{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Course**: BIO-341 [_Systèmes dynamiques en biologie_](https://moodle.epfl.ch/course/info.php?id=14291)\n",
    "\n",
    "**Professor**: _Julian Shillcock_ & _Felix Naef_\n",
    "\n",
    "SSV, BA5, 2025\n",
    "\n",
    "Note that this document is primarily aimed at being consulted as a Jupyter notebook, the PDF rendering being not optimal."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from ipywidgets import interact"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Growth models in 1D"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Linear model for population growth"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider a population of $N$ birds with birth and death rates $n$ and $m$. Arrival of new individuals through migrations occurs at rate $a > 0$. This can be translated into the simple model:\n",
    "\n",
    "\\begin{equation}\n",
    "\\frac{dN}{dt} = F(N)=(n-m)N+a\n",
    "\\end{equation}\n",
    "\n",
    "**1) Write down what type of equation this is, e.g. first order, second order, linear, non-linear, etc.**\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    ">The equation is a linear first order ODE."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**2) Solve this equation analytically using the Ansatz**: $N(t)=Ae^{\\lambda t} + B$. Express $A,B$ and $\\lambda$ in function of the rates and the population size $N_{0} =N(t=0)$; explicitly write the solution $N(t)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    ">Differentiating the Ansatz gives: $\\frac{dN}{dt} = A \\lambda e^{\\lambda t}$  \n",
    "If the Ansatz is a solution of the differential equation, then $\\frac{dN}{dt}$ is also equal to $(n-m)N + a$, so:\n",
    "$$A \\lambda e^{\\lambda t} = k(Ae^{\\lambda t} + B) + a\\text{, with }k = (n-m)$$\n",
    "By separating terms which depends on time and terms which are time-independent, we can deduce that $kB + a = 0 \\Rightarrow B=- \\frac{a}{k}$. In addition, by comparison of the left-hand side and right-hand side it follows that $\\lambda = k$.  \n",
    "To find A, use the initial conditions in the equation: $N(t=0)=N_{0} = A-\\frac{a}{k} \\Rightarrow A=N_{0}+\\frac{a}{k}$.\n",
    "Finally, $N(t)=(N_{0} + \\frac{a}{k})e^{kt} -\\frac{a}{k}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**3) Solve the equation using an alternative method, such as the separation of variables.**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    ">Solving an equation like $\\frac{dN}{dt}=kN$ is trivial, so we seek for a change of variable that could get rid of the $+ a$ term in the original equation. Taking $N=u- \\frac{a}{k} \\Rightarrow u=N+\\frac{a}{k}$ we have:   \n",
    "$dN=du$, since $a$ and $k$ are two constants.  \n",
    "$\\frac{du}{dt}=ku \\Rightarrow u=u_{0}e^{kt}$   \n",
    "Then, you can substitute back: $N(t)=(N_{0} + \\frac{a}{k})e^{kt} - \\frac{a}{k}$ where $u_{0}=N_{0}+ \\frac{a}{k}$  is found with the initial conditions.\n",
    "\n",
    ">Another way to solve this equation is by a direct separation of variables.  \n",
    "$\\int_{N_{0}}^{N} \\frac{dM}{kM+a}= \\int_{0}^{t}d \\tau \\Rightarrow  \\frac{1}{k} \\ln (kM + a) \\Big|_{N_{0}}^N  = t \\Rightarrow \\ln (\\frac{kN + a}{kN_{0} + a}) = kt \\Rightarrow N(t)=(N_{0} + \\frac{a}{k})e^{kt} -\\frac{a}{k}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**4) Qualitative analysis**\n",
    "\n",
    "1) Draw $F(N)$ in function of $N$ for the two cases (i) $n<m$ and (ii) $n>m$. *Note: you are also expected to draw and solve such simple problems by hand.*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    ">$F(N)$ is a linear function with intercept $a$ and slope $n-m$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def Ndot(N,n,m,a):\n",
