scholarly journals Spatiotemporal Complexity of a Leslie-Gower Predator-Prey Model with the Weak Allee Effect

2013 ◽  
Vol 2013 ◽  
pp. 1-16 ◽  
Author(s):  
Yongli Cai ◽  
Caidi Zhao ◽  
Weiming Wang
Keyword(s):  
Steady State ◽  
Allee Effect ◽  
Steady State Solution ◽  
Turing Instability ◽  
Dimensional System ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Nonexistence And Existence ◽  
The Impact

We investigate a diffusive Leslie-Gower predator-prey model with the additive Allee effect on prey subject to the zero-flux boundary conditions. Some results of solutions to this model and its corresponding steady-state problem are shown. More precisely, we give the stability of the positive constant steady-state solution, the refineda prioriestimates of positive solution, and the nonexistence and existence of the positive nonconstant solutions. We carry out the analytical study for two-dimensional system in detail and find out the certain conditions for Turing instability. Furthermore, we perform numerical simulations and show that the model exhibits a transition from stripe-spot mixtures growth to isolated spots and also to stripes. These results show that the impact of the Allee effect essentially increases the model spatiotemporal complexity.

Spatio-Temporal Patterns of Predator–Prey Model with Allee Effect and Constant Stocking Rate for Predator

2021 ◽  
Vol 31 (16) ◽  
Author(s):  
Xuan Tian ◽  
Shangjiang Guo
Keyword(s):  
Steady State ◽  
Allee Effect ◽  
Sufficient Conditions ◽  
Steady State Solution ◽  
Turing Pattern ◽  
Stocking Rate ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Spatio Temporal

A diffusive predator–prey model with Allee effect and constant stocking rate for predator is investigated and it is shown that Allee effect is the decisive factor driving the formation of Turing pattern. Furthermore, it is observed that Turing pattern appears only when the diffusion rate of the prey is faster than that of the predator, which is just opposite to the condition of Turing pattern in the classical predator–prey system. Some sufficient conditions are obtained to ensure the asymptotical stability of a spatially homogeneous steady-state solution. The existence and nonexistence of positive nonconstant steady-state solutions are investigated to understand the mechanisms of generating spatiotemporal patterns. Furthermore, Hopf and steady-state bifurcations are analyzed in detail by using Lyapunov–Schmidt reduction.

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.

Chaos control in a discrete-time predator–prey model with weak Allee effect

2018 ◽  
Vol 11 (07) ◽  
pp. 1850089 ◽  
Author(s):  
Saheb Pal ◽  
Sourav Kumar Sasmal ◽  
Nikhil Pal
Keyword(s):  
Discrete Time ◽  
Allee Effect ◽  
Sufficient Conditions ◽  
Equilibrium Analysis ◽  
Flip Bifurcation ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
The Stability ◽  
The Impact

The stability of the predator–prey model subject to the Allee effect is an interesting topic in recent times. In this paper, we investigate the impact of weak Allee effect on the stability of a discrete-time predator–prey model with Holling type-IV functional response. The mathematical features of the proposed model are analyzed with the help of equilibrium analysis, stability analysis, and bifurcation theory. We provide sufficient conditions for the flip bifurcation by considering Allee parameter as the bifurcation parameter. We observe that the model becomes stable from chaotic dynamics as the Allee parameter increases. Further, we observe bi-stability behavior of the model between only prey existence equilibrium and the coexistence equilibrium. Our analytical findings are illustrated through numerical simulations.

scholarly journals Complex Dynamics of a Diffusive Holling-Tanner Predator-Prey Model with the Allee Effect

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Zongmin Yue ◽  
Xiaoqin Wang ◽  
Haifeng Liu
Keyword(s):  
Allee Effect ◽  
Complex Dynamics ◽  
Initial Conditions ◽  
Turing Instability ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Dynamical Behaviors ◽  
Highly Sensitive ◽  
The Impact

We investigate the complex dynamics of a diffusive Holling-Tanner predation model with the Allee effect on prey analytically and numerically. We examine the existence of the positive equilibria and the related dynamical behaviors of the model and find that when the model is with weak Allee effect, the solutions are local and global stability for some conditions around the positive equilibrium. In contrast, when the model is with strong Allee effect, this may lead to the phenomenon of bistability; that is to say, there is a separatrix curve that separates the behavior of trajectories of the system, implying that the model is highly sensitive to the initial conditions. Furthermore, we give the conditions of Turing instability and determine the Turing space in the parameters space. Based on these results, we perform a series of numerical simulations and find that the model exhibits complex pattern replication: spots, spots-stripes mixtures, and stripes patterns. The results show that the impact of the Allee effect essentially increases the models spatiotemporal complexity.

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.

Turing Instability and Bifurcation in a Diffusion Predator–Prey Model with Beddington–DeAngelis Functional Response

2018 ◽  
Vol 28 (09) ◽  
pp. 1830029 ◽  
Author(s):  
Wei Tan ◽  
Wenwu Yu ◽  
Tasawar Hayat ◽  
Fuad Alsaadi ◽  
Habib M. Fardoun
Keyword(s):  
Hopf Bifurcation ◽  
Functional Response ◽  
Steady State Solution ◽  
Turing Instability ◽  
Rigorous Analysis ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Global Positive Solution ◽  
The Stability

In this paper, we consider a predator–prey model with Beddington–DeAngelis functional response with or without diffusion. For this system, we give a complete and rigorous analysis of the dynamics including the existence of a global positive solution, the stability/Turing instability and the Hopf bifurcation. In the meanwhile, we show, via numerical simulations, that there appears Hopf bifurcation, steady state solution and Turing–Hopf bifurcation with the changes of some parameters of the system.

An Eco-Epidemic Predator–Prey Model with Allee Effect in Prey

2020 ◽  
Vol 30 (13) ◽  
pp. 2050194
Author(s):  
Absos Ali Shaikh ◽  
Harekrishna Das
Keyword(s):  
Allee Effect ◽  
Predator Population ◽  
Bifurcation And Chaos ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Disease In Predator ◽  
The Stability ◽  
The Impact ◽  
Existence Of Equilibria

This article describes the dynamics of a predator–prey model with disease in predator population and prey population subject to Allee effect. The positivity and boundedness of the solutions of the system have been determined. The existence of equilibria of the system and the stability of those equilibria are analyzed when Allee effect is present. The main objective of this study is to investigate the impact of Allee effect and it is observed that the system experiences Hopf bifurcation and chaos due to Allee effect. The results obtained from the model may be useful for analyzing the real-world ecological and eco-epidemiological systems.

scholarly journals The Influence of Additive Allee Effect and Periodic Harvesting to the Dynamics of Leslie-Gower Predator-Prey Model

2020 ◽  
Vol 2 (2) ◽  
pp. 87-96
Author(s):  
Hasan S. Panigoro ◽  
Emli Rahmi ◽  
Novianita Achmad ◽  
Sri Lestari Mahmud
Keyword(s):  
Numerical Simulations ◽  
Autonomous System ◽  
Allee Effect ◽  
Equilibrium Points ◽  
Dimensional System ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Theoretical Results

In this paper, the influence of additive Allee effect in prey and periodic harvesting in predator to the dynamics of the Leslie-Gower predator-prey model is proposed. We first simplify the model to the non-dimensional system by scaling the variable and transform the model into an autonomous system. If the effect Allee is weak, we have at most two equilibrium points, else if the Allee effect is strong, at most four equilibrium points may exist. Furthermore, the behavior of the system around equilibrium points is investigated. In the end, we give numerical simulations to support theoretical results.

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