Asymptotic behaviour of a reaction–diffusion model with a quiescent stage

2007 ◽  
Vol 463 (2080) ◽  
pp. 1029-1043 ◽  
Author(s):  
Kate Fang Zhang ◽  
Xiao-Qiang Zhao
Keyword(s):  
Asymptotic Behaviour ◽  
Diffusion Model ◽  
Wave Speed ◽  
Global Attractivity ◽  
Travelling Waves ◽  
Reaction Diffusion ◽  
Reaction Diffusion Model ◽  
Asymptotic Speed ◽  
Minimal Wave Speed ◽  
Quiescent Stage

This paper is devoted to the investigation of the asymptotic behaviour for a reaction–diffusion model with a quiescent stage. We first establish the existence of the asymptotic speed of spread and show that it coincides with the minimal wave speed for monotone travelling waves. Then we obtain a threshold result on the global attractivity of either zero or positive steady state in the case where the spatial domain is bounded.

scholarly journals Propagation dynamics of a nonlocal time-space periodic reaction-diffusion model with delay

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Ning Wang ◽  
Zhi-Cheng Wang
Keyword(s):  
Diffusion Model ◽  
Critical Speed ◽  
Wave Speed ◽  
Upper And Lower Solutions ◽  
Reaction Diffusion ◽  
Linear Operators ◽  
Time Space ◽  
Reaction Diffusion Model ◽  
Periodic Reaction ◽  
Asymptotic Speed

<p style='text-indent:20px;'>This paper is concerned with a nonlocal time-space periodic reaction diffusion model with age structure. We first prove the existence and global attractivity of time-space periodic solution for the model. Next, by a family of principal eigenvalues associated with linear operators, we characterize the asymptotic speed of spread of the model in the monotone and non-monotone cases. Furthermore, we introduce a notion of transition semi-waves for the model, and then by constructing appropriate upper and lower solutions, and using the results of the asymptotic speed of spread, we show that transition semi-waves of the model in the non-monotone case exist when their wave speed is above a critical speed, and transition semi-waves do not exist anymore when their wave speed is less than the critical speed. It turns out that the asymptotic speed of spread coincides with the critical wave speed of transition semi-waves in the non-monotone case. In addition, we show that the obtained transition semi-waves are actually transition waves in the monotone case. Finally, numerical simulations for various cases are carried out to support our theoretical results.</p>

A periodic reaction–diffusion system modelling man–environment–man epidemics

2017 ◽  
Vol 10 (05) ◽  
pp. 1750074 ◽  
Author(s):  
Zhenguo Bai
Keyword(s):  
Wave Speed ◽  
Global Attractivity ◽  
Travelling Waves ◽  
Spatial Dynamics ◽  
Reaction Diffusion ◽  
Positive Periodic Solution ◽  
System Modelling ◽  
Periodic Reaction ◽  
Minimal Wave Speed ◽  
Disease Spreading

The purpose of this work is to study the spatial dynamics of a periodic reaction–diffusion epidemic model arising from the spread of oral–faecal transmitted diseases. We first show that the disease spreading speed is coincident with the minimal wave speed for monotone periodic travelling waves. Then we obtain a threshold result on the global attractivity of either zero or the positive periodic solution in a bounded spatial domain.

scholarly journals Existence of Traveling Wave Solutions for Cholera Model

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Tianran Zhang ◽  
Qingming Gou ◽  
Xiaoli Wang
Keyword(s):  
Traveling Wave ◽  
Wave Speed ◽  
Shooting Method ◽  
Reproduction Number ◽  
Traveling Wave Solutions ◽  
Reaction Diffusion ◽  
Traveling Wave Solution ◽  
Reaction Diffusion Model ◽  
Minimal Wave Speed ◽  
The Basic Reproduction Number

To investigate the spreading speed of cholera, Codeço’s cholera model (2001) is developed by a reaction-diffusion model that incorporates both indirect environment-to-human and direct human-to-human transmissions and the pathogen diffusion. The two transmission incidences are supposed to be saturated with infective density and pathogen density. The basic reproduction numberR0is defined and the formula for minimal wave speedc*is given. It is proved by shooting method that there exists a traveling wave solution with speedcfor cholera model if and only ifc≥c*.

Asymptotic behaviour, uniqueness and stability of coexistence states of a non-cooperative reaction–diffusion model of nuclear reactors

2010 ◽  
Vol 140 (1) ◽  
pp. 189-201 ◽  
Author(s):  
Rui Peng ◽  
Dong Wei ◽  
Guoying Yang
Keyword(s):  
Asymptotic Behaviour ◽  
Diffusion Model ◽  
Blow Up ◽  
Nuclear Reactors ◽  
Reaction Diffusion ◽  
Fast Neutrons ◽  
Fuel Temperature ◽  
Reaction Diffusion Model ◽  
Coexistence States ◽  
Spatial Dimensions

We investigate a non-cooperative reaction-diffusion model arising in the theory of nuclear reactors and are concerned with the associated steady-state problem. We determine the asymptotic behaviour of the coexistence states near the point of bifurcation from infinity, which exhibits the following very interesting spatial blow-up pattern: when the fuel temperature reaches a certain value, the free fast neutrons undergoing nuclear reaction will blow up in each spatial point of the interior of the reactor. Without any restriction on spatial dimensions, we also discuss the uniqueness and stability of the coexistence states. Our results complement and sharpen those derived in two recent works by Arioli and Lóopez-Gómez.

Sign in / Sign up

Export Citation Format

Share Document