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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, \[ \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.

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)
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