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.

Existence of spatiotemporal patterns in the reaction–diffusion predator–prey model incorporating prey refuge

2016 ◽  
Vol 09 (06) ◽  
pp. 1650085 ◽  
Author(s):  
Lakshmi Narayan Guin ◽  
Benukar Mondal ◽  
Santabrata Chakravarty
Keyword(s):  
Pattern Formation ◽  
Spatial Dynamics ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Model System ◽  
Prey Refuge ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model

The pattern formation in reaction–diffusion system has long been the subject of interest to the researchers in the domain of mathematical ecology because of its universal existence and importance. The present investigation deals with a spatial dynamics of the Beddington–DeAngelis predator–prey model in the presence of a constant proportion of prey refuge. The model system representing boundary value problem under study is subjected to homogeneous Neumann boundary conditions. The asymptotic stability including the local and the global stability and the bifurcation as well of the unique positive homogeneous steady state of the corresponding temporal model has been analyzed. The Turing instability region in two-parameter space and the condition of diffusion-driven instability of the spatiotemporal model are investigated. Based on the appropriate numerical simulations, the present model dynamics in Turing space appears to get influenced by prey refuge while it exhibits diffusion-controlled pattern formation growth to spots, stripe-spot mixtures, labyrinthine, stripe-hole mixtures and holes replication. The results obtained appear to enrich the findings of the model system under consideration.

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