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Hopf bifurcation and global stability for a delayed predator-prey system with stage structure for predator. (English) Zbl 1151.34067
Summary: A delayed predator-prey system with stage structure for the predator is studied. It is found that the time delay is harmless for permanence of the stage-structured system. If $$\alpha \beta < 1$$, sufficient conditions which guarantee the global stability of positive equilibrium are given. If $$\alpha \beta > 1$$, we show that the unique positive equilibrium is locally asymptotically stable when the time delay $$\tau ^{*}$$ is sufficiently small, while loss of stability by a Hopf bifurcation can occur as the delay increases.

##### MSC:
 34K60 Qualitative investigation and simulation of models involving functional-differential equations 34K18 Bifurcation theory of functional-differential equations 34K20 Stability theory of functional-differential equations 92D25 Population dynamics (general) 34K13 Periodic solutions to functional-differential equations
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