scholarly journals Emergence of fluctuating traveling front solutions in macroscopic theory of noisy invasion fronts

2013 ◽  
Vol 87 (1) ◽  
Author(s):  
Baruch Meerson ◽  
Arkady Vilenkin ◽  
Pavel V. Sasorov
Keyword(s):  
Macroscopic Theory ◽  
Invasion Fronts ◽  
Front Solutions ◽  
Traveling Front

A SIMPLIFIED MODEL OF PROPAGATION AND DISSIPATION OF EXCITATION FRONTS

2003 ◽  
Vol 13 (12) ◽  
pp. 3605-3619 ◽  
Author(s):  
V. N. BIKTASHEV
Keyword(s):  
Cardiac Tissue ◽  
Simplified Model ◽  
Excitation Wave ◽  
Temporal Gradient ◽  
Transmembrane Voltage ◽  
Front Solutions ◽  
Traveling Front ◽  
Order Of Magnitude ◽  
Na Inactivation ◽  
Simplified Mathematical Model

An excitation wave in nerve or cardiac tissue may fail to propagate if the temporal gradient of the transmembrane voltage at the front becomes too small to excite the tissue ahead of it. A simplified mathematical model is suggested, that reproduces this phenomenon and has exact traveling front solutions. The spectrum of possible propagation speeds is bounded from below. This causes a front to dissipate if it is not allowed to propagate quickly enough. A crucial role is played by the Na inactivation gates, even if their dynamics are by an order of magnitude slower than the dynamics of the voltage.

scholarly journals Global Stability of Traveling Waves for a More General Nonlocal Reaction-Diffusion Equation

2018 ◽  
Vol 2018 ◽  
pp. 1-13
Author(s):  
Rui Yan ◽  
Guirong Liu
Keyword(s):  
Diffusion Equation ◽  
Global Stability ◽  
Reaction Diffusion ◽  
Rates Of Convergence ◽  
Decay Rates ◽  
Reaction Diffusion Equation ◽  
Weighted Energy Method ◽  
Front Solutions ◽  
Traveling Front ◽  
Nonlocal Reaction

The purpose of this paper is to investigate the global stability of traveling front solutions with noncritical and critical speeds for a more general nonlocal reaction-diffusion equation with or without delay. Our analysis relies on the technical weighted energy method and Fourier transform. Moreover, we can get the rates of convergence and the effect of time-delay on the decay rates of the solutions. Furthermore, according to the stability results, the uniqueness of the traveling front solutions can be proved. Our results generalize and improve the existing results.

scholarly journals Entire solutions for a delayed lattice competitive system

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Rui Yan ◽  
Yang Wang ◽  
Meiping Yao
Keyword(s):  
Comparison Theorem ◽  
Lattice System ◽  
Entire Solutions ◽  
Competitive System ◽  
Front Solutions ◽  
Traveling Front

Abstract In this paper, we investigate the existence of entire solutions for a delayed lattice competitive system. Here the entire solutions are the solutions that exist for all $(n,t)\in \mathbb{Z}\times \mathbb{R}$ ( n , t ) ∈ Z × R . In order to prove the existence, we firstly embed the delayed lattice system into the corresponding larger system, of which the traveling front solutions are identical to those of the delayed lattice system. Then based on the comparison theorem and the sup–sub solutions method, we construct entire solutions which behave as two opposite traveling front solutions moving towards each other from both sides of x-axis and then annihilating. Moreover, our conclusions extend the invading way, which the superior species invade the inferior ones from both sides of x-axis and then the inferior ones extinct, into the lattice and delay case.

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