scholarly journals Dynamic Complexities in 2-Dimensional Discrete-Time Predator-Prey Systems with Allee Effect in the Prey

2016 ◽  
Vol 2016 ◽  
pp. 1-14
Author(s):  
Jie Yan ◽  
Chunli Li ◽  
Xueli Chen ◽  
Lishun Ren
Keyword(s):  
Bifurcation Theory ◽  
Allee Effect ◽  
Center Manifold ◽  
Prey Species ◽  
Extinction Risk ◽  
Flip Bifurcation ◽  
Crowding Effect ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Predator Prey Model

The Allee effect is incorporated into a predator-prey model with linear functional response. Compared with the predator-prey which only takes the crowding effect and predator partially dependent on prey into consideration, it is found that the Allee effect of the prey species would increase the extinction risk of both the prey and predator. Moreover, by using a center manifold theorem and bifurcation theory, it is shown that the model with Allee effect undergoes the flip bifurcation and Hopf bifurcation in the interior ofR+2with different Allee effect values. In the two bifurcations, we can come to the conclusion that different Allee effect will have different bifurcation value and the increasing of the Allee effect will increase the value of bifurcation, respectively.

Complex dynamic behaviors of a discrete predator–prey model with stage structure and harvesting

2016 ◽  
Vol 10 (01) ◽  
pp. 1750013 ◽  
Author(s):  
Boshan Chen ◽  
Jiejie Chen
Keyword(s):  
Numerical Simulations ◽  
Period Doubling ◽  
Stage Structure ◽  
Model Parameters ◽  
Flip Bifurcation ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Predator Prey Model ◽  
Dynamical Behaviors ◽  
Prey Model

First, a discrete stage-structured and harvested predator–prey model is established, which is based on a predator–prey model with Type III functional response. Then theoretical methods are used to investigate existence of equilibria and their local properties. Third, it is shown that the system undergoes flip bifurcation and Neimark–Sacker bifurcation in the interior of [Formula: see text], by using the normal form of discrete systems, the center manifold theorem and the bifurcation theory, as varying the model parameters in some range. In particular, the direction and the stability of the flip bifurcation and the Neimark–Sacker bifurcation are showed. Finally, numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, such as cascades of period-doubling bifurcation and chaotic sets. These results reveal far richer dynamics of the discrete model compared with the continuous model. The Lyapunov exponents are numerically computed to confirm further the complexity of the dynamical behaviors. In addition, we show also the stabilizing effect of the harvesting by using numerical simulations.

scholarly journals Hopf Bifurcation Analysis on General Gause-Type Predator-Prey Models with Delay

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Shuang Guo ◽  
Weihua Jiang
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Predator Prey Model ◽  
Normal Form Method ◽  
Predator Prey Models ◽  
The Stability

A class of three-dimensional Gause-type predator-prey model with delay is considered. Firstly, a group of sufficient conditions for the existence of Hopf bifurcation is obtained via employing the polynomial theorem by analyzing the distribution of the roots of the associated characteristic equation. Secondly, the direction of the Hopf bifurcation and the stability of the bifurcated periodic solutions are determined by applying the normal form method and the center manifold theorem. Finally, some numerical simulations are carried out to illustrate the obtained results.

scholarly journals Dynamic complexity and bifurcation analysis of a host–parasitoid model with Allee effect and Holling type III functional response

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Hua Liu ◽  
Kai Zhang ◽  
Yong Ye ◽  
Yumei Wei ◽  
Ming Ma
Keyword(s):  
Local Stability ◽  
Bifurcation Theory ◽  
Allee Effect ◽  
Center Manifold ◽  
Discrete Time System ◽  
Time System ◽  
Dynamic Complexity ◽  
Center Manifold Theorem ◽  
Dynamical Behaviors ◽  
Local Stability Analysis

AbstractIn this paper, we focus on dynamics in a basic discrete-time system of host–parasitoid interaction. We perform local stability analysis of this system. Furthermore, both flip and Neimark–Sacker bifurcations are also analyzed in the interior of $R_{ +}^{2}$R+2 by using center manifold theorem and bifurcation theory. Finally, numerical simulations are deployed to validate our results with theoretical analysis and to exhibit the dynamical behaviors.

Codimension Bifurcation Analysis of a Modified Leslie–Gower Predator–Prey Model with Two Delays

2018 ◽  
Vol 28 (05) ◽  
pp. 1850060 ◽  
Author(s):  
Jianfeng Jiao ◽  
Ruiqi Wang ◽  
Hongcui Chang ◽  
Xia Liu
Keyword(s):  
Time Delays ◽  
Center Manifold ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Predator Prey Model ◽  
Prey Model ◽  
Form Theory ◽  
Two Delays ◽  
Delay Differential

The Bogdanov–Takens (B–T) and triple-zero bifurcations of a modified Leslie–Gower predator–prey model with two time delays are studied in this paper. By generalizing and using the normal form theory and center manifold theorem for delay differential equations, the normal forms of the B–T and triple-zero bifurcations of the model at its interior equilibria are obtained. In addition, some numerical simulations are presented to illustrate our main 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 Bifurcations, Complex Behaviors, and Dynamic Transition in a Coupled Network of Discrete Predator-Prey System

