scholarly journals On the Existence and Stability of Fast Traveling Waves in a Doubly Diffusive FitzHugh--Nagumo System

2018 ◽  
Vol 17 (1) ◽  
pp. 754-787 ◽  
Author(s):  
Paul Cornwell ◽  
Christopher K. R. T. Jones
Keyword(s):  
Traveling Waves ◽  
Existence And Stability

scholarly journals Hyperbolic traveling waves driven by growth

2014 ◽  
Vol 24 (06) ◽  
pp. 1165-1195 ◽  
Author(s):  
Emeric Bouin ◽  
Vincent Calvez ◽  
Grégoire Nadin
Keyword(s):  
Nonlinear Stability ◽  
Traveling Waves ◽  
Hyperbolic Model ◽  
Kpp Equation ◽  
Transition Parameter ◽  
Minimal Speed ◽  
Traveling Fronts ◽  
Traveling Front ◽  
Maximal Speed ◽  
Existence And Stability

We perform the analysis of a hyperbolic model which is the analog of the Fisher-KPP equation. This model accounts for particles that move at maximal speed ϵ-1 (ϵ > 0), and proliferate according to a reaction term of monostable type. We study the existence and stability of traveling fronts. We exhibit a transition depending on the parameter ϵ: for small ϵ the behavior is essentially the same as for the diffusive Fisher-KPP equation. However, for large ϵ the traveling front with minimal speed is discontinuous and travels at the maximal speed ϵ-1. The traveling fronts with minimal speed are linearly stable in weighted L2 spaces. We also prove local nonlinear stability of the traveling front with minimal speed when ϵ is smaller than the transition parameter.

Existence and stability of invasion traveling waves for a competition system with random vs. nonlocal dispersals

2019 ◽  
Vol 12 (01) ◽  
pp. 1950004
Author(s):  
Jiao Wang ◽  
Zhixian Yu ◽  
Yanling Meng
Keyword(s):  
Traveling Waves ◽  
Weighted Spaces ◽  
Nonlocal Dispersal ◽  
Asymptotic Behaviors ◽  
Competition System ◽  
Weighted Energy ◽  
Existence And Stability ◽  
Exponentially Stable ◽  
Exponentially Weighted

The purpose of this paper is to investigate asymptotic behaviors of the solutions for a competition system with random vs. nonlocal dispersal. We first prove the existence of invasion traveling waves via using the theory of asymptotic speeds of spread. Then we prove the invasion traveling waves are exponentially stable as perturbation in some exponentially weighted spaces by using the weighted energy and the squeezing technique.

scholarly journals Periodically kicked feedforward chains of simple excitable FitzHugh-Nagumo neurons

2021 ◽  
Author(s):  
B. Ambrosio ◽  
S.M. Mintchev
Keyword(s):  
Numerical Investigation ◽  
Traveling Waves ◽  
Small Amplitude ◽  
Mixed Mode ◽  
Filtration Property ◽  
Periodic Traveling Waves ◽  
Mixed Mode Oscillations ◽  
Mode Oscillation ◽  
Existence And Stability ◽  
Mixed Mode Oscillation

Abstract This article communicates results on regular depolarization cascades in periodically-kicked feedforward chains of excitable two-dimensional FitzHugh-Nagumo systems driven by sufficiently strong excitatory forcing at the front node. The study documents a parameter exploration by way of changes to the forcing period, upon which the dynamics undergoes a transition from simple depolarization to more complex behavior, including the emergence of mixed-mode oscillations. Both rigorous studies and careful numerical observations are presented. In particular, we provide rigorous proofs for existence and stability of periodic traveling waves of depolarization, as well as the existence and propagation of a simple mixed-mode oscillation that features depolarization and refraction in alternating fashion. Detailed numerical investigation reveals a mechanism for the emergence of complex mixed-mode oscillations featuring a potentially high number of large amplitude voltage spikes interspersed by an occasional small amplitude reset that fails to cross threshold. Further careful numerical investigation provides insights into the propagation of this complex phenomenology in the downstream, where we see an effective filtration property of the network; the latter amounts to a successive reduction in the complexity of mixed-mode oscillations down the chain.

scholarly journals Analytical and numerical investigation of traveling waves for the Allen–Cahn model with relaxation

2016 ◽  
Vol 26 (05) ◽  
pp. 931-985 ◽  
Author(s):  
Corrado Lattanzio ◽  
Corrado Mascia ◽  
Ramon G. Plaza ◽  
Chiara Simeoni
Keyword(s):  
Numerical Investigation ◽  
Nonlinear Stability ◽  
Traveling Waves ◽  
Propagation Speed ◽  
Stable States ◽  
Cahn Equation ◽  
Numerical Studies ◽  
Traveling Fronts ◽  
Parabolic Case ◽  
Existence And Stability

A modification of the parabolic Allen–Cahn equation, determined by the substitution of Fick’s diffusion law with a relaxation relation of Cattaneo–Maxwell type, is considered. The analysis concentrates on traveling fronts connecting the two stable states of the model, investigating both the aspects of existence and stability. The main contribution is the proof of the nonlinear stability of the wave, as a consequence of detailed spectral and linearized analyses. In addition, numerical studies are performed in order to determine the propagation speed, to compare it to the speed for the parabolic case, and to explore the dynamics of large perturbations of the front.

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