Hopf bifurcation analysis of a delayed fractional-order genetic regulatory network model

In this paper, we propose a novel fractional-order two-gene regulatory network model with delays, which can describe the memory and hereditary properties of genetic regulatory networks more suitably. It is the first time that the dynamics of the stability and Hopf bifurcation are investigated for the delayed fractional-order model of two-gene regulatory network. The total delay is chosen as the bifurcation parameter of the network, and the sufficient conditions of the stability and Hopf bifurcation are achieved through analyzing its characteristic equation. It is found that the delayed fractional-order genetic network can generate a Hopf bifurcation when the total delay passes through some critical values, which can be determined exactly by dealing with the characteristic equation of the network. Finally, the validity of our theoretical analysis is illustrated by carrying out the numerical simulation for the example, and some desirable dynamical behaviors of the case are obtained by choosing the appropriate fractional order. It is discovered that the onset of the Hopf bifurcation increases distinctly when the fractional order deceases. Therefore, the stability domain of the network is inversely proportion to the fractional order of the network.