scholarly journals Bifurcations of a predator-prey system with cooperative hunting and Holling III functional response

2021 ◽  
Author(s):  
Yong Yao ◽  
Teng Song ◽  
Zuxiong Li
Keyword(s):  
Functional Response ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Weak Focus ◽  
Cooperative Hunting ◽  
Prey System ◽  
Saddle Node ◽  
Boundary Equilibrium ◽  
Parameter Values ◽  
Holling Iii Functional Response

Abstract In this paper, we consider the dynamics of a predator-prey system of Gause type with cooperative hunting among predators and Holling III functional response. The known work numerically shows that the system exhibits saddle-node and Hopf bifurcations except homoclinic bifurcation for some special parameter values. Our results show that there are a weak focus of multiplicity three and a cusp of codimension two for general parameter conditions and the system can exhibit various bifurcations as perturbing the bifurcation parameters appropriately, such as the transcritical and the pitchfork bifurcations at the degenerate boundary equilibrium, the saddle-node and the Bogdanov-Takens bifurcations at the degenerate positive equilibrium and the Hopf bifurcation around the weak focus. The comparative study demonstrates that the dynamics are far richer and more complex than that of the system without cooperative hunting among predators. The analysis results reveal that appropriate intensity of cooperative hunting among predators is beneficial for the persistence of predators and the diversity of ecosystem.

Higher Codimension Bifurcation Analysis of Predator–Prey Systems with Nonmonotonic Functional Responses

2020 ◽  
Vol 30 (12) ◽  
pp. 2050167
Author(s):  
Jinhui Yao ◽  
Guihua Li ◽  
Gang Guo
Keyword(s):  
Functional Response ◽  
Positive Equilibrium ◽  
Cusp Point ◽  
Functional Responses ◽  
Predator Prey ◽  
Prey System ◽  
Saddle Node ◽  
Boundary Equilibrium ◽  
Nonmonotonic Functional Response ◽  
Nilpotent Saddle

In this paper, we study the dynamic behaviors of a predator–prey system with a general form of nonmonotonic functional response. Through analysis, it is found that the system exists in extinction equilibrium, boundary equilibrium and two positive equilibria, one or no positive equilibrium. Furthermore, the conditions are given such that the boundary equilibrium is a saddle, node or a saddle-node point of codimension 1, 2 or 3. Then, some conditions are obtained so that the unique positive equilibrium of the system is a cusp point of codimension 2, 3 and higher or a saddle-node one of codimension 1 or 3, or a nilpotent saddle-node of codimension 4. When there are two positive equilibria in the system, the equilibrium with a large number of preys is a saddle point. For the other one, the system may undergo Hopf bifurcation. To verify our conclusion, we consider the functional response function in the literature [ Zhu et al., 2002 ; Xiao & Ruan, 2001 ]. Finally, we give a brief discussion and numerical simulation.

scholarly journals Qualitative Analysis for a Predator Prey System with Holling Type III Functional Response and Prey Refuge

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Xia Liu ◽  
Yepeng Xing
Keyword(s):  
Qualitative Analysis ◽  
Global Stability ◽  
Numerical Simulations ◽  
Functional Response ◽  
Limit Cycle ◽  
Positive Equilibrium ◽  
Prey Refuge ◽  
Predator Prey ◽  
Prey System ◽  
Holling Iii Functional Response

A predator prey system with Holling III functional response and constant prey refuge is considered. By using the Dulac criterion, we discuss the global stability of the positive equilibrium of the system. By transforming the system to a Liénard system, the conditions for the existence of exactly one limit cycle for the system are given. Some numerical simulations are presented.

scholarly journals On a Periodic Predator-Prey System with Holling III Functional Response and Stage Structure for Prey

2010 ◽  
Vol 2010 ◽  
pp. 1-12
Author(s):  
Xiangzeng Kong ◽  
Zhiqin Chen ◽  
Li Xu ◽  
Wensheng Yang
Keyword(s):  
Functional Response ◽  
Prey Species ◽  
Necessary Conditions ◽  
Stage Structure ◽  
Predator Prey ◽  
Prey System ◽  
Sufficient And Necessary Conditions ◽  
Holling Iii Functional Response

We propose and study the permanence of the following periodic Holling III predator-prey system with stage structure for prey and both two predators which consume immature prey. Sufficient and necessary conditions which guarantee the predator and the prey species to be permanent are obtained.

