*(English)*Zbl 1145.34042

This paper concerns the following predator-prey system with delay,

Under the assumption that

system ($*$) has a unique positive equilibrium ${E}_{*}=(\frac{{r}_{1}{a}_{22}+{r}_{2}{a}_{12}}{{a}_{11}{a}_{22}+{a}_{12}{a}_{21}},\frac{{r}_{1}{a}_{21}-{r}_{2}{a}_{11}}{{a}_{11}{a}_{22}+{a}_{12}{a}_{21}})$. 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.

##### MSC:

34K18 | Bifurcation theory of functional differential equations |

34K13 | Periodic solutions of functional differential equations |

34K20 | Stability theory of functional-differential equations |

92D25 | Population dynamics (general) |