scholarly journals Global dynamics of a predator-prey system with density-dependent mortality and ratio-dependent functional response

2017 ◽  
Vol 22 (11) ◽  
pp. 0-0
Author(s):  
Xin Jiang ◽  
◽  
Zhikun She ◽  
Shigui Ruan ◽  
Keyword(s):  
Functional Response ◽  
Global Dynamics ◽  
Predator Prey ◽  
Density Dependent ◽  
Density Dependent Mortality ◽  
Prey System ◽  
Ratio Dependent ◽  
Dependent Mortality

Global dynamics of a ratio-dependent predator-prey system

2001 ◽  
Vol 43 (3) ◽  
pp. 268-290 ◽  
Author(s):  
Dongmei Xiao ◽  
Shigui Ruan
Keyword(s):  
Global Dynamics ◽  
Predator Prey ◽  
Prey System ◽  
Ratio Dependent

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.

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