    "    return (n-m)*N+a\n",
    "\n",
    "Ndomain = np.linspace(0,10,100)\n",
    "\n",
    "#plot\n",
    "plt.plot(Ndomain, Ndot(Ndomain,1,2,1))\n",
    "plt.xlabel('$N$')\n",
    "plt.ylabel('$\\dot{N}$')\n",
    "plt.axhline(0, color = 'black', ls = '--')\n",
    "plt.axvline(0, color = 'black', ls = '--')\n",
    "plt.title('n<m')\n",
    "plt.show()\n",
    "\n",
    "plt.plot(Ndomain, Ndot(Ndomain,2,1,1))\n",
    "plt.axhline(0, color = 'black', ls = '--')\n",
    "plt.axvline(0, color = 'black', ls = '--')\n",
    "plt.xlabel('$N$')\n",
    "plt.ylabel('$\\dot{N}$')\n",
    "plt.title('n>m')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2) What is the main qualitative difference between the two cases and how does this affect the long time $t\\to\\infty$ behavior of the solution that you found above?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    ">If the death rate $m$ is bigger than the birth rate $n$ ($k<0$) the number of individuals \"converges\" to the constant $|\\frac{a}{k}|$ at late times. On the contrary, if birth rate $n$ is bigger than death rate $m$ ($k>0$), then the population grows to infinity."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "3) Here you can verify what you answered under 2) numerically. Generate some representative plots for $N(t)$ where you vary the parameters and initial conditions of the model."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "#code the analytical function for N(t)\n",
    "def N(N0, a, k, t):\n",
    "    return (N0+a/k)*np.exp(k*t)-a/k"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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e2ziEdcPgLEVYeIXzE6HsAABgcfsO12pF0/IQNwwOryEsibIDAIDlLfpyrwxDGtY1SdlJ1l8e4liUHQAALMwwDL3aNIQVbicmH0HZAQDAwr7Yc1g7y6oV67Dru2GyPMSxKDsAAFjYkROTx+d0UJwzfObWORplBwAAi6qr9+mt9Y3LQ4TrEJZE2QEAwLL+u7lxbp2s9jEa1jW85tY5GmUHAACL+teaxiGs68Nwbp2jUXYAALCgPQdrtGJ7mWw2aeKQ8B3CkkwuO88++6wGDBigxMREJSYmKi8vT++8807gecMwNHPmTGVmZiomJkajRo3S5s2bm/0Oj8eje+65RykpKYqLi9O1116roqKitn4rAAAElVfW7JEkXdozVVntw29unaOZWnaysrL08MMPa82aNVqzZo2uuOIKXXfddYFC8+ijj+qJJ57QnDlztHr1amVkZGjs2LGqrKwM/I78/HwtWrRICxcu1IoVK1RVVaWrr75aPp/PrLcFAICpGnz+wBDWTUOzTU5jPpthGIbZIY6WlJSkxx57TD/+8Y+VmZmp/Px83X///ZIaj+Kkp6frkUce0e233y63263U1FS9+OKLuvHGGyVJ+/btU3Z2thYvXqwrr7zyjP5mRUWFXC6X3G63EhMTW+29AQDQFpZsLtFPX1yr5DiHPp0xWo5Ia561cqbf30Hz7n0+nxYuXKjq6mrl5eWpoKBAJSUlGjduXGAfp9Opyy67TCtXrpQkrV27VvX19c32yczMVE5OTmAfAADCzSurCyU1Xm5u1aJzNkyfXWjjxo3Ky8tTXV2d4uPjtWjRIvXt2zdQVtLT05vtn56ert27d0uSSkpK5HA41L59++P2KSkpOenf9Hg88ng8gccVFRUt9XYAADBVsbtWH24tlSTdyBCWpCA4stOrVy+tW7dOq1at0s9+9jNNmTJFW7ZsCTxvszW/VM4wjOO2Het0+8yePVsulytwy87mHwYAgDX8a02R/E2LfnZLjTc7TlAwvew4HA716NFDQ4YM0ezZszVw4EA9/fTTysjIkKTjjtCUlpYGjvZkZGTI6/WqvLz8pPucyIwZM+R2uwO3wsLCFn5XAAC0Pb/fCAxh/fCiTianCR6ml51jGYYhj8ejrl27KiMjQ0uXLg085/V6tWzZMo0YMUKSlJubq6ioqGb7FBcXa9OmTYF9TsTpdAYudz9yAwAg1H28vUx7D9cqMTpS38nJMDtO0DD1nJ1f/epXGj9+vLKzs1VZWamFCxfqo48+0rvvviubzab8/HzNmjVLPXv2VM+ePTVr1izFxsZq0qRJkiSXy6WpU6dq+vTpSk5OVlJSku699171799fY8aMMfOtAQDQ5hZ+3ji3zvWDsxQdZTc5TfAwtezs379fN998s4qLi+VyuTRgwAC9++67Gjt2rCTpvvvuU21tre68806Vl5dr2LBhWrJkiRISEgK/48knn1RkZKQmTpyo2tpajR49WvPmzZPdzv/IAIDwcaDSo6Vb9kuSbrqIc1GPFnTz7JiBeXYAAKHujx9u12P/3aoLO7XTojsvNjtOmwi5eXYAAMC58fkN/WNV47QsNw/vbHKa4EPZAQAgxH3wdan2uevUPjZK3+3fwew4QYeyAwBAiHux6ajOxKHZnJh8ApQdAABC2K6yai3/5oBsNmnyRQxhnQhlBwCAEPaPzxqP6oy6IFWdkmNNThOcKDsAAISounqf/rmmSJJ0cx5HdU6GsgMAQIh6c/0+uWvrldU+RpddkGZ2nKBF2QEAIES91HRi8uRhnWWPOPUi2eGMsgMAQAhaX3hY64vcctgjNHFIltlxghplBwCAEPT3TxuP6lw1oIOS450mpwlulB0AAELMgUqP3ly/TxInJp8Jyg4AACFmwWd75PX5NTC7nQZ3am92nKBH2QEAIIR4GnyBGZN/fHEXc8OECMoOAAAh5O0NxSqr8ig90ck6WGeIsgMAQIgwDEN/+6RAknRLXhdF2fkaPxN8SgAAhIg1u8u1aW+FnJER+uFFncyOEzIoOwAAhIi/rWg8qjNhUEclxTlMThM6KDsAAISAovIa/XdziSTpfy7pYm6YEEPZAQAgBPz9093yG9KI7snqnZFodpyQQtkBACDIVXsatPDzPZKkH1/c1eQ0oYeyAwBAkPvnmkJV1DWoc3KsrujN6uZni7IDAEAQa/D59demE5N/cmk3RbC6+Vmj7AAAEMQWbypRUXmtkuIc+kEuq5ufC8oOAABByjAMPb98hyTplrzOio6ym5woNFF2AAAIUp/uOKhNeysUHRWhW/K6mB0nZFF2AAAIUs8t3ylJ+kFuNpMIngfKDgAAQeir4got/+aAImzSTy7lcvPzQdkBACAI/bnpqM74nA7qnBxncprQRtkBACDI7Dtcq/+s3ydJ+unIbianCX2UHQAAgswLnxSowW9oWNckDcxuZ3ackEfZAQAgiJRXe/WPzxqXhrjjsu4mp7EGyg4AAEHkhU8KVOP1qV9mokb1SjU7jiVQdgAACBKVdfWat3KXJOmuy3vIZmNpiJZA2QEAIEi8uGq3Kuoa1D01Tt/pl2F2HMug7AAAEARqvT799ePGBT/vHNWDBT9bEGUHAIAgsHD1Hh2s9io7KUbXDso0O46lUHYAADCZp8Gn55smEbzjsu6KsvP13JL4NAEAMNlrX+xVsbtO6YlOfT83y+w4lkPZAQDARA0+v579aIck6bZLu8kZaTc5kfVQdgAAMNEb6/Zpz6EaJcU5NGlYJ7PjWBJlBwAAkzT4/Hrmg22SGtfAinVEmpzImig7AACY5LUv92r3wRolxzl0S15ns+NYFmUHAAAT1Pv8+kPTUZ07LuvOUZ1WdM6f7K5du/Txxx9r165dqqmpUWpqqi688ELl5eUpOjq6JTMCAGA5r64tUuGhWqXEO/Wj4RzVaU1nXXYWLFigZ555Rp9//rnS0tLUsWNHxcTE6NChQ9qxY4eio6M1efJk3X///ercmf/xAAA4lrfBrz98sF2S9LNR3RXj4Aqs1nRWZWfw4MGKiIjQrbfeqn/+85/q1Kn5WeMej0effvqpFi5cqCFDhmju3Ln6wQ9+0KKBAQAIdf9aW6i9h2uVluDUZK7AanU2wzCMM9357bff1lVXXXVG+5aVlamgoEBDhw4953BtpaKiQi6XS263W4mJiWbHAQBYmKfBp1GPfaRid51mXtNXt17c1exIIetMv7/P6sjOmRYdSUpJSVFKSsrZ/HoAACzvn6sLVeyuU0ZitG66iKM6beGcr8ay2+0qLS09bvvBgwdltzP2CADAsWq9Pj3TdK7OnZd3V3QU35dt4ZzLzslGvzwejxwOxzkHAgDAqv72SYEOVHqU1T5GNw3lqE5bOeursZ555hlJks1m01/+8hfFx8cHnvP5fFq+fLl69+7dcgkBALCAwzVePbescQ2s6eMukCOSqe7aylmXnSeffFJS45Gd5557rtmQlcPhUJcuXfTcc8+1XEIAACzg2WU7VFnXoN4ZCbpuYEez44SVsy47BQUFkqTLL79cr732mtq3b9/ioQAAsJJid63mfbJLknTfd3opIsJmbqAwc84zKH/44YctmQMAAMt65v1t8jT4dVGXJF3eK83sOGHnrAYMH374YVVXV5/Rvp999pnefvvtcwoFAIBV7DhQpX+uKZIk3T++l2w2juq0tbMqO1u2bFHnzp31s5/9TO+8844OHDgQeK6hoUEbNmzQ3LlzNWLECN10001M0AcACHuPL9kqn9/QmD7pyu2cZHacsHRWw1h///vftWHDBv3xj3/U5MmT5Xa7Zbfb5XQ6VVNTI0m68MIL9dOf/lRTpkyR0+lsldAAAISC9YWHtXhjiWw26ZdX9jI7Ttg6q+UijmYYhtavX6/du3ertrZWKSkpGjRoUEjOmsxyEQCAlmYYhn7451VatfOQbhicpccnDjQ7kuW0ynIRR7PZbBo0aJAGDRp0rr8CAADLWrplv1btPCRnZIR+Mban2XHC2lmXnYiIiNOeXGWz2dTQ0HDOoQAACGXeBr9mv/O1JOknl3ZVVvtYkxOFt7MuO4sWLTrpcytXrtQf/vCHky4lAQBAOHhp1W4VlFUrJd6pn43qYXacsHfWZee66647btvXX3+tGTNm6M0339TkyZP129/+tkXCAQAQag7XePX0+9skSfeOu0DxznM+YwQt5LwW5ti3b59uu+02DRgwQA0NDVq3bp3mz5+vTp1Y3AwAEJ6efn+b3LX16p2RoB8MyTY7DnSOZcftduv+++9Xjx49tHnzZr3//vt68803lZOT09L5AAAIGTsPVOnFT3dLkv7PVX1lZ1mIoHDWx9YeffRRPfLII8rIyNDLL798wmEtAADC0ex3vlaD39AVvdN0Sc/Qm4rFqs76yM4DDzyguro69ejRQ/Pnz9f1119/wtuZmD17toYOHaqEhASlpaVpwoQJ2rp1a7N9DMPQzJkzlZmZqZiYGI0aNUqbN29uto/H49E999yjlJQUxcXF6dprr1VRUdHZvjUAAM7Zyh1lWrplv+wRNv3qu33MjoOjnHXZueWWWzRx4kQlJSXJ5XKd9HYmli1bprvuukurVq3S0qVL1dDQoHHjxjVbf+vRRx/VE088oTlz5mj16tXKyMjQ2LFjVVlZGdgnPz9fixYt0sKFC7VixQpVVVXp6quvls/nO9u3BwDAWWvw+fXQm1skSZOHdVKPtHiTE+Fo5zyDcms4cOCA0tLStGzZMo0cOVKGYSgzM1P5+fm6//77JTUexUlPT9cjjzyi22+/XW63W6mpqXrxxRd14403Smo8cTo7O1uLFy/WlVdeedq/ywzKAIDzMe+TAs18c4vaxUbpw+mj1D7OYXaksHCm39/ndTVWS3O73ZKkpKTGhdIKCgpUUlKicePGBfZxOp267LLLtHLlSknS2rVrVV9f32yfzMxM5eTkBPY5lsfjUUVFRbMbAADn4kClR48v/UZS4/pXFJ3gEzRlxzAMTZs2TZdcckngqq6SkhJJUnp6erN909PTA8+VlJTI4XCoffv2J93nWLNnz2425JadzaWBAIBz8+i7X6uyrkH9O7p001CmXglGQVN27r77bm3YsEEvv/zycc8duzyFYRinXbLiVPvMmDFDbrc7cCssLDz34ACAsLV2d7n+tbbxgpgHr+vHpeZBKijKzj333KP//Oc/+vDDD5WVlRXYnpGRIUnHHaEpLS0NHO3JyMiQ1+tVeXn5Sfc5ltPpVGJiYrMbAABnw+c39Jv/bJIk/SA3S4M7tT/NK2AWU8uOYRi6++679dprr+mDDz5Q165dmz3ftWtXZWRkaOnSpYFtXq9Xy5Yt04gRIyRJubm5ioqKarZPcXGxNm3aFNgHAICWtnD1Hm3aW6GE6EjdP7632XFwCqYu2HHXXXdpwYIFeuONN5SQkBA4guNyuRQTEyObzab8/HzNmjVLPXv2VM+ePTVr1izFxsZq0qRJgX2nTp2q6dOnKzk5WUlJSbr33nvVv39/jRkzxsy3BwCwqPJqrx77b+O8cNPHXqCUeKfJiXAqppadZ599VpI0atSoZttfeOEF3XrrrZKk++67T7W1tbrzzjtVXl6uYcOGacmSJUpISAjs/+STTyoyMlITJ05UbW2tRo8erXnz5slut7fVWwEAhJFH//u1Dtc0rn/1o+GdzY6D0wiqeXbMwjw7AIAz9XnBIU3806eSpH/dkaehXZJMThS+QnKeHQAAgpmnwacZr22QJP3womyKToig7AAAcIae+2indhyoVkq8Uw98h/WvQgVlBwCAM7DjQJX++OF2SdL/u6avXLFRJifCmaLsAABwGoZh6NeLNsrr8+uyC1J1zYAOZkfCWaDsAABwGv9aW6RVOw8pOipCv5uQc9pZ/BFcKDsAAJzCwSqPZi3+SpL0izEXKDsp1uREOFuUHQAATuHBN7focE29+nRI1I8v6Xr6FyDoUHYAADiJ/24u0X/W71OETXr4+v6KsvO1GYr4Xw0AgBMor/bq14saF/q8/bLuGpjdztxAOGeUHQAATuDBNzerrMqjHmnx+vnonmbHwXmg7AAAcIwlm0v0+rrG4avHvj9A0VGstRjKKDsAABzlcI1Xv369cfjqtpHddGGn9iYnwvmi7AAAcJSH3tyiA5UedU+N0y/GXGB2HLQAyg4AAE3e27Jfr325t3H46gcDGb6yCMoOAACSyqo8eqBpRfOpl3TVYIavLIOyAwAIe4Zh6IFXN6isyqte6QmaPq6X2ZHQgig7AICw9/LnhXrvq1I57BF66qZBDF9ZDGUHABDWdh6o0m/f2iJJ+uWVvdSnQ6LJidDSKDsAgLBV7/PrF6+sU229T3ndkjWVta8sibIDAAhbf/hgu9YXuZUYHanHJw5URITN7EhoBZQdAEBYWrv7kOZ8sE2S9Pvv9VdmuxiTE6G1UHYAAGHHXVOv/315nfyGNGFQpq4ZmGl2JLQiyg4AIKwYhqFf/nu99h6uVefkWP12Qo7ZkdDKKDsAgLAyf+UuLdmyXw57hP44abASoqPMjoRWRtkBAISNjUVuzVr8tSTpV9/trZyOLpMToS1QdgAAYaGyrl53v/yFvD6/xvVN15QRXcyOhDZC2QEAWJ5hGJrx2kbtPlijju1i9Nj3B8pm4zLzcEHZAQBY3j8+26O3NhQrMsKmP0y6UK5YztMJJ5QdAIClfbmnXA++uVlS43IQrGYefig7AADLKqvy6GcvfaF6n6Hv9MvQT0d2MzsSTEDZAQBYUoPPr3sWfKmSijp1T43TYz8YwHk6YYqyAwCwpMf+u1Wf7jyoOIddf7o5l/l0whhlBwBgOe9sLNaflu+UJD36/YHqkZZgciKYibIDALCU7aWVuvdf6yVJPx3ZTVcN6GByIpiNsgMAsIzyaq+mzl+jaq9Pw7sl6b4re5kdCUGAsgMAsIR6n193LfhCuw/WKKt9jP44abAi7XzNgbIDALCIB9/crJU7Gk9I/suUIUqOd5odCUGCsgMACHkvfrpLL63aI5tNeuqmC9U7I9HsSAgilB0AQEj7ZHuZZr65RZJ035W9NbZvusmJEGwoOwCAkLXzQJXu/McX8vkNfe/CjrrjMmZIxvEoOwCAkFRW5dGtL6yWu7Zeg7Lbafb1/ZkhGSdE2QEAhJxar08/mb9Gew7VKDspRn++ZYiio+xmx0KQouwAAEKKz2/o5wu/1LrCw2oXG6V5/3ORUhO48gonR9kBAIQMwzD027e2aMmW/XJERujPtwxR99R4s2MhyFF2AAAh468rCjRv5S5J0hMTB2polyRzAyEkUHYAACHhzfX79PvFX0mSfvXd3rp6QKbJiRAqKDsAgKD30dZSTfvnOhmGdEteZ912KZeY48xRdgAAQW3t7kO646W1qvcZumZgpmZe049LzHFWKDsAgKD1VXGF/ueF1aqr9+uyC1L1+A8GKiKCooOzQ9kBAASl3QerdcvfPldFXYNyO7fXcz/KlSOSry2cPf6pAQAEndKKOt381891oNKj3hkJ+tuUoYpxMGkgzg1lBwAQVMqqPPrRXz/TnkM16pQUq7//+CK5YqPMjoUQRtkBAASNQ9Ve/egvn+mb/VVKT3TqpanDlJYYbXYshDjKDgAgKByu8WryXz7T1yWVSktw6uXbhqtTcqzZsWABlB0AgOncNfX60V8/01fFFUqJd2rBbcPVjWUg0EIoOwAAU7lr63Xz3z7Tpr0VSo5z6OXbhqlHGkUHLYeyAwAwjbu2XlP+9rk2FLmVFOfQgtuGq2d6gtmxYDGRZgcAAISnQ9Ve3dJ0RKddbJRemjpMvTIoOmh5lB0AQJsrrawLXHWVHOfQi1OHqW9motmxYFGUHQBAm9p7uFaT/7xKuw7WKD3RqX/8ZDjn6KBVUXYAAG1m98FqTfrzZ9p7uFZZ7WO04CdcXo7WR9kBALSJbfsrNfkvn6m00qNuKXF66SfDlNkuxuxYCAOUHQBAq1u7+5B+PG+N3LX16pWeoBd/cpHSEpgZGW2DsgMAaFVLt+zX3Qu+kKfBr0HZ7fTCrUPVPs5hdiyEEcoOAKDVLPx8j361aKP8hnRF7zTNmXShYh189aBtmTqp4PLly3XNNdcoMzNTNptNr7/+erPnDcPQzJkzlZmZqZiYGI0aNUqbN29uto/H49E999yjlJQUxcXF6dprr1VRUVEbvgsAwLEMw9Af3t+mB15rLDo/yM3Sn27OpejAFKaWnerqag0cOFBz5sw54fOPPvqonnjiCc2ZM0erV69WRkaGxo4dq8rKysA++fn5WrRokRYuXKgVK1aoqqpKV199tXw+X1u9DQDAURp8fv3fNzbp8aXfSJLuvryHHv3+AEXZmbQf5rAZhmGYHUKSbDabFi1apAkTJkhq/H8FmZmZys/P1/333y+p8ShOenq6HnnkEd1+++1yu91KTU3Viy++qBtvvFGStG/fPmVnZ2vx4sW68sorz+hvV1RUyOVyye12KzGRSa0A4FxV1tXr7gVfatk3B2SzSTOv6acpI7qYHQsWdabf30FbswsKClRSUqJx48YFtjmdTl122WVauXKlJGnt2rWqr69vtk9mZqZycnIC+5yIx+NRRUVFsxsA4PwUldfo+89+qmXfHFB0VISenTyYooOgELRlp6SkRJKUnp7ebHt6enrguZKSEjkcDrVv3/6k+5zI7Nmz5XK5Arfs7OwWTg8A4eWLPeWa8MdPtHV/pdISnPrn7Xn6Tk4Hs2MBkoK47Bxhs9maPTYM47htxzrdPjNmzJDb7Q7cCgsLWyQrAISjN9fv003Pr1JZlVd9OiTqjbsv1oCsdmbHAgKCtuxkZGRI0nFHaEpLSwNHezIyMuT1elVeXn7SfU7E6XQqMTGx2Q0AcHb8fkNPLNmqe17+Ut4Gv0b3TtO/78hTBxezIiO4BG3Z6dq1qzIyMrR06dLANq/Xq2XLlmnEiBGSpNzcXEVFRTXbp7i4WJs2bQrsAwBoee7aev3k72v0zAfbJUlTL+mq528Zojgnl5Yj+Jj6T2VVVZW2b98eeFxQUKB169YpKSlJnTp1Un5+vmbNmqWePXuqZ8+emjVrlmJjYzVp0iRJksvl0tSpUzV9+nQlJycrKSlJ9957r/r3768xY8aY9bYAwNK27a/UT19cq4KyajkjIzTre/11Q26W2bGAkzK17KxZs0aXX3554PG0adMkSVOmTNG8efN03333qba2VnfeeafKy8s1bNgwLVmyRAkJCYHXPPnkk4qMjNTEiRNVW1ur0aNHa968ebLb7W3+fgDA6t7ZWKx7/7Ve1V6fOraL0XM/ylX/LJfZsYBTCpp5dszEPDsAcGo+v6HHl2zV3I92SJLyuiVrzqQLlRzvNDkZwtmZfn8