Traveling wave solutions of a nonlocal dispersal predator–prey model with spatiotemporal delay

2018 ◽  
Vol 69 (6) ◽  
Author(s):  
Zhihong Zhao ◽  
Rui Li ◽  
Xiangkui Zhao ◽  
Zhaosheng Feng
Keyword(s):  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Nonlocal Dispersal ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Wave Solutions ◽  
Prey Model ◽  
Spatiotemporal Delay

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 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.

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