Stability and Bifurcation of Ring-Structured Genetic Regulatory Networks With Time Delays

Recently nonlinear differential equations with time delays have been proposed to model genetic regulatory networks, which provide a powerful tool for understanding gene regulatory processes in living organisms. In this paper we study the stability and bifurcation of a class of genetic regulatory networks with ring topology and multiple time delays at its equilibrium state. We first present necessary and sufficient conditions for delay-independently local stability of such genetic regulatory networks in the parameter space. Then we investigate their bifurcation when such networks lose their stability. Although such networks may have multiple delays and different connection strengths among individual nodes, their stability and bifurcation depends on the sum of all time delays among all elements (including both mRNAs and proteins) and the product of the connection strengths between all elements. An autoregulatory network and a repressilatory network are employed to illustrate the theorems developed in this study.