Generalized traveling waves for time-dependent reaction–diffusion systems

2020 ◽  
Author(s):  
Benjamin Ambrosio ◽  
Arnaud Ducrot ◽  
Shigui Ruan
Keyword(s):  
Traveling Waves ◽  
Reaction Diffusion ◽  
Time Dependent ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems

Turing and Benjamin–Feir instability mechanisms in non-autonomous systems

2020 ◽  
Vol 476 (2238) ◽  
pp. 20200003 ◽  
Author(s):  
Robert A. Van Gorder
Keyword(s):  
Autonomous Systems ◽  
Reaction Diffusion ◽  
Reaction Rates ◽  
Time Dependent ◽  
Model Parameters ◽  
Time Varying ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Dependent Model ◽  
Spatio Temporal

The Turing and Benjamin–Feir instabilities are two of the primary instability mechanisms useful for studying the transition from homogeneous states to heterogeneous spatial or spatio-temporal states in reaction–diffusion systems. We consider the case when the underlying reaction–diffusion system is non-autonomous or has a base state which varies in time, as in this case standard approaches, which rely on temporal eigenvalues, break down. We are able to establish respective criteria for the onset of each instability using comparison principles, obtaining inequalities which involve the in general time-dependent model parameters and their time derivatives. In the autonomous limit where the base state is constant in time, our results exactly recover the respective Turing and Benjamin–Feir conditions known in the literature. Our results make the Turing and Benjamin–Feir analysis amenable for a wide collection of applications, and allow one to better understand instabilities emergent due to a variety of non-autonomous mechanisms, including time-varying diffusion coefficients, time-varying reaction rates, time-dependent transitions between reaction kinetics and base states which change in time (such as heteroclinic connections between unique steady states, or limit cycles), to name a few examples.

scholarly journals Non-cooperative Fisher–KPP systems: Asymptotic behavior of traveling waves

2018 ◽  
Vol 28 (06) ◽  
pp. 1067-1104 ◽  
Author(s):  
Léo Girardin
Keyword(s):  
Traveling Waves ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Reaction Diffusion ◽  
Positive Cone ◽  
Precise Description ◽  
Kpp Equation ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Null State

This paper is concerned with non-cooperative parabolic reaction–diffusion systems which share structural similarities with the scalar Fisher–KPP equation. In a previous paper, we established that these systems admit traveling wave solutions whose profiles connect the null state to a compact subset of the positive cone. The main object of this paper is the investigation of a more precise description of these profiles. Non-cooperative KPP systems can model various phenomena where the following three mechanisms occur: local diffusion in space, linear cooperation and superlinear competition.

Computational Study of Traveling Wave Solutions of Isothermal Chemical Systems

2016 ◽  
Vol 19 (5) ◽  
pp. 1461-1472 ◽  
Author(s):  
Yuanwei Qi ◽  
Yi Zhu
Keyword(s):  
Chemical Reaction ◽  
Traveling Waves ◽  
Flow Reactor ◽  
Computational Study ◽  
Chemical Species ◽  
Traveling Wave Solutions ◽  
Reaction Diffusion ◽  
Autocatalytic Reaction ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems

AbstractThis article studies propagating traveling waves in a class of reaction-diffusion systems which model isothermal autocatalytic chemical reactions as well as microbial growth and competition in a flow reactor. In the context of isothermal autocatalytic systems, two different cases will be studied. The first is autocatalytic chemical reaction of order m without decay. The second is chemical reaction of order m with a decay of order n, where m and n are positive integers and m>n≥1. A typical system in autocatalysis is A+2B→3B and B→C involving two chemical species, a reactant A and an auto-catalyst B and C an inert chemical species.The numerical computation gives more accurate estimates on minimum speed of traveling waves for autocatalytic reaction without decay, providing useful insight in the study of stability of traveling waves.For autocatalytic reaction of order m = 2 with linear decay n = 1, which has a particular important role in chemical waves, it is shown numerically that there exist multiple traveling waves with 1, 2 and 3 peaks with certain choices of parameters.

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