Local Hopf bifurcation analysis and global existence of periodic solutions in a gene expression model with delays

In this paper, local Hopf bifurcation of a gene expression model with three delays is investigated by applying the frequency domain approach. It is shown that Hopf bifurcation will occur as the bifurcation parameter, the sum of all delays, passes through a sequence of critical values. The direction and the stability of bifurcating periodic solutions are determined by the Nyquist criterion and the graphical Hopf bifurcation theorem. By using the Bendixson’s criterion for high-dimensional ordinary differential equations and global Hopf bifurcation theorem for functional differential equations, the global existence of periodic solutions is established. Numerical results are also included to give a verification test of the theoretical analysis.