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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, $$ \cases \frac {dx(t)}{dt} = x(t)[r_1-a_{11}x(t)-a_{12}y(t)], \\ \frac{dy(t)}{dt} = y(t)[-r_2+a_{21}x(t)-a_{22}y(t-\tau)]. \endcases\tag$*$ $$ 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, {\it Wendi Wang} and {\it 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 [{\it 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 {\it 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.

34K18Bifurcation theory of functional differential equations
34K13Periodic solutions of functional differential equations
34K20Stability theory of functional-differential equations
92D25Population dynamics (general)
Full Text: DOI
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