Delay-driven pattern formation in a reaction–diffusion predator–prey model incorporating a prey refuge

2013 ◽  
Vol 2013 (04) ◽  
pp. P04006 ◽  
Author(s):  
Xinze Lian ◽  
Hailing Wang ◽  
Weiming Wang
Keyword(s):  
Pattern Formation ◽  
Reaction Diffusion ◽  
Prey Refuge ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model

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.

scholarly journals Corrigendum to “Pattern Formation in a Reaction-Diffusion Predator-Prey Model with Weak Allee Effect and Delay”

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-5
Author(s):  
Hua Liu ◽  
Yong Ye ◽  
Yumei Wei ◽  
Weiyuan Ma ◽  
Ming Ma ◽  
...  
Keyword(s):  
Pattern Formation ◽  
Allee Effect ◽  
Reaction Diffusion ◽  
Predator Prey ◽  
Predator Prey Model ◽  
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.

Bifurcation Analysis and Spatiotemporal Patterns in a Diffusive Predator–Prey Model

2014 ◽  
Vol 24 (06) ◽  
pp. 1450081 ◽  
Author(s):  
Guangping Hu ◽  
Xiaoling Li ◽  
Shiping Lu ◽  
Yuepeng Wang
Keyword(s):  
Pattern Formation ◽  
Spiral Wave ◽  
Reaction Diffusion ◽  
Spatiotemporal Chaos ◽  
Diffusion System ◽  
Target Pattern ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
The Impact

In this paper, we consider a species predator–prey model given a reaction–diffusion system. It incorporates the Holling type II functional response and a quadratic intra-predator interaction term. We focus on the qualitative analysis, bifurcation mechanisms and pattern formation. We present the results of numerical experiments in two space dimensions and illustrate the impact of the diffusion on the Turing pattern formation. For this diffusion system, we also observe non-Turing structures such as spiral wave, target pattern and spatiotemporal chaos resulting from the time evolution of these structures.

scholarly journals Pattern Formation in a Reaction-Diffusion Predator-Prey Model with Weak Allee Effect and Delay

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-14 ◽  
Author(s):  
Hua Liu ◽  
Yong Ye ◽  
Yumei Wei ◽  
Weiyuan Ma ◽  
Ming Ma ◽  
...  
Keyword(s):  
Numerical Simulation ◽  
Pattern Formation ◽  
Allee Effect ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Point Of View ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Pattern Formations

In this paper, we establish a reaction-diffusion predator-prey model with weak Allee effect and delay and analyze the conditions of Turing instability. The effects of Allee effect and delay on pattern formation are discussed by numerical simulation. The results show that pattern formations change with the addition of weak Allee effect and delay. More specifically, as Allee effect constant and delay increases, coexistence of spotted and stripe patterns, stripe patterns, and mixture patterns emerge successively. From an ecological point of view, we find that Allee effect and delay play an important role in spatial invasion of populations.

scholarly journals Allee-Effect-Induced Instability in a Reaction-Diffusion Predator-Prey Model

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Weiming Wang ◽  
Yongli Cai ◽  
Yanuo Zhu ◽  
Zhengguang Guo
Keyword(s):  
Pattern Formation ◽  
Allee Effect ◽  
Reaction Diffusion ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Diffusion Controlled ◽  
Model Dynamics ◽  
And Diffusion

We investigate the spatiotemporal dynamics induced by Allee effect in a reaction-diffusion predator-prey model. In the case without Allee effect, there is nonexistence of diffusion-driven instability for the model. And in the case with Allee effect, the positive equilibrium may be unstable under certain conditions. This instability is induced by Allee effect and diffusion together. Furthermore, via numerical simulations, the model dynamics exhibits both Allee effect and diffusion controlled pattern formation growth to holes, stripes-holes mixture, stripes, stripes-spots mixture, and spots replication, which shows that the dynamics of the model with Allee effect is not simple, but rich and complex.

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