Existence and asymptotic behavior of invasion wave solutions in temporally discrete diffusion systems with delays

2018 ◽  
Vol 11 (02) ◽  
pp. 1850016 ◽  
Author(s):  
Hui Xue ◽  
Jianhua Huang ◽  
Zhixian Yu
Keyword(s):  
Fixed Point ◽  
Asymptotic Behavior ◽  
Fixed Point Theorem ◽  
Traveling Wave ◽  
Wave Solution ◽  
Traveling Wave Solutions ◽  
Competitive System ◽  
Wave Solutions ◽  
Diffusion Systems ◽  
Invasion Wave

This paper is concerned with the existence and asymptotic behavior of invasion wave solutions for a time-discrete delayed diffusion competitive system with non-quasi-monotone conditions. The existence of invasion wave solution is investigated by applying upper–lower solutions method and Schauder’s fixed point theorem. Further, with the help of Ikeharaś theorem, we establish the exponential decay asymptotic behavior of traveling wave solutions at the minus/plus infinity.

scholarly journals Traveling Wave Solutions for Some Three-Species Predator-Prey Systems

2021 ◽  
Vol 52 (1) ◽  
pp. 25-36
Author(s):  
Jong-Shenq Guo
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Traveling Waves ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Predator Prey ◽  
Wave Solutions ◽  
Recent Developments ◽  
Wave Profiles ◽  
Predator Prey Models

In this paper, we present some recent developments on the application of Schauder’s fixed point theorem to the existence of traveling waves for some three-species predator-prey systems. The existence of traveling waves of predator-prey systems is closely related to the invasion phenomenon of some alien species to the habitat of aboriginal species. Three different three-species predator-prey models with different invaded and invading states are presented. In this paper, we focus on the methodology of deriving the convergence of stale tail of wave profiles.

Traveling wave solutions of a delayed prey–predator system with diffusion

2016 ◽  
Vol 10 (01) ◽  
pp. 1750001
Author(s):  
Junli Liu ◽  
Tailei Zhang
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Schauder’S Fixed Point Theorem ◽  
Delayed Reaction ◽  
Fusion Systems ◽  
Wave Solutions ◽  
Predator System

This paper discusses the existence of traveling wave solutions of delayed reaction–dif-fusion systems with partial quasi-monotonicity. By using the Schauder’s fixed point theorem, the existence of traveling wave solutions is obtained by the existence of a pair of upper–lower solutions. We study the existence of traveling wave solutions in a delayed prey–predator system.

The existence of traveling wave solutions in an integro-difference competition–cooperation system

2018 ◽  
Vol 11 (01) ◽  
pp. 1850009
Author(s):  
Kun Li ◽  
Xiong Li
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Traveling Wave ◽  
Upper And Lower Solutions ◽  
Traveling Wave Solutions ◽  
Wave Solutions ◽  
The Cross ◽  
Partial Monotonicity ◽  
Cooperation System

In this paper, we are devoted to establishing the existence of traveling wave solutions in an integro-difference competition–cooperation system with partial monotonicity by using Schauder’s fixed point theorem, the cross-iteration and the upper and lower solutions method. To illustrate our result, we present an application to integro-difference competition–cooperation system with a special kernel by constructing a pair of upper and lower solutions, while the verification of upper and lower solutions is nontrivial.

scholarly journals Single peaked traveling wave solutions to a generalized μ-Novikov Equation

2020 ◽  
Vol 10 (1) ◽  
pp. 66-75
Author(s):  
Byungsoo Moon
Keyword(s):  
Linear Combination ◽  
Traveling Wave ◽  
Wave Solution ◽  
Traveling Wave Solutions ◽  
Traveling Wave Solution ◽  
Wave Solutions ◽  
Quadratic Nonlinearities ◽  
Novikov Equation

Abstract In this paper, we study the existence of peaked traveling wave solution of the generalized μ-Novikov equation with nonlocal cubic and quadratic nonlinearities. The equation is a μ-version of a linear combination of the Novikov equation and Camassa-Hom equation. It is found that the equation admits single peaked traveling wave solutions.

Exact Cuspon and Compactons of the Novikov Equation

2014 ◽  
Vol 24 (03) ◽  
pp. 1450037 ◽  
Author(s):  
Jibin Li
Keyword(s):  
Dynamical Systems ◽  
Qualitative Analysis ◽  
Traveling Wave ◽  
Wave Solution ◽  
Traveling Wave Solutions ◽  
Phase Portraits ◽  
Breaking Wave ◽  
Wave Solutions ◽  
Novikov Equation

In this paper, we apply the method of dynamical systems to the traveling wave solutions of the Novikov equation. Through qualitative analysis, we obtain bifurcations of phase portraits of the traveling system and exact cuspon wave solution, as well as a family of breaking wave solutions (compactons). We find that the corresponding traveling system of Novikov equation has no one-peakon solution.

scholarly journals The Traveling Wave Solutions and Their Bifurcations for the BBM-LikeB(m,n)Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-17 ◽  
Author(s):  
Shaoyong Li ◽  
Zhengrong Liu
Keyword(s):  
Traveling Wave ◽  
Wave Solution ◽  
Blow Up ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Periodic Wave Solution ◽  
Simulation Approach ◽  
Bifurcation Method ◽  
Periodic Wave Solutions ◽  
Wave Solutions

We investigate the traveling wave solutions and their bifurcations for the BBM-likeB(m,n)equationsut+αux+β(um)x−γ(un)xxt=0by using bifurcation method and numerical simulation approach of dynamical systems. Firstly, for BBM-likeB(3,2)equation, we obtain some precise expressions of traveling wave solutions, which include periodic blow-up and periodic wave solution, peakon and periodic peakon wave solution, and solitary wave and blow-up solution. Furthermore, we reveal the relationships among these solutions theoretically. Secondly, for BBM-likeB(4,2)equation, we construct two periodic wave solutions and two blow-up solutions. In order to confirm the correctness of these solutions, we also check them by software Mathematica.

scholarly journals Approximate Damped Oscillatory Solutions for Generalized KdV-Burgers Equation and Their Error Estimates

2011 ◽  
Vol 2011 ◽  
pp. 1-26 ◽  
Author(s):  
Weiguo Zhang ◽  
Xiang Li
Keyword(s):  
Dynamical Systems ◽  
Error Estimates ◽  
Traveling Wave ◽  
Wave Solution ◽  
Burgers Equation ◽  
Traveling Wave Solutions ◽  
Approximate Solutions ◽  
Traveling Wave Solution ◽  
Oscillatory Solutions ◽  
Wave Solutions

We focus on studying approximate solutions of damped oscillatory solutions of generalized KdV-Burgers equation and their error estimates. The theory of planar dynamical systems is employed to make qualitative analysis to the dynamical systems which traveling wave solutions of this equation correspond to. We investigate the relations between the behaviors of bounded traveling wave solutions and dissipation coefficient, and give two critical valuesλ1andλ2which can characterize the scale of dissipation effect, for right and left-traveling wave solution, respectively. We obtain that for the right-traveling wave solution if dissipation coefficientα≥λ1, it appears as a monotone kink profile solitary wave solution; that if0<α<λ1, it appears as a damped oscillatory solution. This is similar for the left-traveling wave solution. According to the evolution relations of orbits in the global phase portraits which the damped oscillatory solutions correspond to, we obtain their approximate damped oscillatory solutions by undetermined coefficients method. By the idea of homogenization principle, we give the error estimates for these approximate solutions by establishing the integral equations reflecting the relations between exact and approximate solutions. The errors are infinitesimal decreasing in the exponential form.

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