2019 ◽  
Vol 2019 ◽  
pp. 1-22 ◽  
Author(s):  
Tousheng Huang ◽  
Huayong Zhang ◽  
Shengnan Ma ◽  
Ge Pan ◽  
Zhaodeng Wang ◽  
...  
Keyword(s):  
Center Manifold ◽  
Period Doubling ◽  
Flip Bifurcation ◽  
Closed Curves ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Prey System ◽  
Period 2 ◽  
New Perspective ◽  
Matrix Center

The nonlinear dynamics of predator-prey systems coupled into network is an important issue in recent biological advances. In this research, we consider each node of the coupled network represents a discrete predator-prey system, and the network dynamics is investigated. By applying Jacobian matrix, center manifold theorem and bifurcation theorems, stability of fixed points, flip bifurcation and Neimark-Sacker bifurcation of the discrete predator-prey system are analyzed. Via the method of Lyapunov exponents, the nonchaos-chaos transition of the coupled network along the routes to chaos induced by bifurcations is determined. Numerical simulations are performed to demonstrate the bifurcations, various attractors and dynamic transitions of the coupled network. Via comparison, we find that the coupled network exhibits far richer and more complex behaviors than single predator-prey system, including period-doubling cascades in orbits of period-2, period-4, period-8, invariant closed curves, dynamic windows for periodic orbits and invariant curves, quasiperiodic orbits, tori, and chaotic sets. Moreover, the attractors of the coupled network show more diverse and complicated structures. These results may provide a new perspective on the predator-prey dynamics in complex networks.

scholarly journals Codimension two 1:1 strong resonance bifurcation in a discrete predator-prey model with Holling Ⅳ functional response

2021 ◽  
Vol 7 (2) ◽  
pp. 3150-3168
Author(s):  
Mianjian Ruan ◽  
◽  
Chang Li ◽  
Xianyi Li ◽  
Keyword(s):  
Functional Response ◽  
Center Manifold ◽  
Discrete Version ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Codimension Two ◽  
Predator Prey Model ◽  
Strong Resonance ◽  
Prey Model ◽  
Resonance Bifurcation

<abstract><p>In this paper we revisit a discrete predator-prey model with Holling Ⅳ functional response. By using the method of semidiscretization, we obtain new discrete version of this predator-prey model. Some new results, besides its stability of all fixed points and the transcritical bifurcation, mainly for codimension two 1:1 strong resonance bifurcation, are derived by using the center manifold theorem and bifurcation theory, showing that this system possesses complicate dynamical properties.</p></abstract>

scholarly journals A Holling–Tanner Predator–Prey Model with Strong Allee Effect

2019 ◽  
Vol 29 (11) ◽  
pp. 1930032 ◽  
Author(s):  
Claudio Arancibia-Ibarra ◽  
José D. Flores ◽  
Graeme Pettet ◽  
Peter van Heijster
Keyword(s):  
Allee Effect ◽  
Phase Plane ◽  
Prey Species ◽  
Basins Of Attraction ◽  
Extended Model ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Strong Allee Effect ◽  
Species Production

We analyze a modified Holling–Tanner predator–prey model where the predation functional response is of Holling type II and we incorporate a strong Allee effect associated with the prey species production. The analysis complements the results of previous articles by Saez and González-Olivares [1999] and Arancibia-Ibarra and González-Olivares [2015] discussing Holling–Tanner models which incorporate a weak Allee effect. The extended model exhibits rich dynamics and we prove the existence of separatrices in the phase plane separating basins of attraction related to coexistence and extinction of the species. We also show the existence of a homoclinic curve that degenerates to form a limit cycle and discuss numerous potential bifurcations such as saddle-node, Hopf, and Bogdanov–Takens bifurcations.

scholarly journals Multiple Bifurcations and Chaos in a Discrete Prey-Predator System with Generalized Holling III Functional Response

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Xia Liu ◽  
Yanwei Liu ◽  
Qiaoping Li
Keyword(s):  
Numerical Methods ◽  
Functional Response ◽  
Bifurcation Theory ◽  
Allee Effect ◽  
Center Manifold ◽  
Step Size ◽  
Center Manifold Theorem ◽  
Strong Allee Effect ◽  
Holling Iii Functional Response ◽  
Predator System

A prey-predator system with the strong Allee effect and generalized Holling type III functional response is presented and discretized. It is shown that the combined influences of Allee effect and step size have an important effect on the dynamics of the system. The existences of Flip and Neimark-Sacker bifurcations and strange attractors and chaotic bands are investigated by using the center manifold theorem and bifurcation theory and some numerical methods.

BIFURCATION ANALYSIS OF A DISCRETE-TIME KALDOR MODEL OF BUSINESS CYCLE

2012 ◽  
Vol 22 (08) ◽  
pp. 1250186 ◽  
Author(s):  
XING HE ◽  
CHUANDONG LI ◽  
YONGLU SHU
Keyword(s):  
Business Cycle ◽  
Discrete Time ◽  
Bifurcation Analysis ◽  
Bifurcation Theory ◽  
Center Manifold ◽  
Flip Bifurcation ◽  
Numerical Examples ◽  
Center Manifold Theorem ◽  
Codimension Two ◽  
Codimension Two Bifurcation

This paper reports bifurcation dynamics of a discrete-time Kaldor model of business cycle. By using center manifold theorem and bifurcation theory, it is shown that the model not only undergoes flip bifurcation and Neimark–Sacker bifurcation, but also 1 : 1 resonance of codimension two bifurcation occurs. Some numerical examples are given to support the analytic results.

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