Hopf bifurcation in a Volterra prey--predator model with strong kernel
type: Item
instance of: scholarly article
title: Hopf bifurcation in a Volterra prey--predator model with strong kernel
zbMATH Open document ID: 1073.34086
author: Shaowen Li
author: Xiaofeng Liao
author: Chunguang Li
published in: Chaos, Solitons and Fractals
publication date: 2005-04-21T00:00:00Z
review text: In the prey-predator model with distributed delays and strong kernels \(F_i(u)=a_i^2u\text{e}^{-a_iu}\), \(a_i>0\), \(i=1, 2\), \[ \begin{aligned} \dot{N}_1 &=b_1N_1 \biggl((1-c_{11}N_1-c_{12}\int_0^{+\infty}N_1(t-u)F_1(u)\,du\biggr),\\ \dot{N}_2 &=b_2N_2\biggl(-1+c_{21}\int_0^{+\infty}N_1(t-u)F_2(u)\,du\biggr), \end{aligned} \] \(b_i>0\), \(c_{ij}>0\), suggested by \textit{J. M. Cushing} [J. Math. Biol. 3, 369--380 (1976; Zbl 0342.92012)], the stability of the bifurcating periodic solutions is investigated taking the density coefficient \(c_{11}\) of the prey as bifurcation parameter. For the determination of the direction and stability of the Hopf bifurcation, the Nyquist criterion and graphical Hopf bifurcation theorem are used. Numerical simulation results are presented.
Mathematics Subject Classification ID: 34K18
Mathematics Subject Classification ID: 92D25
Mathematics Subject Classification ID: 34C60
zbMATH DE Number: 2160091
zbMATH Keywords: prey-predator model
zbMATH Keywords: distributed delays
zbMATH Keywords: Hopf bifurcation
reviewed by: Boris V. Loginov
MaRDI profile type: Publication
full work available at URL: j.chaos.2004.02.048
OpenAlex ID: W2049069733
Wikidata QID: Q115580062
cites work: Bifurcations and chaos in a predator-prey model with delay and a laser-diode system with self-sustained pulsations.
cites work: Stability and bifurcation for a delayed predator-prey model and the effect of diffusion
cites work: Hopf bifurcation and stability analysis in a harvested one-predator-two-prey model
cites work: Predator prey interactions with time delays
cites work: Periodic orbits arising from Hopf bifurcations in a Volterra prey-predator model
cites work: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields
cites work: Q3930015
cites work: Harmonic balance and the Hopf bifurcation
cites work: The Hopf bifurcation theorem and its applications to nonlinear oscillations in circuits and systems
cites work: Q4379546
cites work: Computations of limit cycles via higher-order harmonic balance approximation
cites work: FREQUENCY DOMAIN APPROACH TO COMPUTATION AND ANALYSIS OF BIFURCATIONS AND LIMIT CYCLES: A TUTORIAL
cites work: Hopf bifurcation on a two-neuron system with distributed delays: a frequency domain approach
DOI: 10.1016/J.CHAOS.2004.02.048
description: scientific article; zbMATH DE number 2160091