Stability for Monostable Wave Fronts of Delayed Lattice Differential Equations

Hsu, Cheng Hsiung; Lin, Ja-Hon; Yang, Tsung-Ying

doi:10.1007/s10884-015-9447-9

Cited by 12 publications

(3 citation statements)

References 23 publications

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“…In particular, the problem of determining local asymptotic stability of a known solution to (1.1) greatly increases in complexity when moving from the finite-dimensional to the infinite-dimensional setting. In the case when Λ = Z some authors have circumvented this difficulty through the use of comparison principles to obtain stability of traveling waves solutions to LDSs [7,21,29,37,44,45]. The problem with this method is that it requires a number of assumptions on the model and can only capture a specific class of initial conditions that converge back to the given solution.…”

Section: Introductionmentioning

confidence: 99%

Stable periodic solutions to Lambda-Omega lattice dynamical systems

Bramburger

2020

Journal of Differential Equations

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In this manuscript we consider the stability of periodic solutions to Lambda-Omega lattice dynamical systems. In particular, we show that an appropriate ansatz casts the lattice dynamical system as an infinite-dimensional fast-slow differential equation. In a neighbourhood of the periodic solution an invariant slow manifold is proven to exist, and that this slow manifold is uniformly exponentially attracting. The dynamics of solutions on the slow manifold become significantly more complicated and require a more delicate treatment. We present sufficient conditions to guarantee convergence on the slow manifold which is algebraic, as opposed to exponential, in the slow-time variable. Of particular interest to our work in this manuscript is the stability of a rotating wave solution, recently found to exist in the Lambda-Omega systems studied herein.

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Section: Introductionmentioning

confidence: 99%

Stable periodic solutions to Lambda-Omega lattice dynamical systems

Bramburger

2020

Journal of Differential Equations

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“…Lattice differential equations (LDEs) are systems of ordinary differential equations with a discrete spatial structure, which can naturally arise in various fields, such as image processing, neural networks, patten recognition and chemical reaction theory. These can be seen in [9,14,29,37] and the references therein. Recently, there is a particular interest on studying the species population living in a patchy environment consisting of all integer nodes, see [7,8,34,35].…”

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confidence: 97%

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

Cheng

2021

era

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<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>

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“…To the best of our knowledge, there only have limited papers considering the stability problems of traveling wave solutions for lattice differential systems, cf. [11,13,25]. Hence, this gives us the main motivation to consider the stability problem for the general system (1).…”

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confidence: 99%

Stability analysis of traveling wave solutions for lattice reaction-diffusion equations

Hsu¹,

Lin²

2020

Discrete &Amp; Continuous Dynamical Systems - B

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In this work, we establish a framework to study the stability of traveling wave solutions for some lattice reaction-diffusion equations. The systems arise from epidemic, biological and many other applied models. Applying different kinds of comparison theorems, we show that all solutions of the Cauchy problem for the lattice differential equations converge exponentially to the traveling wave solutions provided that the initial perturbations around the traveling wave solutions belonging to suitable spaces. Our results can be applied to various discrete reaction-diffusion systems, e.g., the discrete multi-species Lotka-Volterra cooperative model, discrete epidemic model, three-species Lotka-Volterra competitive model, etc.

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Stability for Monostable Wave Fronts of Delayed Lattice Differential Equations

Cited by 12 publications

References 23 publications

Stable periodic solutions to Lambda-Omega lattice dynamical systems

Stable periodic solutions to Lambda-Omega lattice dynamical systems

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

Stability analysis of traveling wave solutions for lattice reaction-diffusion equations

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