Linear Stability Analysis of a Delay Differential Integral Equation Motivated by Genetic Networks

This work presents a computational study on the linear stability analysis of a genetic network with time delay. The network is modeled as a continuous system and takes the form of a nonlinear delay differential integral equation coupled to an ordinary differential equation. Analysis of the stable equilibria shows the existence of a critical time delay beyond which limit cycle oscillations are born in a Hopf bifurcation. We confirm our results by discretizing the continuous model into an N-dimensional system and showing that the associated critical time delays for the discrete system approach the critical time delay for the continuous system as N becomes large.