Stability and bifurcation analysis of arbitrarily high-dimensional genetic regulatory networks with hub structure and bidirectional coupling

This paper mainly studies the stability and Hopf bifurcation criteria of hub-based genetic regulatory networks with multiple delays and bidirectional couplings. The hub-structured network is an important motif in complex networks, which provides a new view angle on structure to describe the regulation mechanism between genes (including both mRNAs and proteins). It is well known that hubs play a leading role in characterizing the network dynamical behaviors. However, the dynamics of hubs or star-coupled systems is not well understood. In this paper, we first examine the existence of the positive equilibria in this type of genetic networks. By analyzing the associated characteristic equation, we present sufficient conditions of biochemical parameters for delay-independent local stability in hub-coupled genetic regulatory networks. Then we investigate their Hopf bifurcation when such networks lose their stability. Specific conditions for delay-dependent stability and Hopf bifurcations in genetic networks with hub structure are derived. It is found that the dynamics of hub-structured genetic regulatory networks has no direct relationship with single time delay or individual connection, but instead depends on the sum of all delays among all genes and the product of the connection strengths between all genes. Finally, some simulation examples are provided to substantiate our analysis.