{
 "metadata": {
  "name": "",
  "signature": "sha256:0e2ac0dc493fad138cf4f54e91ba4c80dae6eb22e770fd0b02f90e75a708d6be"
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%pylab inline\n",
      "import odespy"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Populating the interactive namespace from numpy and matplotlib\n"
       ]
      }
     ],
     "prompt_number": 63
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print odespy.list_available_solvers()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "['AdamsBashMoulton2', 'AdamsBashMoulton3', 'AdamsBashforth2', 'AdamsBashforth3', 'AdamsBashforth4', 'Backward2Step', 'BackwardEuler', 'BogackiShampine', 'CashKarp', 'CrankNicolson', 'Dop853', 'Dopri5', 'DormandPrince', 'Euler', 'EulerCromer', 'Fehlberg', 'ForwardEuler', 'Heun', 'Leapfrog', 'LeapfrogFiltered', 'Lsoda', 'Lsodar', 'Lsode', 'Lsodes', 'Lsodi', 'Lsodis', 'Lsoibt', 'MidpointImplicit', 'MidpointIter', 'RK2', 'RK3', 'RK4', 'RKC', 'RKF45', 'RKFehlberg', 'Radau5', 'Radau5Explicit', 'Radau5Implicit', 'RungeKutta1', 'RungeKutta2', 'RungeKutta3', 'RungeKutta4', 'ThetaRule', 'Trapezoidal', 'Vode', 'lsoda_scipy', 'odefun_sympy']\n"
       ]
      }
     ],
     "prompt_number": 64
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$$\n",
      "\\frac{dy}{dt} = r y (1-\\frac{y}{K})\n",
      "$$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def f(y,t,r,K):\n",
      "    return r*y*(1-y/K)\n",
      "r,K=0.1,1\n",
      "solver = odespy.Vode(f, f_args=(r,K))\n",
      "solver.set_initial_condition(0.1)\n",
      "t = linspace(0,100)\n",
      "y,t = solver.solve(t)\n",
      "plot(t,y);"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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7lbuvAJYTMrZe2Q74jD0kle8SLZWjgFeBfd19beKjtUCxLGsyDhhGGIjfphiv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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x56d5210>"
       ]
      }
     ],
     "prompt_number": 74
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "t"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 75,
       "text": [
        "array([   0.        ,    2.04081633,    4.08163265,    6.12244898,\n",
        "          8.16326531,   10.20408163,   12.24489796,   14.28571429,\n",
        "         16.32653061,   18.36734694,   20.40816327,   22.44897959,\n",
        "         24.48979592,   26.53061224,   28.57142857,   30.6122449 ,\n",
        "         32.65306122,   34.69387755,   36.73469388,   38.7755102 ,\n",
        "         40.81632653,   42.85714286,   44.89795918,   46.93877551,\n",
        "         48.97959184,   51.02040816,   53.06122449,   55.10204082,\n",
        "         57.14285714,   59.18367347,   61.2244898 ,   63.26530612,\n",
        "         65.30612245,   67.34693878,   69.3877551 ,   71.42857143,\n",
        "         73.46938776,   75.51020408,   77.55102041,   79.59183673,\n",
        "         81.63265306,   83.67346939,   85.71428571,   87.75510204,\n",
        "         89.79591837,   91.83673469,   93.87755102,   95.91836735,\n",
        "         97.95918367,  100.        ])"
       ]
      }
     ],
     "prompt_number": 75
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$$\n",
      "\\frac{dy_1}{dt} = r_1 y_1 \\Big(1-\\frac{y_1+y_2}{K_1}\\Big) \\\\\n",
      "\\frac{dy_2}{dt} = r_2 y_2 \\Big(1-\\frac{y_1+y_2}{K_2}\\Big)\n",
      "$$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def f(y, t, r, K):\n",
      "    return r[0]*y[0]*(1-y.sum()/K[0]), r[1]*y[1]*(1-y.sum()/K[1])\n",
      "\n",
      "r,K = (0.1,0.11),(1,1.1)\n",
      "solver = odespy.Vode(f, f_args=(r,K))\n",
      "solver.set_initial_condition((0.1,0.1))\n",
      "t = linspace(0,1000,10000)\n",
      "y,t = solver.solve(t)#, lambda u, t, step_no: any(u[step_no]<1e-6))\n",
      "plot(t,y)\n",
