Stability and Bifurcation Analysis of Cyclic Genetic Regulatory Networks with Mixed Time Delays

Abstract

This paper analyzes the stability and bifurcation criteria of cyclic genetic regulatory networks with mixed time delays (discrete and Gamma-type distributed delays). It is of more realistic biological background to use Gamma-type distributed delay kernels to describe the delays in the process between genes. The key aim of our research is to reveal more about the dynamic functions played by biochemical parameters and different mechanisms of discrete and distributed time delays. The existence of positive equilibria in this kind of genetic network is verified. With the mathematical tools of subharmonic function and complex theory, exact conditions of biochemical parameters for stability and bifurcations in cyclic genetic regulatory networks with both positive and negative gains are deduced, respectively. Finally, several simulation examples are adopted to deepen our analysis.