Hopf bifurcation and global stability of density‐dependent model with discrete delays involving Beddington‐DeAngelis functional response

2021 ◽  
Author(s):  
Anuraj Singh ◽  
Ankit Parwaliya ◽  
Ajay Kumar
Keyword(s):  
Hopf Bifurcation ◽  
Global Stability ◽  
Functional Response ◽  
Density Dependent ◽  
Discrete Delays ◽  
Dependent Model

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.

THE EFFECTS OF HARVESTING AND TIME DELAY ON PREDATOR-PREY SYSTEMS WITH BEDDINGTON–DEANGELIS FUNCTIONAL RESPONSE

2012 ◽  
Vol 05 (01) ◽  
pp. 1250008 ◽  
Author(s):  
XINYOU MENG ◽  
HAIFENG HUO ◽  
XIAOBING ZHANG
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Global Stability ◽  
Functional Response ◽  
Local Stability ◽  
Positive Equilibrium ◽  
Combined Effects ◽  
Predator Prey ◽  
Region Of Stability ◽  
Two Parameter

The combined effects of harvesting and time delay on predator-prey systems with Beddington–DeAngelis functional response are studied. The region of stability in model with harvesting of the predator, local stability of equilibria and the existence of Hopf bifurcation are obtained by analyzing the associated characteristic equation due to the two-parameter geometric criteria developed by Ma, Feng and Lu [Discrete Contin. Dyn. Syst. Ser B9 (2008) 397–413]. The global stability of the positive equilibrium is investigated by the comparison theorem. Furthermore, local stability of steady states and the existence of Hopf bifurcation for prey harvesting are also considered. Numerical simulations are given to illustrate our theoretical findings.

Sign in / Sign up

Export Citation Format

Share Document