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    "# Problem 9.3: Linear stability analysis of the repressilator with mRNA\n",
    "\n",
    "<hr>"
   ]
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    "The dimensionless dynamical equations we used for the repressilator system including mRNA are\n",
    "\n",
    "\\begin{align}\n",
    "\\frac{\\mathrm{d}m_i}{\\mathrm{d}t} &= \\beta\\left(\\rho + \\frac{1}{1 + x_j^n}\\right) - m_i, \\\\[1em]\n",
    "\\gamma^{-1}\\,\\frac{\\mathrm{d}x_i}{\\mathrm{d}t} &= m_i - x_i.\n",
    "\\end{align}\n",
    "\n",
    "**a)** Show that the fixed point is unique and satisfies\n",
    "\n",
    "\\begin{align}\n",
    "(x_0 - \\beta\\rho)(1+x_0^n) = \\beta.\n",
    "\\end{align}\n",
    "\n",
    "\n",
    "**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\n",
    "\n",
    "\\begin{align}\n",
    "\\left(\\sqrt{\\gamma} + \\sqrt{\\gamma^{-1}}\\right)^2 < \\frac{3f_0^2}{4+2f_0},\n",
    "\\end{align}\n",
    "\n",
    "where\n",
    "\n",
    "\\begin{align}\n",
    "f_0 = \\frac{\\beta n x_0^{n-1}}{(1+x_0^n)^2}.\n",
    "\\end{align}\n"
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