predator-prey model with nonlocal effect and delay

Since spatial memory and nonlocal effect have significant impact on animal movement modeling, we construct a pine wilt disease model with memory-based diffusion and nonlocal effect incorporating Holling-II functional response function to study the control of pine wilt disease. We establish the existence conditions for Turing, Hopf and Turing-Hopf bifurcations, and find that the joint effect of memory-based diffusion coefficient and memory delay can give rise to Turing-Hopf bifurcation. Moreover, the multiple time scales method is extended, showing that it is applicable to the reaction-diffusion system with both nonlocal effect and memory-based diffusion, and the methods for eliminating the secular terms and solving high-order terms for such systems are novelly provided as well in the process of derivation. By using this method, the degenerate Hopf-transcritical bifurcation is derived. Via concisely comparing it with the classical Hopf-zero bifurcation, some interesting results are presented, which suggest that the proposed model exhibits various complex spatiotemporal patterns, such as the six-stable phenomenon, that is, a pair of stable spatially inhomogeneous steady states and periodic solutions induced by Turing-Hopf bifurcation and a pair of stable spatially inhomogeneous large amplitude periodic solutions found numerically, which may provide an alternative approach in explaining the periodic outbreaks of pine wilt disease. Numerical simulations are also performed by adopting scientifically obtained parameters with actual biological meaning. Based on this, the corresponding ecological significance and disease control suggestions have more practical reference value.

This paper is devoted to investigating the pattern dynamics of Lotka–Volterra cooperative system with nonlocal effect and finding some new phenomena. Firstly, by discussing the Turing bifurcation, we build the conditions of Turing instability, which indicates the emergence of Turing patterns in this system. Then, by using multiple scale analysis, we obtain the amplitude equations about different Turing patterns. Furthermore, all possible pattern structures of the model are obtained through some transformation and stability analysis. Finally, two new patterns of the system are given by numerical simulation.

Speed of pattern appearance in reaction-diffusion models: implications in the pattern formation of limb bud mesenchyme cells

We formulate and study a one–dimensional single–species diffusive–delay population model. The time delay is the time taken from birth to maturity. Without diffusion, the delay differential model extends the well–known logistic differential equation by allowing delayed constant birth processes and instantaneous quadratically regulated death processes. This delayed model is known to have simple global dynamics similar to that of the logistic equation. Through the use of a sub/supersolution pair method, we show that the diffusive delay model continues to generate simple global dynamics. This has the important biological implication that quadratically regulated death processes dramatically simplify the growth dynamics. We also consider the possibility of travelling wavefront solutions of the scalar equation for the mature population, connecting the zero solution of that equation with the positive steady state. Our main finding here is that our fronts appear to be all monotone, regardless of the size of the delay. This is in sharp contrast to the frequently reported findings that delay causes a loss of monotonicity, with the front developing a prominent hump in some other delay models.

In this paper, we present an exact closed-form solution for the three-dimensional deformation of a layered magnetoelectroelastic simply-supported plate with the nonlocal effect. The solution is achieved by making use of the pseudo-Stroh formalism and propagator matrix method. Our solution shows, for the first time, that for a homogeneous plate with traction boundary condition applied on its top or bottom surface, the induced stresses are independent of the nonlocal length whilst the displacements increase with increasing nonlocal length. Under displacement boundary condition over a homogeneous or layered plate, all the induced displacements and stresses are functions of the nonlocal length. Our solution further shows that regardless of the Kirchoff or Mindlin plate model, the error of the transverse displacements between the thin plate theory and the three-dimensional solution increases with increasing nonlocal length revealing an important feature for careful application of the thin plate theories towards the problem with nonlocal effect. Various other numerical examples are presented for the extended displacements and stresses in homogeneous elastic plate, piezoelectric plate, magnetostrictive plate, and in sandwich plates made of piezoelectric and magnetostrictive materials. These results should be very useful as benchmarks for future development of approximation plate theories and numerical modeling and simulation with nonlocal effect.

Thermal–mechanical and nonlocal elastic vibration of single-walled carbon nanotubes conveying fluid

User behavior directly influences the speed, spread, and content of information diffusion on social networks. Social networks can be regarded as social, complex, dynamic, and networked systems. Mathematical models are a powerful tool for better understanding the system behavior better, identifying key factors, predicting future behaviors, and developing management policies. This paper presents the susceptible‐exposed‐infected‐recovered‐susceptible with quitted and addicted (SEIRS‐QA) model, a dynamic framework that effectively captures user behavior in social networks over time. By incorporating addiction, popularity, and user awareness, this model offers insights into the dynamic nature of social network usage. The analytical model serves as the foundation for characterizing social networks, considering three key factors that shape their impact. The equilibrium points of the model are studied, and their stability is assessed using the Routh–Hurwitz criteria and comparison principle. The model’s dynamic behavior is influenced by the basic reproductive ratio, which is determined using the next‐generation matrix. To validate the theoretical results, numerical simulations are conducted, confirming the accuracy and reliability of the proposed approach. Through analytical and stability analyses, along with numerical simulations, this study provides a theoretical foundation to advance our understanding of information diffusion dynamics in social networks.