Traveling waves and spreading properties for a reaction-diffusion competition model with seasonal succession*

Nonlinearity ◽  
2021 ◽  
Vol 35 (1) ◽  
pp. 134-169
Author(s):  
Mingxin Wang ◽  
Qianying Zhang ◽  
Xiao-Qiang Zhao
Keyword(s):  
Upper And Lower Solutions ◽  
Seasonal Succession ◽  
Traveling Wave Solutions ◽  
Reaction Diffusion ◽  
Positive Periodic Solution ◽  
Competition Model ◽  
Competition Condition ◽  
Whole Space ◽  
Comparison Arguments ◽  
Globally Stable

Abstract In this paper, we investigate the propagation dynamics of a reaction–diffusion competition model with seasonal succession in the whole space. Under the weak competition condition, the corresponding kinetic system admits a globally stable positive periodic solution ( u ^ ( t ) , v ^ ( t ) ) . By the method of upper and lower solutions and the Schauder fixed point theorem, we first obtain the existence and nonexistence of traveling wave solutions connecting (0, 0) to ( u ^ ( t ) , v ^ ( t ) ) . Then we use the comparison arguments to establish the spreading properties for a large class of solutions.

scholarly journals Traveling Wave Solutions for Epidemic Cholera Model with Disease-Related Death

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Tianran Zhang ◽  
Qingming Gou
Keyword(s):  
Laplace Transform ◽  
Traveling Wave ◽  
Upper And Lower Solutions ◽  
Traveling Wave Solutions ◽  
Reaction Diffusion ◽  
Reaction Diffusion Systems ◽  
Wave Solutions ◽  
Diffusion Systems ◽  
Prior Estimate ◽  
Minimal Wave Speed

Based on Codeço’s cholera model (2001), an epidemic cholera model that incorporates the pathogen diffusion and disease-related death is proposed. The formula for minimal wave speedc∗is given. To prove the existence of traveling wave solutions, an invariant cone is constructed by upper and lower solutions and Schauder’s fixed point theorem is applied. The nonexistence of traveling wave solutions is proved by two-sided Laplace transform. However, to apply two-sided Laplace transform, the prior estimate of exponential decrease of traveling wave solutions is needed. For this aim, a new method is proposed, which can be applied to reaction-diffusion systems consisting of more than three equations.

scholarly journals Traveling Wave Solutions in a Reaction-Diffusion Epidemic Model

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Sheng Wang ◽  
Wenbin Liu ◽  
Zhengguang Guo ◽  
Weiming Wang
Keyword(s):  
Traveling Wave ◽  
Epidemic Model ◽  
Wave Speed ◽  
Upper And Lower Solutions ◽  
Traveling Wave Solutions ◽  
Reaction Diffusion ◽  
Monotone Iteration ◽  
Wave Solutions ◽  
Minimal Wave Speed

We investigate the traveling wave solutions in a reaction-diffusion epidemic model. The existence of the wave solutions is derived through monotone iteration of a pair of classical upper and lower solutions. The traveling wave solutions are shown to be unique and strictly monotonic. Furthermore, we determine the critical minimal wave speed.

scholarly journals Traveling Wave Solutions of a Four Dimensional Reaction-Diffusion Model for Porcine Reproductive and Respiratory Syndrome with Time Dependent Infection Rate

Computation ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 30
Author(s):  
Jeerawan Suksamran ◽  
Yongwimon Lenbury ◽  
Sanoe Koonprasert
Keyword(s):  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Reaction Diffusion ◽  
Growing Pigs ◽  
Respiratory Syndrome Virus ◽  
Swine Industry ◽  
Reaction Diffusion Model ◽  
Spatial Dimensions ◽  
Tangent Method ◽  
Spatial Transmission

Porcine reproductive and respiratory syndrome virus (PRRSV) causes reproductive failure in sows and respiratory disease in piglets and growing pigs. The disease rapidly spreads in swine populations, making it a serious problem causing great financial losses to the swine industry. However, past mathematical models used to describe the spread of the disease have not yielded sufficient understanding of its spatial transmission. This work has been designed to investigate a mathematical model for the spread of PRRSV considering both time and spatial dimensions as well as the observed decline in infectiousness as time progresses. Moreover, our model incorporates into the dynamics the assumption that some members of the infected population may recover from the disease and become immune. Analytical solutions are derived by using the modified extended hyperbolic tangent method with the introduction of traveling wave coordinate. We also carry out a stability and phase analysis in order to obtain a clearer understanding of how PRRSV spreads spatially through time.

