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Hopf bifurcation and global periodic solutions in a delayed predator – prey system. (English) Zbl 1090.92052

Summary: This paper is concerned with a delayed predator – prey system with same feedback delays of predator and prey species to their growth, respectively. We prove that a sequence of Hopf bifurcations occur at the positive equilibrium as the delay increases monotonously from zero. By using the theory of normal norm and center manifold reduction, an explicit algorithm for determining the direction of Hopf bifurcations and the stability of bifurcating periodic solutions is derived. In addition, the global existence results of periodic solutions bifurcating from Hopf bifurcations are established by using a global Hopf bifurcation result due to J. Wu [Symmetric functional differential equations and neural networks with memory. Trans. Am. Math. Soc. 350, No. 12, 4799–4838 (1998; Zbl 0905.34034)]. Finally, a numerical example supporting our theoretical predictions is also given.

Our findings are contrasted with recent studies on a delayed predator – prey system with the feedback time delay of prey species to its growth [Y. Song and J. Wei, Local Hopf bifurcation and global periodic solutions in a delayed predator-prey system. J. Math. Anal. Appl. 301, 1–21 (2005; Zbl 1067.34076)]. As the feedback time delay $\tau$ increases monotonously from zero, the positive equilibrium of the latter switches k times from stability to instability to stability. In contrast, the positive equilibrium of our system appears to lose the above property.

##### MSC:
 92D40 Ecology 34K18 Bifurcation theory of functional differential equations 34K20 Stability theory of functional-differential equations 92D25 Population dynamics (general) 37N25 Dynamical systems in biology 34K13 Periodic solutions of functional differential equations
##### Keywords:
predator-prey system; time delay; Hopf bifurcation