Linear Analysis of an Integro-Differential Delay Equation Model

Abstract

This paper presents a computational study of the stability of the steady state solutions of a biological model with negative feedback and time delay. The motivation behind the construction of our system comes from biological gene networks and the model takes the form of an integro-delay differential equation (IDDE) coupled to a partial differential equation. Linear analysis shows the existence of a critical delay where the stable steady state becomes unstable. Closed form expressions for the critical delay and associated frequency are found and confirmed by approximating the IDDE model with a system of $N$ delay differential equations (DDEs) coupled to $N$ ordinary differential equations. An example is then given that shows how the critical delay for the DDE system approaches the results for the IDDE model as $N$ becomes large.