On the spread of ultrafine particulate matter: A mathematical model for motor vehicle emissions and their effects as an asthma trigger

2021 ◽  
Author(s):  
Michelle N. Rosado-Pérez ◽  
Karen Ríos-Soto
Keyword(s):  
Mathematical Model ◽  
Ultrafine Particles ◽  
Motor Vehicle ◽  
Traveling Wave Solutions ◽  
Reaction Diffusion ◽  
Ambient Air ◽  
Air Pollutant ◽  
Pollutant Emissions ◽  
Motor Vehicles ◽  
Diffusion Type

Asthma is a respiratory disease that affects the lungs, with a prevalence of 339.4 million people worldwide [G. Marks, N. Pearce, D. Strachan, I. Asher and P. Ellwood, The Global Asthma Report 2018, globalasthmareport.org (2018)]. Many factors contribute to the high prevalence of asthma, but with the rise of the industrial age, air pollutants have become one of the main Ultrafine particles (UFPs), which are a type of air pollutant that can affect asthmatics the most. These UFPs originate primarily from the combustion of motor vehicles [P. Solomon, Ultrafine particles in ambient air. EM: Air and Waste Management Association’s Magazine for Environmental Managers (2012)] and although in certain places some regulations to control their emission have been implemented they might not be enough. In this work, a mathematical model of reaction–diffusion type is constructed to study how UFPs grow and disperse in the environment and in turn how they affect an asthmatic population. Part of our focus is on the existence of traveling wave solutions and their minimum asymptotic speed of pollutant propagation [Formula: see text]. Through the analysis of the model it was possible to identify the necessary threshold conditions to control the pollutant emissions and consequently reduce the asthma episodes in the population. Analytical and numerical results from this work prove how harmful the UFEs are for the asthmatic population and how they can exacerbate their asthma episodes.

scholarly journals Minimal Wave Speed in a Predator-Prey System with Distributed Time Delay

2018 ◽  
Vol 2018 ◽  
pp. 1-8
Author(s):  
Fuzhen Wu ◽  
Dongfeng Li
Keyword(s):  
Time Delay ◽  
Traveling Wave ◽  
Wave Speed ◽  
Upper And Lower Solutions ◽  
Traveling Wave Solutions ◽  
Predator Prey ◽  
Wave Solutions ◽  
Prey System ◽  
Minimal Wave Speed ◽  
Distributed Time Delay

This paper is concerned with the minimal wave speed of traveling wave solutions in a predator-prey system with distributed time delay, which does not satisfy comparison principle due to delayed intraspecific terms. By constructing upper and lower solutions, we obtain the existence of traveling wave solutions when the wave speed is the minimal wave speed. Our results complete the known conclusions and show the precisely asymptotic behavior of traveling wave solutions.

scholarly journals Some New Traveling Wave Solutions of the Nonlinear Reaction Diffusion Equation by Using the Improved (G′/G)-Expansion Method

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Hasibun Naher ◽  
Farah Aini Abdullah
Keyword(s):  
Diffusion Equation ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Reaction Diffusion ◽  
Expansion Method ◽  
Reaction Diffusion Equation ◽  
Wave Solutions ◽  
Nonlinear Reaction ◽  
Nonlinear Reaction Diffusion Equation ◽  
Constant Coefficients

We construct new exact traveling wave solutions involving free parameters of the nonlinear reaction diffusion equation by using the improved (G′/G)-expansion method. The second-order linear ordinary differential equation with constant coefficients is used in this method. The obtained solutions are presented by the hyperbolic and the trigonometric functions. The solutions become in special functional form when the parameters take particular values. It is important to reveal that our solutions are in good agreement with the existing results.

scholarly journals Global Stability Analysis of a Nonautonomous Stage-Structured Competitive System with Toxic Effect and Double Maturation Delays

2014 ◽  
Vol 2014 ◽  
pp. 1-15 ◽  
Author(s):  
Chao Liu ◽  
Yuanke Li
Keyword(s):  
Global Stability ◽  
Toxic Effect ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Stage Structure ◽  
Positive Periodic Solution ◽  
Competitive System ◽  
Comparison Arguments ◽  
Double Time

We investigate a nonautonomous two-species competitive system with stage structure and double time delays due to maturation for two species, where toxic effect of toxin liberating species on nontoxic species is considered and the inhibiting effect is zero in absence of either species. Positivity and boundedness of solutions are analytically studied. By utilizing some comparison arguments, an iterative technique is proposed to discuss permanence of the species within competitive system. Furthermore, existence of positive periodic solutions is investigated based on continuation theorem of coincidence degree theory. By constructing an appropriate Lyapunov functional, sufficient conditions for global stability of the unique positive periodic solution are analyzed. Numerical simulations are carried out to show consistency with theoretical analysis.

scholarly journals A Lotka-Volterra Competition Model with Cross-Diffusion

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Wenyan Chen ◽  
Ya Chen
Keyword(s):  
Boundary Condition ◽  
Positive Solutions ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Upper And Lower Solutions ◽  
Sufficient Conditions ◽  
Competition Model ◽  
Homogeneous Dirichlet Boundary Condition ◽  
Cross Diffusion ◽  
Existence Of Positive Solutions

A Lotka-Volterra competition model with cross-diffusions under homogeneous Dirichlet boundary condition is considered, where cross-diffusions are included in such a way that the two species run away from each other because of the competition between them. Using the method of upper and lower solutions, sufficient conditions for the existence of positive solutions are provided when the cross-diffusions are sufficiently small. Furthermore, the investigation of nonexistence of positive solutions is also presented.

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