scholarly journals Stability of traveling wavefronts for a 2D lattice dynamical system arising in a diffusive population model

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Haiqin Zhao
Keyword(s):  
Dynamical System ◽  
Existence And Uniqueness ◽  
Population Model ◽  
Wave Speed ◽  
Exponential Convergence ◽  
Nonlinear Anal ◽  
Lattice Dynamical System ◽  
Two Component ◽  
Exponentially Stable ◽  
The Stability

Abstract This paper is concerned with the traveling wavefronts of a 2D two-component lattice dynamical system. This problem arises in the modeling of a species with mobile and stationary subpopulations in an environment in which the habitat is two-dimensional and divided into countable niches. The existence and uniqueness of the traveling wavefronts of this system have been studied in (Zhao and Wu in Nonlinear Anal., Real World Appl. 12: 1178–1191, 2011). However, the stability of the traveling wavefronts remains unsolved. In this paper, we show that all noncritical traveling wavefronts with given direction of propagation and wave speed are exponentially stable in time. In particular, we obtain the exponential convergence rate.

scholarly journals Global stability of traveling wave fronts in a two-dimensional lattice dynamical system with global interaction

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Cui-Ping Cheng ◽  
Ruo-Fan An
Keyword(s):  
Dynamical System ◽  
Traveling Wave ◽  
Single Species ◽  
Wave Fronts ◽  
Dimensional Lattice ◽  
Weighted Energy Method ◽  
Traveling Wave Fronts ◽  
Weighted Energy ◽  
Lattice Dynamical System ◽  
Exponentially Stable

<p style='text-indent:20px;'>This paper is concerned with the traveling wave fronts for a lattice dynamical system with global interaction, which arises in a single species in a 2D patchy environment with infinite number of patches connected locally by diffusion and global interaction by delay. We prove that all non-critical traveling wave fronts are globally exponentially stable in time, and the critical traveling wave fronts are globally algebraically stable by the weighted energy method combined with the comparison principle and the discrete Fourier transform.</p>

scholarly journals Minimal Wave Speed in a Delayed Lattice Dynamical System with Competitive Nonlinearity

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Fuzhen Wu
Keyword(s):  
Dynamical System ◽  
Traveling Wave ◽  
Wave Speed ◽  
Upper And Lower Solutions ◽  
Traveling Wave Solutions ◽  
Partial Ordering ◽  
Wave Solutions ◽  
Lattice Dynamical System ◽  
Lattice Dynamical Systems ◽  
Minimal Wave Speed

This paper deals with the minimal wave speed of delayed lattice dynamical systems without monotonicity in the sense of standard partial ordering in R2. By constructing upper and lower solutions appealing to the exponential ordering, we prove the existence of traveling wave solutions if the wave speed is not smaller than some threshold. The nonexistence of traveling wave solutions is obtained when the wave speed is smaller than the threshold. Therefore, we confirm the threshold is the minimal wave speed, which completes the known results.

Stability of Traveling Wavefronts for a Two-Component Lattice Dynamical System Arising in Competition Models

2018 ◽  
Vol 61 (2) ◽  
pp. 423-437 ◽  
Author(s):  
Guo-Bao Zhang ◽  
Ge Tian
Keyword(s):  
Dynamical System ◽  
Initial Perturbation ◽  
Asymptotically Stable ◽  
One Dimensional ◽  
Weighted Energy ◽  
Competition Models ◽  
Lattice Dynamical System ◽  
Two Component ◽  
Spatial Lattice ◽  
Large Speed

AbstractIn this paper, we study a two-component Lotka–Volterra competition systemon a one-dimensional spatial lattice. By the comparison principle, together with the weighted energy, we prove that the traveling wavefronts with large speed are exponentially asymptotically stable, when the initial perturbation around the traveling wavefronts decays exponentially as j + ct → −∞, where j ∈ , t > 0, but the initial perturbation can be arbitrarily large on other locations. This partially answers an open problem by J.-S. Guo and C.-H.Wu.