zuAoAOKXSijr978IvtWrnIUmN5+fMGN9bkcyIjBBB2QEAnNTH2w7oF6+sU1mVV7EOu2Zf31/XDepodizgrFB2AADHafD59dR72/THj7bLMKTeGQmaM2mweqTFmx0NOGuUHQBAMyXuxmGrzwsah60mDeuk/3d1X0VHceEHQhNlBwAQsGRziR54baMOVXsV74zUrOv769qBmWbHAs4LZQcAoGpPgx56c4teWdO4fE7fDon64+TB6poSZ3Iy4PxRdgAgzK3dXa5p/1yn3QdrZLNJPx3ZTdPGXiBnJMNWsAbKDgCEqXqfX394f5vmfLhdfkPq2C5Gj08cqOHdks2OBrQoyg4AhKGtJZX65b/Xa0ORW5L0vQs76sHr+ikxOsrkZEDLo+wAQBjxNvj17Ec7NOfDbar3GUqMjtTvv9df13ASMiyMsgMAYWJjkVu//Pd6fV1SKUka0yddv/9ejtITo01OBrQuyg4AWFxdvU/PvL9Nf1q+Uz6/ofaxUXrwuhxdM6CDbDab2fGAVkfZAQAL+2R7mf7v65u0s6xaknT1gA6aeW0/pbCAJ8IIZQcALKi0ok6/e/sr/Wf9PklSaoJTv70uR9/JyTA5GdD2KDsAYCE+v6EXP92lx5d8o0pPgyJs0s3DO2v6lb240gphi7IDABaxvvCwfv36Rm3aWyFJGpDl0u8n9Ff/LJfJyQBzUXYAIMSVVtbpsXe36t9fFMkwpIToSN33nd6adFEn2SM4ARmg7ABAiKqr9+mvKwo098Ptqvb6JDVODvir7/ZRagInIANHUHYAIMQYhqF3NpVo1uKvVFReK0kamN1Ov7mmrwZ3am9yOiD4UHYAIISsLzys3y/+Sp8XHJIkZSRG6/7xvXTdwI6KYMgKOCHKDgCEgB0HqvT4kq1avLFEkhQdFaHbR3bX7Zd1U6yD/5QDp8K/IQAQxIrdtXr6vW3619oi+fyGbLbG83LuHddLme1izI4HhATKDgAEocM1Xj370Q7NW7lLnga/pMa1rH55ZS/1ykgwOR0QWig7ABBEyqu9+suKnZq/creqPA2SpIu6JOn+8b2U2znJ5HRAaKLsAEAQOFjl0Z8/LtCLn+4KXEbep0Oi7ruyl0b1SmXBTuA8UHYAwERlVR79eflOvbhqt2qaSk6/zET97+ieGtsnnSusgBZA2QEAExSV1+ivKwq08PNC1dY3lpz+HV3639E9NaZPGkdygBZE2QGANrR5n1vPL9+ptzYUy+c3JEkDs1z6+ZieurwXJQdoDZQdAGhlhmFo5Y6Dem7ZDn28rSyw/eIeyfrpyO4a2TOFkgO0IsoOALSSunqf3t5QrBdWFgRWIo+wSVcNyNTtI7sppyOrkQNtgbIDAC2s2F2rl1bt1sLPC3Ww2itJiomy68ah2Zp6SVdlJ8WanBAIL5QdAGgBhmHo84JDmv/pLv138/7A+TiZrmhNHt5Zky7qpPZxDpNTAuGJsgMA56Girl7/WbdPL63ara9LKgPbh3VN0q0jumhs33RF2iNMTAiAsgMAZ8kwDK3dXa6Fqwv19obiwKXj0VER+t6FWbolr7P6dEg0OSWAIyg7AHCGDlZ5tOjLvVq4ulDbS6sC23umxevGodn6fm6W2sUyVAUEG8oOAJyCt8Gvj7aWatGXe/XeV/tV72s8Fycmyq6rB3TQTRd10uBO7bh0HAhilB0AOIZhGPpiT7kWfblXb20o1uGa+sBzA7JcunFotq4dmKmE6CgTUwI4U5QdAGiyq6xai77cq9fX7dXugzWB7WkJTl03KFPfuzBLfTM5FwcINZQdAGFt98Fqvb2xWIs3Fgcm/pOkWIdd3+mXoe8N7qgR3VNkZ0FOIGRRdgCEnYKyai1uKjib931bcCJs0iU9U3X9hR01rl+6Yh38JxKwAv5NBmB5hmFoW2mVlmwu0dsbS/RV8bcFxx5hU163ZH23fwdd2S9dyfFOE5MCaA2UHQCWVO/z6/OCQ3rvq/1676v9KjxUG3jOHmHTiO5HCk6GkpjZGLA0yg4Ayzhc49VHWw/ova/2a9nWA6r0NASec9gjNKJHssbnZGhc3wyWbgDCCGUHQMjy+w1t2ufWx9vKtOybA1q7uzywJpUkJcc5dEXvNI3uk65Le6Yozsl/8oBwxL/5AEJKsbtWH28r08fbyrRi2wGVHzUHjiT1Sk/Q6D6NBWdQdjuuogJA2QEQ3Crq6rVm1yF9sv2gPt52QN/sr2r2fLwzUnndkzWyZ4pG9UpTdlKsSUkBBCvKDoCgUlFXr9UFh7Rq50F9VnBIm/a6ddTIlGw2aWBWO43smaJLL0jVoOx2imJVcQCnQNkBYCp3Tb1W7/q23Gze17zcSFKX5FgN75asS3um6uIeySy2CeCsUHYAtBm/39COA1X6Yk+51u4u1xd7DjdbPfyIrilxGt4tScO6JmtYtyR1cMWYkBaAVVB2ALSaKk+D1u05HCg3X+4pV0Vdw3H7dUuJ07BuyYGCk+GKNiEtAKui7ABoETXeBm3eV6GNRW5t3OvWhqLD2llWLeOYIanoqAgNzGqn3M7tNbhTe13YqR2zFgNoVZQdAGet1uvTluIKbSw6rA173dq0163tpVXHnWsjSR3bxTQVm3bK7Zyk3h0SOKEYQJui7AA4KcMwVFReq69LKvV1cYW+LqnUVyUV2lVWfcJik57oVP+O7dS/o0sDslzK6ehSagJHbQCYi7IDQJJUWVevb/ZX6euSCn1dXBm4P3rJhaOlxDs1IMsVKDb9O7qUlsi5NgCCD2UHCCOGYehApUfbD1RpR2mVtpdWaceBam0vrVJJRd0JXxNlt6lHWoL6ZCSod4cE9c5IVO+MBKUmOGWzMTsxgOBH2QEsqK7ep6LyGu0qq/m22BxoLDeVJ7ga6oi0BKf6dEhU7w4J6pPReN8tJV6OSM6xARC6KDtAiKr2NGj3wRrtOVStXQdrtPtgtXaV1WjPoRrtc9cedxXUERE2qVNSrLqnxqtHWry6p8U3/pwaL1dsVNu+CQBoA5QdIEh5G/wqdtdqb3mtig433ZfXBsrNgUrPKV8f74xU5+RYdWsqMo3FJk5dkuMUHWVvo3cBAOaj7AAmqfE2NCsye4+6LyqvUWml56RHZ45oHxulzslx6pIcq05N90ceJ8U5OKcGAETZAVqcz2/oYJVH+ys8KqmoU0lFnfa7m+6bbiXuuhPOJHwsZ2SEOraPUcd2Mcpquu+cHKfOybHqnBTHsBMAnAHKDnCGar0+lVV5mm5elVV5dLDKowOVnqYi49H+ijqVVnrkO9EkNCeQ4IxUx/bfFpnGYhMbKDgp8RydAYDzRdlB2Grw+eWurVd5Tb0OVXt1sKnIHGgqMmWVHh2s/vbnaq/vjH93hK1xHpoMV7TSE6OVkRit9ERn48+uxsdpidFyxXBkBgBaG2UHIc8wDNXW+1ReU6/yaq8O19SrvMarwzXexm01324rr6lv3F7tPaNhpGM5IiOUGu9USrxDyU33R0pNWsK3RSYl3qFIlkQAgKBA2YHpDMNQjdenyroGVdTVq6K2vtnPFYGfG1RZ1/S4tj7ws7u2Xt4G/zn//cToSCXFNZaW5KbykhLvVEqCUylxjsb7pmIT74xkWAkAQgxlB+ekwedXtcenam+Dqj0NqvI0qMbra7pvUJXHp2pPg2o8jT83bmvct9rb+Fxl3bfl5UzPcTmVyAib2sU6lBQXpXaxDrWPjVL7WMcxP0epfVzj43axDrWLieIIDABYnGXKzty5c/XYY4+puLhY/fr101NPPaVLL73U7Fim8PsNeRr8qq33Nd68PtXVN96OPK498tjrU229v9njI/sdvX+N90ixaSwqnvM4knIy9gibEqMjlRAdpcSYSCVGRykhuvE+MeZkPzfet4uN4qgLAOCELFF2XnnlFeXn52vu3Lm6+OKL9ac//Unjx4/Xli1b1KlTJ9NyuWsbh2E8DX55G/zyNPia7v3f3vt88tT75fX5j7r3ydPs8Yle6zvmNY3bGktKyxeRk4my2xTnjFScI1JxTvtJfo5UnKPxcbwzUrFNzyU4I5UYExUoNbEOO2UFANDibIZxumnLgt+wYcM0ePBgPfvss4Ftffr00YQJEzR79uzTvr6iokIul0tut1uJiYktluv+f2/QK2sKW+z3nQtHZIRiouyKibIrOipC0VF2xTjs32476ucYh13RTfsduy0m6tuyEue0B0oMayYBAMxypt/fIX9kx+v1au3atXrggQeabR83bpxWrlx5wtd4PB55PN9OtV9RUdEq2RqLQoSckXY5IiPkjIxouv/2sfPY7fYIOaMijrq3yxl14tc2/x32b4uN40i5scsewZESAEB4C/myU1ZWJp/Pp/T09Gbb09PTVVJScsLXzJ49Ww8++GCrZ5t5bT/NvLZfq/8dAABwcpYZgzj2XA/DME56/seMGTPkdrsDt8JCc4eaAABA6wn5IzspKSmy2+3HHcUpLS097mjPEU6nU06nsy3iAQAAk4X8kR2Hw6Hc3FwtXbq02falS5dqxIgRJqUCAADBIuSP7EjStGnTdPPNN2vIkCHKy8vT888/rz179uiOO+4wOxoAADCZJcrOjTfeqIMHD+qhhx5ScXGxcnJytHjxYnXu3NnsaAAAwGSWmGfnfLXWPDsAAKD1nOn3d8ifswMAAHAqlB0AAGBplB0AAGBplB0AAGBplB0AAGBplB0AAGBplB0AAGBplB0AAGBplphB+XwdmVexoqLC5CQAAOBMHfnePt38yJQdSZWVlZKk7Oxsk5MAAICzVVlZKZfLddLnWS5Ckt/v1759+5SQkCCbzdZiv7eiokLZ2dkqLCxkGYpWxmfdNvic2wafc9vgc24brfk5G4ahyspKZWZmKiLi5GfmcGRHUkREhLKyslrt9ycmJvIvUhvhs24bfM5tg8+5bfA5t43W+pxPdUTnCE5QBgAAlkbZAQAAlkbZaUVOp1O/+c1v5HQ6zY5ieXzWbYPPuW3wObcNPue2EQyfMycoAwAAS+PIDgAAsDTKDgAAsDTKDgAAsDTKDgAAsDTKTiuaO3euunbtqujoaOXm5urjjz82O5LlLF++XNdcc40yMzNls9n0+uuvmx3JcmbPnq2hQ4cqISFBaWlpmjBhgrZu3Wp2LEt69tlnNWDAgMDka3l5eXrnnXfMjmVps2fPls1mU35+vtlRLGfmzJmy2WzNbhkZGaZkoey0kldeeUX5+fn69a9/rS+//FKXXnqpxo8frz179pgdzVKqq6s1cOBAzZkzx+wolrVs2TLdddddWrVqlZYuXaqGhgaNGzdO1dXVZkeznKysLD388MNas2aN1qxZoyuuuELXXXedNm/ebHY0S1q9erWef/55DRgwwOwoltWvXz8VFxcHbhs3bjQlB5eet5Jhw4Zp8ODBevbZZwPb+vTpowkTJmj27NkmJrMum82mRYsWacKECWZHsbQDBw4oLS1Ny5Yt08iRI82OY3lJSUl67LHHNHXqVLOjWEpVVZUGDx6suXPn6ne/+50GDRqkp556yuxYljJz5ky9/vrrWrdundlROLLTGrxer9auXatx48Y12z5u3DitXLnSpFRAy3C73ZIav4TRenw+nxYuXKjq6mrl5eWZHcdy7rrrLl111VUaM2aM2VEsbdu2bcrMzFTXrl110003aefOnabkYCHQVlBWViafz6f09PRm29PT01VSUmJSKuD8GYahadOm6ZJLLlFOTo7ZcSxp48aNysvLU11dneLj47Vo0SL17dvX7FiWsnDhQn3xxRdavXq12VEsbdiwYfr73/+uCy64QPv379fvfvc7jRgxQps3b1ZycnKbZqHstCKbzdbssWEYx20DQsndd9+tDRs2aMWKFWZHsaxevXpp3bp1Onz4sF599VVNmTJFy5Yto/C0kMLCQv385z/XkiVLFB0dbXYcSxs/fnzg5/79+ysvL0/du3fX/PnzNW3atDbNQtlpBSkpKbLb7ccdxSktLT3uaA8QKu655x795z//0fLly5WVlWV2HMtyOBzq0aOHJGnIkCFavXq1nn76af3pT38yOZk1rF27VqWlpcrNzQ1s8/l8Wr58uebMmSOPxyO73W5iQuuKi4tT//79tW3btjb/25yz0wocDodyc3O1dOnSZtuXLl2qESNGmJQKODeGYejuu+/Wa6+9pg8++EBdu3Y1O1JYMQxDHo/H7BiWMXr0aG3cuFHr1q0L3IYMGaLJkydr3bp1FJ1W5PF49NVXX6lDhw5t/rc5stNKpk2bpptvvllDhgxRXl6enn/+ee3Zs0d33HGH2dEspaqqStu3bw88Ligo0Lp165SUlKROnTqZmMw67rrrLi1YsEBvvPGGEhISAkcsXS6XYmJiTE5nLb/61a80fvx4ZWdnq7KyUgsXLtRHH32kd9991+xolpGQkHDc+WZxcXFKTk7mPLQWdu+99+qaa65Rp06dVFpaqt/97neqqKjQlClT2jwLZaeV3HjjjTp48KAeeughFRcXKycnR4sXL1bnzp3NjmYpa9as0eWXXx54fGQceMqUKZo3b55JqazlyPQJo0aNarb9hRde0K233tr2gSxs//79uvnmm1VcXCyXy6UBAwbo3Xff1dixY82OBpy1oqIi/fCHP1RZWZlSU1M1fPhwrVq1ypTvQebZAQAAlsY5OwAAwNIoOwAAwNIoOwAAwNIoOwAAwNIoOwAAwNIoOwAAwNIoOwAAwNIoOwAAwNIoOwAsbdSoUcrPzzc7BgATUXYAAIClsVwEAMu69dZbNX/+/GbbCgoK1KVLF3MCATAFZQeAZbndbo0fP145OTl66KGHJEmpqamy2+0mJwPQllj1HIBluVwuORwOxcbGKiMjw+w4AEzCOTsAAMDSKDsAAMDSKDsALM3hcMjn85kdA4CJKDsALK1Lly767LPPtGvXLpWVlcnv95sdCUAbo+wAsLR7771Xdrtdffv2VWpqqvbs2WN2JABtjEvPAQCApXFkBwAAWBplBwAAWBplBwAAWBplBwAAWBplBwAAWBplBwAAWBplBwAAWBplBwAAWBplBwAAWBplBwAAWBplBwAAWBplBwAAWNr/B0iZwSH7e62yAAAAAElFTkSuQmCC",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#simulate it with n>m\n",
    "tspan = np.linspace(0,5,1000)\n",
    "a = 1.\n",
    "n = 2.\n",
    "m = 1.\n",
    "N0 = 2.\n",
    "\n",
    "k = n-m\n",
    "\n",
    "#plot\n",
    "plt.plot(tspan, N(N0,a,k,tspan))\n",
    "plt.xlabel('t')\n",
