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>

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.

Bifurcation analysis and control of a delayed stage-structured predator–prey model with ratio-dependent Holling type III functional response

2019 ◽  
Vol 26 (13-14) ◽  
pp. 1232-1245
Author(s):  
Miao Peng ◽  
Zhengdi Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Functional Response ◽  
Control Method ◽  
Equilibrium Points ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Type Iii ◽  
Predator Prey Model ◽  
Prey Model ◽  
Ratio Dependent

A delayed stage-structured predator–prey model with ratio-dependent Holling type III functional response is proposed and explored in this study. We discuss the positivity and the existence of equilibrium points. By choosing time delay as the bifurcation parameter and analyzing the relevant characteristic equations, the local stability of the trivial equilibrium, the predator-extinction equilibrium, and the coexistence equilibrium of the system is investigated. In accordance with the normal form method and center manifold theorem, the property analysis of Hopf bifurcation of the system is obtained. Furthermore, for the purpose of protecting the stability of such a biological system, a hybrid control method is presented to control the Hopf bifurcation. Finally, numerical examples are given to verify the theoretical findings.

BIFURCATION AND CHAOS IN A DISCRETE PREDATOR–PREY MODEL WITH HOLLING TYPE-III FUNCTIONAL RESPONSE AND HARVESTING EFFECT

2021 ◽  
pp. 1-28
Author(s):  
ANURAJ SINGH ◽  
PREETI DEOLIA
Keyword(s):  
Functional Response ◽  
Sufficient Conditions ◽  
Flip Bifurcation ◽  
Bifurcation Structure ◽  
Bifurcation And Chaos ◽  
Predator Prey ◽  
Type Iii ◽  
Predator Prey Model ◽  
Prey Model ◽  
Wide Range

In this paper, we study a discrete-time predator–prey model with Holling type-III functional response and harvesting in both species. A detailed bifurcation analysis, depending on some parameter, reveals a rich bifurcation structure, including transcritical bifurcation, flip bifurcation and Neimark–Sacker bifurcation. However, some sufficient conditions to guarantee the global asymptotic stability of the trivial fixed point and unique positive fixed points are also given. The existence of chaos in the sense of Li–Yorke has been established for the discrete system. The extensive numerical simulations are given to support the analytical findings. The system exhibits flip bifurcation and Neimark–Sacker bifurcation followed by wide range of dense chaos. Further, the chaos occurred in the system can be controlled by choosing suitable value of prey harvesting.

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