Local Hopf bifurcation and global periodic solutions in a delayed predator–prey system
By Yongli Song, Junjie Wei. 2005
213 citations recorded by the source corpus.
Abstract We consider a delayed predator–prey system. We first consider the existence of local Hopf bifurcations, and then derive explicit formulas which enable us to determine the stability and the direction of periodic solutions bifurcating from Hopf bifurcations, using the normal form theory and center manifold argument. Special attention is paid to the global existence of periodic solutions bifurcating from Hopf bifurcations. By using a global Hopf bifurcation result due to Wu [Trans. Amer. Math. Soc. 350 (1998) 4799], we show that the local Hopf bifurcation implies the global Hopf bifurcation after the second critical value of delay. Finally, several numerical simulations supporting the theoretical analysis are also given.
Subfields: Genetics, Geometry and Topology, Public Health, Environmental and Occupational Health
View Paper: https://www.semanticscholar.org/paper/f33314f9dec62ba9c23cac8552564702b4908fe8
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Genetics
In Geometry and Topology
In Public Health, Environmental and Occupational Health
Citations in the corpus
- Bifurcations for a predator-prey system with two delays ✩ (2008, background): “E-mail addresses: syl.mail@163.com (Y. Song), weijj@hit.edu.cn (J. Wei).”
- On Nonlinear Dynamics of Predator-Prey Models with Discrete Delay ⁄ (2009, unknown): “The presentation here is mainly adapted from Ruan [74] and Ruan and Wei [76].”
- RETRACTED: Stability and Hopf bifurcations in a delayed Leslie–Gower predator–prey system (2009, methodology): “We would like to mention that in [17,24,25], the authors investigated not only local Hopf bifurcations but also the global continuation of local Hopf bifurcations for the simplest delayed predator–prey system.”
- Dynamics of the density dependent predator–prey system with Beddington–DeAngelis functional response (2011, background): “(4.2) Let px∗ + s2y∗ = a1 > 0, s1y∗ = a2 > 0, (ps1 − qr)x∗y∗ = a4 > 0, ps2x∗y∗ = a3 > 0, then equation (4.2) becomes λ2 + a1λ + a2λe −λτ + a3 + a4e−λτ = 0, (4.3) which has been extensively studied by many researchers (see, for example Ruan [18], Bellman and Cooke [2], Song and Wei [20]).”
- Stability and Hopf bifurcation in a three-species system with feedback delays (2011, background): “They found that the unique positive equilibrium of system (1.1) will no longer be absolutely stable and the switches from stability to instability to stability disappear, which had been obtained for system (1.1 )b y Song and Wei [ 10 ].”
- Global existence of periodic solutions in a tri-neuron network model with delays (2004, methodology): “Then, from Lemma 2.3 of Li and Wei [14], we have the following transversality condition.”
- Stability and Hopf bifurcation in a ratio-dependent predator–prey system with stage structure (2008, background): “Recently, great attention has been received and a large body of work has been carried out on the existence of Hopf bifurcations in population models (see, for example, [1,7,8,12,13,18–21,23–27,30–32] and references cited therein).”