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Local and global Hopf bifurcation in a delayed hematopoiesis model. (English) Zbl 1090.37547

Summary: We consider the following nonlinear differential equation

$\frac{dx}{dt}=x\left(t\right)\left[\frac{q}{r+{x}^{n}\left(t-\tau \right)}-p\right]·\phantom{\rule{2.em}{0ex}}\left(1\right)$

We first consider the existence of local Hopf bifurcations, and then derive explicit formulas which determine the stability, direction and other properties of bifurcating periodic solutions, using the normal form theory and center manifold reduction. Further, particular attention is focused on the existence of the global Hopf bifurcation. By using the global Hopf bifurcation theory due to Wu [1998], we show that the local Hopf bifurcation of (1) implies the global Hopf bifurcation after the second critical value of the delay $t$. Finally, numerical simulation results are given to support the theoretical predictions.

MSC:
 37G15 Bifurcations of limit cycles and periodic orbits 34K18 Bifurcation theory of functional differential equations 37N25 Dynamical systems in biology 92C50 Medical applications of mathematical biology