    "plt.ylabel('N(t)')\n",
    "plt.title('n>m')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#simulate it with n<m\n",
    "m = 3.\n",
    "k = n-m\n",
    "\n",
    "#plot\n",
    "plt.plot(tspan, N(2,a,k,tspan), label = 'N0 = 2')\n",
    "plt.plot(tspan, N(0.5,a,k,tspan), label = 'N0 = 0.5')\n",
    "plt.xlabel('t')\n",
    "plt.ylabel('N(t)')\n",
    "plt.title('n<m')\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "9fb3226b86604e3fb5627c00fdb008aa",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "interactive(children=(FloatSlider(value=1.0, description='k', max=3.0, min=-3.0, step=0.01), Output()), _dom_c…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<function __main__.f(k=1.0)>"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Create a widget to play with the parameters\n",
    "# The widget does not work if you are using Jupyter Lab, but it works if you are using Jupyter Notebooks.\n",
    "def f(k=1.):\n",
    "    plt.plot(tspan, N(N0,a,k,tspan))\n",
    "    plt.xlabel('t')\n",
    "    plt.ylabel('N(t)')\n",
    "    plt.title('Interactive plot: n-m=' + str(n-m))\n",
    "    plt.show()\n",
    "\n",
    "a = 1.\n",
    "N0 = 10.\n",
    "    \n",
    "interact(f, k = (-3.,3.,0.01))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**5) Discuss why this equation is good or bad at describing real populations.**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    ">Limitations in resources are not taken into account in this equation, thus a population can grow to infinity in this model."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Integration by separation of variables"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**1) Solve the following differential equations to obtain $x(t)$ by the method of separation of variables. Use the initial condition $x(t)=x_{0}$ when $t=0$.**\n",
    "\n",
    "1. $\\frac{dx}{dt}= xe^{-2t}$\n",
    "2. $\\frac{dx}{dt}= 4x^{2}-1$\n",
    "\n",
    "*Reminder*: The explicit solution of a differential equation $\\frac{dx}{dt} = F(x)$ can be written as a function of $t, x(t)=g(t)$,  while the implicit form is written $h(x(t))=g(t)$, where the function $h(x)$ might not be easily invertible."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    ">1. Separating the variables gives :   \n",
    "$\\int_{x_{0}}^{x} \\frac{1}{x} dx = \\int_{0}^{t} e^{-2 t}d t$  \n",
    "After integration you obtain the implicit form:   \n",
    "$\\ln(x(t))= - \\frac{1}{2}e^{-2t} + C$   \n",
    "With $C= \\frac{1}{2} + \\ln(x_{0})$  \n",
    "The explicit form is:  \n",
    "$x(t)=e^{\\frac{1}{2}}x_{0}e^{-\\frac{1}{2}e^{-2t}}$  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    ">2. Separating the variables gives :   \n",
    "$\\int_{x_{0}}^{x} \\frac{1}{4x^{2}-1} dx = \\int_{0}^{t}d t$  \n",
    "And from partial fractions: $\\frac{1}{4x^{2}-1} = \\frac{1}{2}(\\frac{1}{2x-1} - \\frac{1}{2x+1} )$\n",
    "So after integration you get:  \n",
    "$x(t)=\\frac{1}{2}(\\frac{2x_0 + 1 \\:+\\: (2x_0 - 1)e^{4t})}{2x_0 + 1 \\:-\\: (2x_0 - 1)e^{4t}})$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The non-autonomous Gompertz model for tumor growth"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A surprisingly accurate model for the growth of a tumor of volume $N$ is given by the following differential equation\n",
    "\\begin{equation}\n",
    "\\frac{dN}{dt}=r(t)N(t) \n",
    "\\end{equation}\n",
    "with $r(t)=r_{0}e^{-at}$ and initial size $N(0)=N_{0}$. In other words, the population grows with a time dependent rate $r(t)$, which decreases exponentially in time with a rate $a$.  \n",
    "\n",
