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 Hopf Bifurcation Analysis for a Semiratio-Dependent Predator-Prey System with Two Delays

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Ming Zhao
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Center Manifold ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Prey System ◽  
Two Delays ◽  
Normal Form Method ◽  
Nonmonotonic Functional Response

This paper is concerned with a semiratio-dependent predator-prey system with nonmonotonic functional response and two delays. It is shown that the positive equilibrium of the system is locally asymptotically stable when the time delay is small enough. Change of stability of the positive equilibrium will cause bifurcating periodic solutions as the time delay passes through a sequence of critical values. The properties of Hopf bifurcation such as direction and stability are determined by using the normal form method and center manifold theorem. Numerical simulations confirm our theoretical findings.

scholarly journals Hopf bifurcation and stability in a Beddington-DeAngelis predator-prey model with stage structure for predator and time delay incorporating prey refuge

2019 ◽  
Vol 17 (1) ◽  
pp. 141-159 ◽  
Author(s):  
Zaowang Xiao ◽  
Zhong Li ◽  
Zhenliang Zhu ◽  
Fengde Chen
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Prey Refuge ◽  
Predator Species ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Bifurcation And Stability

Abstract In this paper, we consider a Beddington-DeAngelis predator-prey system with stage structure for predator and time delay incorporating prey refuge. By analyzing the characteristic equations, we study the local stability of the equilibrium of the system. Using the delay as a bifurcation parameter, the model undergoes a Hopf bifurcation at the coexistence equilibrium when the delay crosses some critical values. After that, by constructing a suitable Lyapunov functional, sufficient conditions are derived for the global stability of the system. Finally, the influence of prey refuge on densities of prey species and predator species is discussed.

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