      "plot(t,y.sum(axis=1), label='sum')\n",
      "legend(labels=['y0','y1'], loc='center right')\n",
      "axhline(y=1, color='k', ls='--')\n",
      "axhline(y=0.95, color='k', ls='--')\n",
      "ylim(0,1.1);"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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xAU30FZUVjJs5jqzMLCV6ERFCIdkXFkL79gE95cOLHqZNizbcesatAT2viEi48r83TlFR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\nXkQkCijZi4hEgf8PvtYowufYgU4AAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x74ff050>"
       ]
      }
     ],
     "prompt_number": 95
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot(t,y)\n",
      "xlim(0,50)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 79,
       "text": [
        "(0, 50)"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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mWJl5wARgdvSPwOHigV7eSomIyMW5YKi7e76ZTQAWAvHA8+6ebmbjo+9Pc/cF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\nXUQkRBTqIiIholAXEQmR/w/QeHEBVBE6xQAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x56df210>"
       ]
      }
     ],
     "prompt_number": 79
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# Leveau 2001\n",
      "\n",
      "$$\n",
      "\\frac{\\partial N}{\\partial OD} = \\frac{P - m N /OD -D_n}{r(1-OD/K)} \\\\\n",
      "\\frac{\\partial F}{\\partial OD} = \\frac{m N /OD - D_f}{r(1-OD/K)} \\\\\n",
      "D_x = D_{max} \\frac{x}{N+F+K_m OD}\n",
      "$$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print odespy.list_available_solvers()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "['AdamsBashMoulton2', 'AdamsBashMoulton3', 'AdamsBashforth2', 'AdamsBashforth3', 'AdamsBashforth4', 'Backward2Step', 'BackwardEuler', 'BogackiShampine', 'CashKarp', 'CrankNicolson', 'Dop853', 'Dopri5', 'DormandPrince', 'Euler', 'EulerCromer', 'Fehlberg', 'ForwardEuler', 'Heun', 'Leapfrog', 'LeapfrogFiltered', 'Lsoda', 'Lsodar', 'Lsode', 'Lsodes', 'Lsodi', 'Lsodis', 'Lsoibt', 'MidpointImplicit', 'MidpointIter', 'RK2', 'RK3', 'RK4', 'RKC', 'RKF45', 'RKFehlberg', 'Radau5', 'Radau5Explicit', 'Radau5Implicit', 'RungeKutta1', 'RungeKutta2', 'RungeKutta3', 'RungeKutta4', 'ThetaRule', 'Trapezoidal', 'Vode', 'lsoda_scipy', 'odefun_sympy']\n"
       ]
      }
     ],
     "prompt_number": 117
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def f(y, OD, r, K, P, m, Dmax, Km):\n",
      "    N,F = y\n",
      "    Dn = Dmax * N/(N + F + Km * OD)\n",
      "    Df = Dmax * F/(N + F + Km * OD)\n",
      "    dOD = r * (1 - OD/K)\n",
      "    dN = P - m*N/OD - Dn\n",
      "    dF = m*N/OD - Df   \n",
      "    return dN/dOD, dF/dOD\n",
      "\n",
      "r, K, P, m, Dmax, Km = 0.1, 1, 250, 100, 230, 35\n",
      "solver = odespy.Dopri5(f, f_args=(r, K, P, m, Dmax, Km))\n",
      "solver.set_initial_condition((0.,0.))\n",
      "OD = linspace(0.1,K*0.999,100)\n",
      "y,OD = solver.solve(OD)#, lambda u, OD, step_no: any(u[step_no]<1e-6))\n",
      "plot(OD,y)\n",
      "legend(labels=['N','F'], loc='upper left')\n",
      "xlabel('OD')\n",
      "ylabel('F');"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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mcPzM9++HOXMa5li1/XxLSuDJJ+u+v2COFcx2ZWXwyCMn37ahf5bl5TB7dnh+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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x76f4bd0>"
       ]
      }
     ],
     "prompt_number": 133
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def FvsOD(OD, r, K, P, m, Dmax, Km):\n",
      "    solver = odespy.Dopri5(f, f_args=(r, K, P, m, Dmax, Km))\n",
      "    solver.set_initial_condition((0.1,0.))\n",
      "    y,OD = solver.solve(OD)\n",
      "    N,F = y\n",
      "    return F"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 127
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}