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 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

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 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.

scholarly journals Pattern Formation in Predator-Prey Model with Delay and Cross Diffusion

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Xinze Lian ◽  
Shuling Yan ◽  
Hailing Wang
Keyword(s):  
Time Delay ◽  
Pattern Formation ◽  
Turing Instability ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Prey Model ◽  
Diffusion Controlled ◽  
The Stability ◽  
Self Replication

We consider the effect of time delay and cross diffusion on the dynamics of a modified Leslie-Gower predator-prey model incorporating a prey refuge. Based on the stability analysis, we demonstrate that delayed feedback may generate Hopf and Turing instability under some conditions, resulting in spatial patterns. One of the most interesting findings is that the model exhibits complex pattern replication: the model dynamics exhibits a delay and diffusion controlled formation growth not only to spots, stripes, and holes, but also to spiral pattern self-replication. The results indicate that time delay and cross diffusion play important roles in pattern formation.

Turing Instability and Hopf Bifurcation in a Modified Leslie–Gower Predator–Prey Model with Cross-Diffusion

2018 ◽  
Vol 28 (07) ◽  
pp. 1850089 ◽  
Author(s):  
Walid Abid ◽  
R. Yafia ◽  
M. A. Aziz-Alaoui ◽  
Ahmed Aghriche
Keyword(s):  
Spatial Dynamics ◽  
Real Life ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Spatiotemporal Pattern ◽  
Predator Prey ◽  
Interior Equilibrium ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Prey Model

This paper is concerned with some mathematical analysis and numerical aspects of a reaction–diffusion system with cross-diffusion. This system models a modified version of Leslie–Gower functional response as well as that of the Holling-type II. Our aim is to investigate theoretically and numerically the asymptotic behavior of the interior equilibrium of the model. The conditions of boundedness, existence of a positively invariant set are proved. Criteria for local stability/instability and global stability are obtained. By using the bifurcation theory, the conditions of Hopf and Turing bifurcation critical lines in a spatial domain are proved. Finally, we carry out some numerical simulations in order to support our theoretical results and to interpret how biological processes affect spatiotemporal pattern formation which show that it is useful to use the predator–prey model to detect the spatial dynamics in the real life.

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.

Pattern Dynamics in a Predator–Prey Model with Schooling Behavior and Cross-Diffusion

2019 ◽  
Vol 29 (11) ◽  
pp. 1950146
Author(s):  
Wen Wang ◽  
Shutang Liu ◽  
Zhibin Liu ◽  
Da Wang
Keyword(s):  
Spatial Dynamics ◽  
Multiple Scale ◽  
Turing Instability ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Schooling Behavior ◽  
Pattern Dynamics ◽  
Prey Model ◽  
Pattern Formations

In this paper, a diffusive predator–prey model is considered in which the predator and prey populations both exhibit schooling behavior. The system’s spatial dynamics are captured via a suitable threshold parameter, and a sequence of spatiotemporal patterns such as hexagons, stripes and a mixture of the two are observed. Specifically, the linear stability analysis is applied to obtain the conditions for Hopf bifurcation and Turing instability. Then, employing the multiple-scale analysis, the amplitude equations near the critical point of Turing bifurcation are derived, through which the selection and stability of pattern formations are investigated. The theoretical results are verified by numerical simulations.

Stability and Turing Patterns in a Predator–Prey Model with Hunting Cooperation and Allee Effect in Prey Population

2020 ◽  
Vol 30 (09) ◽  
pp. 2050137
Author(s):  
Danxia Song ◽  
Yongli Song ◽  
Chao Li
Keyword(s):  
Spatial Patterns ◽  
Allee Effect ◽  
Turing Instability ◽  
Local System ◽  
Positive Equilibrium ◽  
Predator Population ◽  
Predator Prey ◽  
Turing Bifurcation ◽  
Predator Prey Model ◽  
Prey Model

In this paper, we are concerned with a diffusive predator–prey model where the functional response follows the predator cooperation in hunting and the growth of the prey obeys the Allee effect. Firstly, the existence and stability of the positive equilibrium are explicitly determined by the local system parameters. It is shown that the ability of the hunting cooperation can affect the existence of the positive equilibrium and stability, and the intrinsic growth rate of the predator population does not affect the existence of the positive equilibrium, but affects the stability. Then the diffusion-driven Turing instability is investigated and the Turing bifurcation value is obtained, and we conclude that when the ability of the cooperation in hunting is weaker than some critical value, there is no Turing instability. The standard weakly nonlinear analysis method is employed to derive the amplitude equations of the Turing bifurcation, which is used to predict the types of the spatial patterns. And it is found that in the Turing instability region, with the parameter changing from approaching Turing bifurcation value to approaching Hopf bifurcation value, spatial patterns emerge from spot, spot-stripe to stripe. Finally, the numerical simulations are used to support the analytical results.

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