scholarly journals Traveling wave solutions of a reaction-diffusion predator-prey model

2017 ◽  
Vol 10 (5) ◽  
pp. 1063-1078
Author(s):  
Jiang Liu ◽  
◽  
Xiaohui Shang ◽  
Zengji Du
Keyword(s):  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Reaction Diffusion ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Wave Solutions ◽  
Prey Model

scholarly journals Traveling Waves in a Diffusive Predator-Prey Model Incorporating a Prey Refuge

2014 ◽  
Vol 2014 ◽  
pp. 1-13
Author(s):  
Xiujuan Wu ◽  
Yong Luo ◽  
Yizheng Hu
Keyword(s):  
Traveling Wave ◽  
Wave Train ◽  
Three Dimensional ◽  
Traveling Wave Solutions ◽  
Reaction Diffusion ◽  
Prey Refuge ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Manifold Theory

We establish the existence of traveling wave solutions and small amplitude traveling wave train solutions for a reaction-diffusion system based on a predator-prey model incorporating a prey refuge. By using the shooting argument, invariant manifold theory, and the Hopf bifurcation theorem, we analyze the dynamic behavior of this model in the three-dimensional phase space. Numerical results are also presented to illustrate the theoretical results.

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.

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