Stability and Hopf Bifurcation of a Delayed Density-Dependent Predator–Prey System with Beddington–DeAngelis Functional Response

2016 ◽  
Vol 26 (10) ◽  
pp. 1650165 ◽  
Author(s):  
Haiyin Li ◽  
Gang Meng ◽  
Zhikun She
Keyword(s):  
Hopf Bifurcation ◽  
Density Dependence ◽  
Functional Response ◽  
Positive Equilibrium ◽  
Stability Switches ◽  
Predator Prey ◽  
Geometric Criterion ◽  
Density Dependent ◽  
Prey System ◽  
The Stability

In this paper, we investigate the stability and Hopf bifurcation of a delayed density-dependent predator–prey system with Beddington–DeAngelis functional response, where not only the prey density dependence but also the predator density dependence are considered such that the studied predator–prey system conforms to the realistically biological environment. We start with the geometric criterion introduced by Beretta and Kuang [2002] and then investigate the stability of the positive equilibrium and the stability switches of the system with respect to the delay parameter [Formula: see text]. Especially, we generalize the geometric criterion in [Beretta & Kuang, 2002] by introducing the condition [Formula: see text] which can be assured by the condition [Formula: see text], and adopting the technique of lifting to define the function [Formula: see text] for alternatively determining stability switches at the zeroes of [Formula: see text]s. Afterwards, by the Poincaré normal form for Hopf bifurcation in [Kuznetsov, 1998] and the bifurcation formulae in [Hassard et al., 1981], we qualitatively analyze the properties for the occurring Hopf bifurcations of the system (3). Finally, an example with numerical simulations is given to illustrate the obtained results.

STABILITY AND HOPF BIFURCATION FOR AN ECO-EPIDEMIOLOGY MODEL WITH HOLLING-III FUNCTIONAL RESPONSE AND DELAYS

2008 ◽  
Vol 01 (03) ◽  
pp. 377-389 ◽  
Author(s):  
WEI ZOU ◽  
JIEHUA XIE ◽  
ZUOLIANG XIONG
Keyword(s):  
Hopf Bifurcation ◽  
Functional Differential Equations ◽  
Positive Equilibrium ◽  
Asymptotically Stable ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Retarded Functional Differential Equations ◽  
Boundary Equilibrium ◽  
Functional Differential ◽  
Holling Iii Functional Response

In this paper, a system of retarded functional differential equations is proposed as a predator-prey model with disease in the prey. The invariance of non-negativity, nature of boundary equilibrium and global stability are analyzed. It also shows that positive equilibrium is locally asymptotically stable when time delay τ = τ1 + τ2 is suitable small, while a loss of stability by a Hopf bifurcation can occur around the positive equilibrium as the delays increase.

scholarly journals Dynamics of a Predator-Prey System with Beddington-DeAngelis Functional Response and Delays

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Nai-Wei Liu ◽  
Ting-Ting Kong
Keyword(s):  
Asymptotic Stability ◽  
Functional Response ◽  
Global Asymptotic Stability ◽  
Stage Structure ◽  
Necessary Condition ◽  
Positive Equilibrium ◽  
Sufficient Condition ◽  
Predator Prey ◽  
Prey System ◽  
Hyperbolic Curve

We consider a predator-prey system with Beddington-DeAngelis functional response and delays, in which not only the stage structure on prey but also the delay due to digestion is considered. First, we give a sufficient and necessary condition for the existence of a unique positive equilibrium by analyzing the corresponding locations of a hyperbolic curve and a line. Then, by constructing an appropriate Lyapunov function, we prove that the positive equilibrium is locally asymptotically stable under a sufficient condition. Finally, by using comparison theorem and theω-limit set theory, we study the global asymptotic stability of the boundary equilibrium and the positive equilibrium, respectively. Also, we obtain a sufficient condition to assure the global asymptotic stability.

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