Problem 9.3: Linear stability analysis of the repressilator with mRNA

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The dimensionless dynamical equations we used for the repressilator system including mRNA are

\begin{align} \frac{\mathrm{d}m_i}{\mathrm{d}t} &= \beta\left(\rho + \frac{1}{1 + x_j^n}\right) - m_i, \\[1em] \gamma^{-1}\,\frac{\mathrm{d}x_i}{\mathrm{d}t} &= m_i - x_i. \end{align}

a) Show that the fixed point is unique and satisfies

\begin{align} (x_0 - \beta\rho)(1+x_0^n) = \beta. \end{align}

b) Perform linear stability analysis to show that when the fixed point is unstable, it is an oscillatory instability and that the instability occurs when

\begin{align} \left(\sqrt{\gamma} + \sqrt{\gamma^{-1}}\right)^2 < \frac{3f_0^2}{4+2f_0}, \end{align}

where

\begin{align} f_0 = \frac{\beta n x_0^{n-1}}{(1+x_0^n)^2}. \end{align}