Global stability of non-monotone traveling wave solutions for a nonlocal dispersal equation with time delay

2019 ◽  
Vol 475 (1) ◽  
pp. 605-627 ◽  
Author(s):  
Guo-Bao Zhang
Keyword(s):  
Time Delay ◽  
Global Stability ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Nonlocal Dispersal ◽  
Wave Solutions

scholarly journals Spreading Speed in A Nonmonotone Equation with Dispersal and Delay

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 291 ◽  
Author(s):  
Xi-Lan Liu ◽  
Shuxia Pan
Keyword(s):  
Time Delay ◽  
Traveling Wave ◽  
Wave Speed ◽  
Traveling Wave Solutions ◽  
Spreading Speed ◽  
Limit Function ◽  
Nonlocal Dispersal ◽  
Initial Value ◽  
Wave Solutions ◽  
Nonmonotone Equation

This paper is concerned with the estimation of spreading speed of a nonmonotone equation, which involves time delay and nonlocal dispersal. Due to the time delay, this equation does not generate monotone semiflows when the positive initial value is given. By constructing an auxiliary monotone equation, we obtain the spreading speed when the initial value admits nonempty compact support. Moreover, by passing to a limit function, we confirm the existence of traveling wave solutions if the wave speed equals to the spreading speed, which states the minimal wave speed of traveling wave solutions and improves the known results.

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 Traveling Waves in a Nonlocal Dispersal SIR Model with Standard Incidence Rate and Nonlocal Delayed Transmission

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 641 ◽  
Author(s):  
Kuilin Wu ◽  
Kai Zhou
Keyword(s):  
Incidence Rate ◽  
Traveling Wave ◽  
Epidemic Model ◽  
Reaction System ◽  
Traveling Wave Solutions ◽  
Sir Epidemic Model ◽  
Nonlocal Dispersal ◽  
Sir Epidemic ◽  
Wave Solutions ◽  
Minimal Wave Speed

In this paper, we study the traveling wave solutions for a nonlocal dispersal SIR epidemic model with standard incidence rate and nonlocal delayed transmission. The existence and nonexistence of traveling wave solutions are determined by the basic reproduction number of the corresponding reaction system and the minimal wave speed. To prove these results, we apply the Schauder’s fixed point theorem and two-sided Laplace transform. The main difficulties are that the complexity of the incidence rate in the epidemic model and the lack of regularity for nonlocal dispersal operator.

A spatial echinococcosis transmission model with time delays: Stability and traveling waves

2017 ◽  
Vol 10 (06) ◽  
pp. 1750081 ◽  
Author(s):  
Zhiting Xu ◽  
Cuihua Ai
Keyword(s):  
Global Stability ◽  
Traveling Wave ◽  
Time Delays ◽  
Traveling Wave Solutions ◽  
Spatial Domain ◽  
Transmission Model ◽  
Initial Value ◽  
Wave Solutions ◽  
Number Formula ◽  
Disease Free

In this paper, we derive a time-delayed and diffusive echinococcosis transmission model. We first address the well-posedness to the initial-value problem for the model and give the basic reproduction number [Formula: see text]. In the case of a bounded spatial domain, we establish the local stability as well as the global stability of the disease-free and disease equilibria of the model. The methods to prove the local and the global stability are to analyze the corresponding characteristic equations and construct Lyapunov functionals, respectively. In the case of an unbounded spatial domain, by applying Schauder’s fixed point theorem and the limiting arguments, we show that when [Formula: see text], there exists a constant [Formula: see text] such that the model admits positive traveling wave solutions connecting the disease-free and endemic equilibrium for [Formula: see text], and when [Formula: see text] and [Formula: see text], the model has no positive traveling wave solutions connecting them.

Traveling wave solutions of a nonlocal dispersal SIRS model with spatio-temporal delay

2017 ◽  
Vol 10 (05) ◽  
pp. 1750071 ◽  
Author(s):  
Zhaohai Ma ◽  
Rong Yuan
Keyword(s):  
Fixed Point ◽  
Traveling Wave ◽  
Wave Speed ◽  
Traveling Wave Solutions ◽  
Temporal Delay ◽  
Nonlocal Dispersal ◽  
Wave Solutions ◽  
Sirs Model ◽  
Existence And Nonexistence ◽  
Spatio Temporal

This paper is mainly concerned with the existence and nonexistence of traveling wave solutions of a nonlocal dispersal SIRS model with nonlocal delayed transmissions. We find that the existence and nonexistence of traveling wave solutions are determined by the critical wave speed [Formula: see text]. More specifically, we establish the existence of traveling wave solutions for every wave speed [Formula: see text] and [Formula: see text] by means of upper-lower solutions and Schauder’s fixed point theorem. Nonexistence of traveling wave solutions is obtained by Laplace transform for any wave speed [Formula: see text] and [Formula: see text].

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