HYPERBOLIC HOMOCLINIC POINTS OF ℤd-ACTIONS IN LATTICE DYNAMICAL SYSTEMS

1996 ◽  
Vol 06 (06) ◽  
pp. 1059-1075 ◽  
Author(s):  
V. S. AFRAIMOVICH ◽  
SHUI-NEE CHOW ◽  
WENXIAN SHEN
Keyword(s):  
Dynamical System ◽  
Homoclinic Solution ◽  
Homoclinic Point ◽  
Equilibrium Solutions ◽  
Homoclinic Points ◽  
Lattice Dynamical System ◽  
Spatial Coordinates ◽  
The Stability ◽  
Homoclinic And Heteroclinic Points ◽  
Heteroclinic Points

We study ℤd action on a set of equilibrium solutions of a lattice dynamical system, i.e., a system with discrete spatial variables, and the stability and hyperbolicity of the equilibrium solutions. Complicated behavior of ℤd-action corresponds to the existence of an infinite number of equilibrium solutions which are randomly situated along spatial coordinates. We prove that the existence of a homoclinic point of a ℤd-action implies complicated behavior, provided the hyperbolicity of the homoclinic solution with respect to the lattice dynamical system (this is a generalization of the previous work of the first two authors). Similar result holds for hyperbolic partially homoclinic and heteroclinic points. We show the equivalence of stability for any equilibrium solutions and the equivalence of hyperbolicity for homoclinic points under various norms.

Global stability analysis of fractional-order gene regulatory networks with time delay

2019 ◽  
Vol 12 (06) ◽  
pp. 1950067 ◽  
Author(s):  
Zhaohua Wu ◽  
Zhiming Wang ◽  
Tiejun Zhou
Keyword(s):  
Time Delay ◽  
Fractional Order ◽  
Gene Regulatory Networks ◽  
Existence And Uniqueness ◽  
Regulatory Networks ◽  
Lyapunov Method ◽  
Regulation Mechanism ◽  
Global Stability Analysis ◽  
Gene Regulatory ◽  
The Stability

Fractional-order gene regulatory networks with time delay (DFGRNs) have proven that they are more suitable to model gene regulation mechanism than integer-order. In this paper, a novel DFGRN is proposed. The existence and uniqueness of the equilibrium point for the DFGRN are proved under certain conditions. On this basis, the conditions on the global asymptotic stability are established by using the Lyapunov method and comparison theorem for the DFGRN, and the stability conditions are dependent on the fractional-order [Formula: see text]. Finally, numerical simulations show that the obtained results are reasonable.

scholarly journals A Fractional SAIDR Model in the Frame of Atangana–Baleanu Derivative

2021 ◽  
Vol 5 (2) ◽  
pp. 32
Author(s):  
Esmehan Uçar ◽  
Sümeyra Uçar ◽  
Fırat Evirgen ◽  
Necati Özdemir
Keyword(s):  
Existence And Uniqueness ◽  
Laplace Transformation ◽  
Cell Phones ◽  
Mobile Network ◽  
The Other ◽  
Banach Fixed Point Theorem ◽  
Propagation Models ◽  
Computer Worms ◽  
Computer Viruses ◽  
The Stability

It is possible to produce mobile phone worms, which are computer viruses with the ability to command the running of cell phones by taking advantage of their flaws, to be transmitted from one device to the other with increasing numbers. In our day, one of the services to gain currency for circulating these malignant worms is SMS. The distinctions of computers from mobile devices render the existing propagation models of computer worms unable to start operating instantaneously in the mobile network, and this is particularly valid for the SMS framework. The susceptible–affected–infectious–suspended–recovered model with a classical derivative (abbreviated as SAIDR) was coined by Xiao et al., (2017) in order to correctly estimate the spread of worms by means of SMS. This study is the first to implement an Atangana–Baleanu (AB) derivative in association with the fractional SAIDR model, depending upon the SAIDR model. The existence and uniqueness of the drinking model solutions together with the stability analysis are shown through the Banach fixed point theorem. The special solution of the model is investigated using the Laplace transformation and then we present a set of numeric graphics by varying the fractional-order θ with the intention of showing the effectiveness of the fractional derivative.

scholarly journals Upper semicontinuity of the attractor for lattice dynamical systems of partly dissipative reaction diffusion systems

2005 ◽  
Vol 2005 (3) ◽  
pp. 273-288 ◽  
Author(s):  
Ahmed Y. Abdallah
Keyword(s):  
Dynamical System ◽  
Hilbert Space ◽  
Upper Semicontinuity ◽  
Reaction Diffusion ◽  
Dimensional Lattice ◽  
Reaction Diffusion Systems ◽  
Infinite Dimensional ◽  
Diffusion Systems ◽  
Lattice Dynamical System ◽  
Lattice Dynamical Systems

We investigate the existence of a global attractor and its upper semicontinuity for the infinite-dimensional lattice dynamical system of a partly dissipative reaction diffusion system in the Hilbert spacel2×l2. Such a system is similar to the discretized FitzHugh-Nagumo system in neurobiology, which is an adequate justification for its study.

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