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Local Hopf bifurcation and global periodic solutions in a delayed predator–prey system. (English) Zbl 1067.34076
The article considers a delayed predator-prey system of the form \[ \begin{aligned}\dot x(t)&=x(t)[r_1-a_{11}x(t-\tau)-a_{12}y(t)],\\ \dot y(t)&=y(t)[-r_2+a_{21}x(t)-a_{22}y(t)],\end{aligned} \] where all constants are positive. First, the authors discuss the existence of local Hopf bifurcations, deriving explicit formulas for the stability and direction of the branch of periodic solutions emerging from the Hopf bifurcation. This is achieved using normal form theory and center manifold theory. Next, the authors consider the global existence of periodic solutions bifurcating from the Hopf bifurcation. Using a result from J. Wu [Trans. Am. Math. Soc. 350, No. 12, 4799–4838 (1998; Zbl 0905.34034)], they prove that, for delays greater than a critical value, there always exist periodic solutions. Finally, several numerical simulations supporting the theoretical analysis are given.

MSC:
34K18 Bifurcation theory of functional-differential equations
34K13 Periodic solutions to functional-differential equations
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