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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,

dx(t) dt=x(t)[r 1 -a 11 x(t)-a 12 y(t)],dy(t) dt=y(t)[-r 2 +a 21 x(t)-a 22 y(t-τ)]·(*)

Under the assumption that

r 1 a 21 -r 2 a 11 >0,(H)

system (*) has a unique positive equilibrium E * =(r 1 a 22 +r 2 a 12 a 11 a 22 +a 12 a 21 ,r 1 a 21 -r 2 a 11 a 11 a 22 +a 12 a 21 ). When τ=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 τ=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:
34K18Bifurcation theory of functional differential equations
34K13Periodic solutions of functional differential equations
34K20Stability theory of functional-differential equations
92D25Population dynamics (general)