    "**1) Give a plausible explanation for the proposed behavior of $r(t)$. Why should the growth rate decrease with time?**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    ">This is due to the fact that resources (i.e. nutrients) become limiting : as the population grows, the surface of the tumor grows slower than its volume, so the nutrients intake per cell decreases in time. The lack of space also limits the growth."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**2) What is the meaning of $r_{0}$?**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    ">$r_{0}$ is the population *relative* growth rate at time zero $(t=0)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**3) Show that the solution for $N(t)$ in function of the 3 parameters $N_{0}, r_{0}, a$  can be written as $N(t)=N_{0}e^{\\frac{r_{0}}{a}(1-e^{-at})}$.**  \n",
    "*Hint*: Use the method of separation of variables."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    ">$\\int_{N_{0}}^{N} \\frac{1}{N} dN= r_{0} \\int_{0}^{t}e^{-at} dt$\n",
    ">\n",
    "Integration gives the solution."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**4) Study the solution:**\n",
    "\n",
    "1. Show that for very short times the population grows linearly like $N(t)= N_0 (1 + r_0 t)$.\n",
    "2. Show that for very long times $N(t)\\cong N_{max}(1-\\frac{r_{0}}{a}e^{-at})$.\n",
    "\n",
    "\n",
    "*Hint*: Use the Taylor approximation $e^{x} \\approx 1 + x$ (valid for small x) for the inner or the outer exponential when appropriate."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    ">1. When $t$ is close to zero, the condition for the Taylor approximation is valid for the inner exponential $e^{-at}$.  \n",
    "Using $e^{-at} \\approx 1 - at$, we find that:  \n",
    "$N(t)=N_{0}e^{\\frac{r_{0}}{a}(1-1+at)}=N_{0}e^{r_{0}t}$  \n",
    "Then, we apply the Taylor approximation again, the growth is *linear* for small times:\n",
    "$$N(t) \\approx N_0 e^{r_{0}t} \\approx N_0 (1 + r_0 t)$$\n",
    "\n",
    ">2. Rewriting the solution like:  \n",
    "$N(t)=N_{0}e^{\\frac{r_{0}}{a}}e^{-\\frac{r_{0}e^{-at}}{a}}$  \n",
    "We see that $-\\frac{r_{0}}{a}e^{-at}$ becomes small when $t$ is large, so we can use the Taylor expansion to the first order:  \n",
    "$N(t)=N_{0}e^{\\frac{r_{0}}{a}}(1- \\frac{r_{0}}{a}e^{-at})$ <br> or $N(t)=N_{max}(1- \\frac{r_{0}}{a}e^{-at})$ with $N_{max}=N_{0}e^{\\frac{r_{0}}{a}}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**5) Sketch the solution. Indicate $N_{max}$. How does the $N$ approaches $N_{max}$?**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here is the plot of the solution for arbitrary values of the constants $N_{0}, r_{0}$ and $a$. $N_{max}$ is the asymptotic value of $N(t)$ for long times. $N$ approaches $N_{max}$ exponentially fast."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#define analytical function for N\n",
    "def N2(N0, r0, a, t):\n",
    "    return N0*np.exp((r0/a)*(1-np.exp(-a*t)))\n",
    "\n",
    "#define parameters\n",
    "N0 = 2\n",
    "r0 = 3\n",
    "a = 0.7\n",
    "Nmax = N0*np.exp(r0/a)\n",
    "tspan = np.linspace(0,11,1000)\n",
    "\n",
    "#plot\n",
    "plt.plot(tspan, N2(N0,r0,a,tspan))\n",
    "plt.axhline(Nmax, color = 'red', ls = '--')\n",
    "plt.text(0.5, Nmax-5.5, 'Nmax', color = 'red')\n",
    "plt.xlabel('t')\n",
    "plt.ylabel('N(t)')\n",
    "plt.show()"
   ]
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.11.5"
  },
  "title": "Exercise 1: Growth models in 1D",
  "widgets": {
   "state": {
    "24aa0b8d62014553b6fdce05a067bb36": {
     "views": [
      {
       "cell_index": 18
      }
     ]
    }
   },
   "version": "1.2.0"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}