scholarly journals Multiple Focus and Hopf Bifurcations in a Predator-Prey System with Nonmonotonic Functional Response

2006 ◽  
Vol 66 (3) ◽  
pp. 802-819 ◽  
Author(s):  
Dongmei Xiao ◽  
Huaiping Zhu
Keyword(s):  
Functional Response ◽  
Hopf Bifurcations ◽  
Predator Prey ◽  
Prey System ◽  
Multiple Focus ◽  
Nonmonotonic Functional Response

THE EFFECT OF DIFFUSION FOR A PREDATOR–PREY SYSTEM WITH NONMONOTONIC FUNCTIONAL RESPONSE

2004 ◽  
Vol 14 (12) ◽  
pp. 4309-4316 ◽  
Author(s):  
ZHIHUA LIU ◽  
RONG YUAN
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Functional Response ◽  
Bifurcation Analysis ◽  
Predator Prey ◽  
Prey System ◽  
Nonmonotonic Functional Response

We consider the delayed predator–prey system with diffusion. The bifurcation analysis of the model shows that Hopf bifurcation can occur under some conditions and the system has a Bogdanov–Takens singularity for any time delay value.

scholarly journals Permanence of a Semi-Ratio-Dependent Predator-Prey System with Nonmonotonic Functional Response and Time Delay

2009 ◽  
Vol 2009 ◽  
pp. 1-6 ◽  
Author(s):  
Xuepeng Li ◽  
Wensheng Yang
Keyword(s):  
Time Delay ◽  
Functional Response ◽  
Negative Feedback ◽  
Time Delays ◽  
Sufficient Conditions ◽  
Prey Population ◽  
Predator Prey ◽  
Prey System ◽  
Ratio Dependent ◽  
Nonmonotonic Functional Response

Sufficient conditions for permanence of a semi-ratio-dependent predator-prey system with nonmonotonic functional response and time delay    are obtained, where and stand for the density of the prey and the predator, respectively, and is a constant. stands for the time delays due to negative feedback of the prey population.

scholarly journals Stability and Bifurcation Analysis of a Delayed Leslie-Gower Predator-Prey System with Nonmonotonic Functional Response

2013 ◽  
Vol 2013 ◽  
pp. 1-19 ◽  
Author(s):  
Jiao Jiang ◽  
Yongli Song
Keyword(s):  
Functional Response ◽  
Limit Cycle ◽  
Upper And Lower Solutions ◽  
Heteroclinic Orbits ◽  
Positive Equilibrium ◽  
Hopf Bifurcations ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Nonmonotonic Functional Response

A delayed Leslie-Gower predator-prey model with nonmonotonic functional response is studied. The existence and local stability of the positive equilibrium of the system with or without delay are completely determined in the parameter plane. Using the method of upper and lower solutions and monotone iterative scheme, a sufficient condition independent of delay for the global stability of the positive equilibrium is obtained. Hopf bifurcations induced by the ratio of the intrinsic growth rates of the predator and prey and by delay, respectively, are found. Employing the normal form theory, the direction and stability of Hopf bifurcations can be explicitly determined by the parameters of the system. Some numerical simulations are given to support and extend our theoretical results. Two limit cycles enclosing an equilibrium, one limit cycle enclosing three equilibria and different types of heteroclinic orbits such as connecting two equilibria and connecting a limit cycle and an equilibrium are also found by using analytic and numerical methods.

scholarly journals Complexity Analysis of a Modified Predator-Prey System with Beddington–DeAngelis Functional Response and Allee-Like Effect on Predator

2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Shuangte Wang ◽  
Hengguo Yu
Keyword(s):  
Functional Response ◽  
Theoretical Basis ◽  
Complexity Analysis ◽  
Future Research ◽  
Hopf Bifurcations ◽  
Predator Prey ◽  
Dynamical Behaviors ◽  
Prey System ◽  
Threshold Conditions ◽  
Theoretical Results

In this paper, complex dynamical behaviors of a predator-prey system with the Beddington–DeAngelis functional response and the Allee-like effect on predator were studied by qualitative analysis and numerical simulations. Theoretical derivations have given some sufficient and threshold conditions to guarantee the occurrence of transcritical, saddle-node, pitchfork, and nondegenerate Hopf bifurcations. Computer simulations have verified the feasibility and effectiveness of the theoretical results. In short, we hope that these works could provide a theoretical basis for future research of complexity in more predator-prey ecosystems.

scholarly journals Stability and Coexistence of a Diffusive Predator-Prey System with Nonmonotonic Functional Response and Fear Effect

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Xiaozhou Feng ◽  
Hao Sun ◽  
Yangfan Xiao ◽  
Feng Xiao
Keyword(s):  
Functional Response ◽  
Sufficient Conditions ◽  
Degree Theory ◽  
Upper And Lower Bounds ◽  
Positive Equilibrium ◽  
Equilibrium Solutions ◽  
Predator Prey ◽  
Fear Effect ◽  
Prey System ◽  
Nonmonotonic Functional Response

This paper investigates the diffusive predator-prey system with nonmonotonic functional response and fear effect. Firstly, we discussed the stability of the equilibrium solution for a corresponding ODE system. Secondly, we established a priori positive upper and lower bounds for the positive solutions of the PDE system. Thirdly, sufficient conditions for the local asymptotical stability of two positive equilibrium solutions of the system are given by using the method of eigenvalue spectrum analysis of linearization operator. Finally, the existence and nonexistence of nonconstant positive steady states of this reaction-diffusion system are established by the Leray–Schauder degree theory and Poincaré inequality.

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