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

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

$\left\{\begin{array}{c}\frac{dx\left(t\right)}{dt}=x\left(t\right)\left[{r}_{1}-{a}_{11}x\left(t\right)-{a}_{12}y\left(t\right)\right],\hfill \\ \frac{dy\left(t\right)}{dt}=y\left(t\right)\left[-{r}_{2}+{a}_{21}x\left(t\right)-{a}_{22}y\left(t-\tau \right)\right]·\hfill \end{array}\right\\phantom{\rule{2.em}{0ex}}\left(*\right)$

Under the assumption that

${r}_{1}{a}_{21}-{r}_{2}{a}_{11}>0,\phantom{\rule{2.em}{0ex}}\left(\text{H}\right)$

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