scholarly journals Traveling waves in cooperative predation: relaxation of sublinearity

2021 ◽  
pp. 1-10
Author(s):  
Srijana Ghimire ◽  
Xiang-Sheng Wang
Keyword(s):  
Traveling Wave ◽  
Wave Solution ◽  
Wave Speed ◽  
Traveling Wave Solutions ◽  
Traveling Wave Solution ◽  
Positive Equilibrium ◽  
Open Problems ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Wave Solutions

In this paper, we investigate traveling wave solutions of a diffusive predator-prey model which takes into consideration hunting cooperation. Sublinearity condition is violated for the function of cooperative predation. When the basic reproduction number for the diffusion-free model is greater than one, we find a critical wave speed below which no positive traveling wave solution shall exist. On the other hand, if the wave speed exceeds this critical value, we prove the existence of a positive traveling wave solution connecting the predator-free equilibrium to the unique positive equilibrium under a technical assumption of weak cooperative predation. The key idea of the proof contains two major steps: (i) we construct a suitable pentahedron and find inside it a trajectory connecting the predator-free equilibrium; and (ii) we construct a suitable Lyapunov function and use LaSalle invariance principle to prove that the trajectory also connects the positive equilibrium. In the end of this paper, we propose five open problems related to traveling wave solutions in cooperative predation.

Traveling wave solutions for a diffusive predator–prey model with predator saturation and competition

2017 ◽  
Vol 10 (06) ◽  
pp. 1750086 ◽  
Author(s):  
Lin Zhu ◽  
Shi-Liang Wu
Keyword(s):  
Traveling Wave ◽  
Shooting Method ◽  
Traveling Wave Solutions ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Wave Solutions ◽  
Prey Model ◽  
Boundary Equilibrium ◽  
Zero Equilibrium

The purpose of this paper is to study the traveling wave solutions of a diffusive predator–prey model with predator saturation and competition functional response. The system admits three equilibria: a zero equilibrium [Formula: see text], a boundary equilibrium [Formula: see text] and a positive equilibrium [Formula: see text] under some conditions. We establish the existence of two types of traveling wave solutions which connect [Formula: see text] and [Formula: see text] and [Formula: see text] and [Formula: see text], respectively. Our main arguments are based on a simplified shooting method, a sandwich method and constructions of appropriate Lyapunov functions. Our particular interest is to investigate the oscillation of both types of traveling wave solutions when they approach the positive equilibrium.

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.

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 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.

Traveling wave solutions in predator–prey models with competition

2021 ◽  
Author(s):  
Guo Lin ◽  
Yibing Xing
Keyword(s):  
Traveling Wave ◽  
Wave Speed ◽  
Traveling Wave Solutions ◽  
Initial Value ◽  
Predator Prey ◽  
Wave Solutions ◽  
Minimal Wave Speed ◽  
Predator Prey Models ◽  
Scalar Equations ◽  
Asymptotic Spreading

This paper studies the minimal wave speed of traveling wave solutions in predator–prey models, in which there are several groups of predators that compete among different groups. We investigate the existence and nonexistence of traveling wave solutions modeling the invasion of predators and coexistence of these species. When the positive solution of the corresponding kinetic system converges to the unique positive steady state, a threshold that is the minimal wave speed of traveling wave solutions is obtained. To finish the proof, we construct contracting rectangles and upper–lower solutions and apply the asymptotic spreading theory of scalar equations. Moreover, multiple propagation thresholds in the corresponding initial value problem are presented by numerical examples, and one threshold may be the minimal wave speed of traveling wave solutions.

scholarly journals Existence of traveling wave solutions with critical speed in a delayed diffusive epidemic model

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Yueling Cheng ◽  
Dianchen Lu ◽  
Jiangbo Zhou ◽  
Jingdong Wei
Keyword(s):  
Traveling Wave ◽  
Epidemic Model ◽  
Traveling Wave Solutions ◽  
Traveling Wave Solution ◽  
Sir Epidemic Model ◽  
Open Problems ◽  
Sir Epidemic ◽  
Nonlinear Anal ◽  
Wave Solutions ◽  
Saturated Incidence

AbstractIn this paper, we prove the existence of a critical traveling wave solution for a delayed diffusive SIR epidemic model with saturated incidence. Moreover, we establish the nonexistence of traveling wave solutions with nonpositive wave speed for this model. Our results solve some open problems left in the recent paper (Z. Xu in Nonlinear Anal. 111:66–81, 2014).

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