##
**Global qualitative analysis for a predator-prey system with delay.**
*(English)*
Zbl 1145.34042

This paper concerns the following predator-prey system with delay,
\[
\begin{cases} \frac {dx(t)}{dt} = x(t)[r_1-a_{11}x(t)-a_{12}y(t)], \\ \frac{dy(t)}{dt} = y(t)[-r_2+a_{21}x(t)-a_{22}y(t-\tau)]. \end{cases}\tag{\(*\)}
\]
Under the assumption that
\[
r_1a_{21}-r_2a_{11}>0, \tag{\text{H}}
\]
system (\(*\)) has a unique positive equilibrium \(E_{*}=(\frac {r_1a_{22}+r_2a_{12}}{a_{11}a_{22}+a_{12}a_{21}}, \frac {r_1a_{21}-r_2a_{11}}{a_{11}a_{22}+a_{12}a_{21}})\). When \(\tau=0\), it is known that \(E_{*}\) is globally asymptotically stable. Also, Wendi Wang and Zhien Ma showed that solutions of (\(*\)) are bounded, uniformly persistent and the delay is harmless for the uniform persistence [J.Math.Anal.Appl.158, 256–268 (1991; Zbl 0731.34085)].

First, by constructing suitable Lyapunov functionals, the authors provide a set of sufficient conditions on the global stability of \(E_{*}\). Notice that the conditions can not be reduced to those for the case where \(\tau=0\). Then, it was shown that local Hopf bifurcation can occur. Moreover, the direction, stability and period of the periodic solution bifurcating from \(E_{*}\) was investigates through the normal form theorem and center manifold argument [R. D. Nussbaum, Ann. Mat. Pura Appl. IV. Ser. 101, 263–306 (1974; Zbl 0323.34061)]. Finally, using the global Hopf bifurcation theorem due to J. Wu [Trans. Am. Math. Soc. 350, 4799–4838 (1998; Zbl 0905.34034)], the authors also investigates the global existence of a periodic solution of (\(*\)). This may be the most important contribution of this paper as results on global existence of periodic solutions are scarce in the literature.

First, by constructing suitable Lyapunov functionals, the authors provide a set of sufficient conditions on the global stability of \(E_{*}\). Notice that the conditions can not be reduced to those for the case where \(\tau=0\). Then, it was shown that local Hopf bifurcation can occur. Moreover, the direction, stability and period of the periodic solution bifurcating from \(E_{*}\) was investigates through the normal form theorem and center manifold argument [R. D. Nussbaum, Ann. Mat. Pura Appl. IV. Ser. 101, 263–306 (1974; Zbl 0323.34061)]. Finally, using the global Hopf bifurcation theorem due to J. Wu [Trans. Am. Math. Soc. 350, 4799–4838 (1998; Zbl 0905.34034)], the authors also investigates the global existence of a periodic solution of (\(*\)). This may be the most important contribution of this paper as results on global existence of periodic solutions are scarce in the literature.

Reviewer: Yuming Chen (Waterloo)

### MSC:

34K18 | Bifurcation theory of functional-differential equations |

34K13 | Periodic solutions to functional-differential equations |

34K20 | Stability theory of functional-differential equations |

92D25 | Population dynamics (general) |

### Keywords:

predator-prey system; delay; global stability; Hopf bifurcation; normal form; center manifold; global Hopf bifurcation
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\textit{C. Sun} et al., Chaos Solitons Fractals 32, No. 4, 1582--1596 (2007; Zbl 1145.34042)

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