Insights from a qualitative analysis of a gene expression model with delays

Delays appear in the dynamics of many systems due to non-vanishing reaction times control systems. In biochemical systems, long sequences repeated steps, especially biopolymerization processes, can be modeled by delays. However, modelling with delays is often complicated physical constraints, such as requirement that solutions representing concentrations chemical species remain positive. this work, we consider a model for detoxifying enzyme whose synthesis controlled its substrate. The includes binding-site clearance delays, caused time required an RNA polymerase or ribosome clear binding site before another machine bind. existence positive equilibrium and positivity boundedness corresponding delay-differential equations are proven. addition, stability studied using “small-gain” theorem.