On the existence of oscillatory solutions in negative feedback cellular control processes

Abstract

It is shown that the differential equation $$\frac{{d^3 Z}}{{dt^3 }} + (\alpha + \beta + \gamma )\frac{{d^2 Z}}{{dt^2 }} + (\alpha \beta + \beta \gamma + \gamma \alpha )\frac{{dZ}}{{dt}} + \alpha \beta \gamma Z = (1 + Z^m )^{ - 1}$$ has at least one periodic solution past the instability of the stationary state solution, Z=Z0, the unique real positive root of $$\beta \gamma Z = (1 + Z^m )^{ - 1}$$