Wave propagation for a reaction-diffusion model with a quiescent stage on a 2D spatial lattice
type: Item
instance of: scholarly article
title: Wave propagation for a reaction-diffusion model with a quiescent stage on a 2D spatial lattice
zbMATH Open document ID: 1243.34013
author: Haiqin Zhao
author: Shi-Liang Wu
published in: Nonlinear Analysis. Real World Applications
publication date: 2011-01-18T00:00:00Z
review text: The following 2D lattice spatially discrete reaction-diffusion system with a quiescent stage is considered \[ \begin{cases} {\mathrm d u_{i,j}(t) \over \mathrm d t} = (\Delta_2 u)_{i,j} + f(u_{i,j}(t)) - \gamma_1 u_{i,j}(t) + \gamma_2 v_{i,j}(t), \\ {\mathrm d v_{i,j}(t) \over \mathrm d t} = \gamma_1 u_{i,j}(t) - \gamma_2 v_{i,j}(t), \end{cases} \] where \(u_{i,j}(t), v_{i,j}(t) \in \mathbb{R}, i,j \in \mathbb{Z}, t>0\) and \((\Delta_2 u)_{i,j}:= d[u_{i+1,j} + u_{i-1,j} + u_{i,j+1} + u_{i, j-1} - 4u_{i,j}]\), and finally \(f\) is a mono-stable reaction term. The aim is to investigate the phenomenon of the biological invasion of a single-species population with two separate mobile and stationary parts. To this aim, for every direction \(\theta \in \mathbb R\), the existence of a minimal wave speed \(c^*(\theta)\) is proved, such that traveling wave solutions exist if and only if the wave speed is greater or equal than \(c^*(\theta)\). Moreover, the asymptotic behavior, monotonicity and uniqueness of traveling wave solutions are also studied. Finally, the effects of the direction of propagation \(\theta\) and the transfer rates \(\gamma_1, \gamma_2\) on the minimal wave speed are discussed.
reviewed by: Cristina Marcelli
Mathematics Subject Classification ID: 34A33
Mathematics Subject Classification ID: 34B40
Mathematics Subject Classification ID: 35C07
Mathematics Subject Classification ID: 92D25
zbMATH DE Number: 5838196
zbMATH Keywords: spatially discrete reaction-diffusion systems
zbMATH Keywords: traveling wave fronts
zbMATH Keywords: quiescent stage
Wikidata QID: Q56763676
MaRDI profile type: Publication
full work available at URL: j.nonrwa.2010.09.011
OpenAlex ID: W1987536149
cites work: Q3738973
cites work: Q5474668
cites work: Mathematical biology. Vol. 2: Spatial models and biomedical applications.
cites work: Q4659859
cites work: Biological invasion of an organism with separate mobile and stationary states: Modelling and analysis
cites work: Q3378908
cites work: Asymptotic behaviour of a reaction–diffusion model with a quiescent stage
cites work: Monotonicity and uniqueness of traveling waves for a reaction-diffusion model with a quiescent stage
cites work: Q4440650
cites work: Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics
cites work: Existence and uniqueness of traveling waves for a monostable 2-D lattice dynamical system
cites work: Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem
cites work: Q4440653
cites work: Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction--diffusion models.
cites work: Traveling fronts in monostable equations with nonlocal delayed effects
cites work: Asymptotic speed of propagation of wave fronts in a lattice delay differential equation with global interaction
cites work: Traveling wave fronts of reaction-diffusion systems with delay
cites work: Global asymptotic stability of bistable traveling fronts in reaction-diffusion systems and their applications to biological models
cites work: Uniqueness of non-monotone traveling waves for delayed reaction-diffusion equations
cites work: Oscillatory waves in reaction-diffusion equations with nonlocal delay and crossing-monostability
cites work: Asymptotic stability of traveling wave fronts in nonlocal reaction-diffusion equations with delay
cites work: Existence and uniqueness of traveling waves for non-monotone integral equations with application
cites work: Fisher waves in an epidemic model
cites work: Uniqueness of travelling waves for nonlocal monostable equations
DOI: 10.1016/J.NONRWA.2010.09.011
description: scientific article; zbMATH DE number 5838196