NA01_Dodos.ipynb

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    "colab": {
      "name": "NA01_Dodos.ipynb",
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  "cells": [
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      "cell_type": "markdown",
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        "id": "view-in-github",
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      "source": [
        "<a href=\"https://colab.research.google.com/github/john-s-butler-dit/CaseStudy_PredatorPrey/blob/master/NA01_Dodos.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
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      "source": [
        "# Numerical Assignment 1 Dodos\n",
        "\n",
        "Name and Student ID: John Butler 999999999"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "zSk_lK9vWoTr"
      },
      "source": [
        "Adapt the python code to model the initial value problem, which describes the Dodo population (p) in the 17th century the year (t=0) that the Dodo are exposed to man and subsequent interactions modeled by the over 10 years using a stepsize of h=0.1.\n",
        "To learn more about the dodo look here [2]."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "R2ETLv-mWoTr"
      },
      "source": [
        "## Library\n",
        "import numpy as np\n",
        "import matplotlib.pyplot as plt"
      ],
      "execution_count": 1,
      "outputs": []
    },
    {
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      "source": [
        "## Dodo Population Equation before man\n",
        "The Dodo population prior to being found by man is described by the non-linear, \n",
        "\\begin{equation}\n",
        "\\frac{d p}{dt}=\\alpha_{Birth} p-\\alpha_{Death}p^2,\n",
        "\\end{equation}\n",
        "where $p$ is the population at time $t$,\n",
        "for $a\\leq t \\leq b$ years, where  $\\alpha_{Birth}$ is the birth rate and $\\alpha_{Death}$ is the death rate, with the initial condition\n",
        "$$ p(a)=A. $$\n",
        "TIn this simulation we set the parameters of the different equations:\n",
        "*  as $\\alpha_{Birth}=0.3$, and $\\alpha_{Death}=0.01$ with the inital condition $p(0)=30$, over the time period $0\\leq t \\leq 10$."
      ]
    },
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      "metadata": {
        "id": "aRMI-CG1WoTs"
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      "source": [
        "alpha_birth=0.3\n",
        "alpha_death=0.01"
      ],
      "execution_count": 2,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "BtRQ0x_mWoTs"
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      "source": [
        "## Discrete Time Domain\n",
        "To numerically approximate the solution we discretise the continuous domain using a step size, $h=0.1$, \n",
        "which gives\n",
        "\\begin{equation}\n",
        "t_i=0+i h,\n",
        "\\end{equation}\n",
        "for $i=0,...100$.\n",
        "\n",
        "The Figure below illustrate the discrete domain."
      ]
    },
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          "height": 248
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        "outputId": "77bffa53-2108-4ea9-d73a-beb0385aa6ae"
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      "source": [
        "h=0.1\n",
        "t_end=10\n",
        "t=np.arange(0,t_end+h/2,h)\n",
        "fig = plt.figure(1,figsize=(14,4))\n",
        "ax = fig.add_subplot(1,1,1)\n",
        "ax.plot(t,0*t,'o:',color='k')\n",
        "ax.set_xlabel('Time (yrs)')\n",
        "plt.title('Discrete Time')\n",
        "plt.show()"
      ],
      "execution_count": 3,
      "outputs": [
        {
          "output_type": "display_data",
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\n",
            "text/plain": [
              "<Figure size 1008x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "96bMXPF_WoTu"
      },
      "source": [
        "N=int(t_end/h) # Number of Steps\n",
        "P=np.zeros(N+1) # Numerical Solution P\n",
        "P[0]=30"
      ],
      "execution_count": 4,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "hRi5vrn5WoTu"
      },
      "source": [
        "We numerically approximate the differential equation using Eulers method to give the difference equation\n",
        "\\begin{equation}\n",
        "P_{i}=P_{i-1}+h \\big(\\alpha_{Birth} P_{i-1}-\\alpha_{Death}P_{i-1}^2 \\big),\n",
        "\\end{equation}\n",
        "where $h$ is the stepsize and where $P_i$ is the numerical approximation of $p(t_i)$ at distance $t_i$, for $i=0,...,10.$\n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Dey6FlcXWoTv"
      },
      "source": [
        "for i in range (1,N+1):\n",
        "    P[i]=P[i-1]+h*(alpha_birth*P[i-1]-alpha_death*P[i-1]*P[i-1])\n",
        "\n"
      ],
      "execution_count": 5,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "dvbG7skkWoTv"
      },
      "source": [
        "## Results\n",
        "The plot below shows the numerical solution $P_i$ for the Dodo population as a function of time $t$."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 249
        },
        "id": "k7xn-jxDWoTv",
        "outputId": "5d3287cc-889c-4f75-e1d2-77a85efaeb49"
      },
      "source": [
        "## Plotting Figure\n",
        "fig = plt.figure(figsize=(14,4))\n",
        "plt.plot(t,P,'o:',color='red',label='Population')\n",
        "plt.legend()\n",
        "plt.xlabel('Time (yrs)')\n",
        "plt.ylabel('Population (thousands)')\n",
        "#ax.legend(loc='best')\n",
        "plt.title('Dodo Population, P')\n",
        "plt.ylim(0,40)\n",
        "plt.show()"
      ],
      "execution_count": 9,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 1008x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "jaElZK2wWoTv"
      },
      "source": [
        "## Dodo Population Equation after man\n",
        "The Dodo population prior to being found by man is described by the non-linear, \n",
        "\\begin{equation}\n",
        "\\frac{d p}{dt}=\\alpha_{Birth} p-\\alpha_{Death}p^2-p(e^{0.1t}-1),\n",
        "\\end{equation}\n",
        "where $p$ is the population at time $t$,\n",
        "for $a\\leq t \\leq b$ years, where  $\\alpha_{Birth}$ is the birth rate and $\\alpha_{Death}$ is the death rate, with the initial condition\n",
        "$$ p(a)=A. $$\n",
        "The introduction of humans to the island is simulated with the inclusion of the terms\n",
        "$$-p(e^{0.1t}-1).$$\n",
        "\n",
        "Like above the parameters are set:\n",
        "*  as $\\alpha_{Birth}=0.3$, and $\\alpha_{Death}=0.01$ with the inital condition $p(0)=30$, over the time period $0\\leq t \\leq 10$."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "UQ_Mg369WoTw"
      },
      "source": [
        "## Problem 1\n",
        "Adapt the text in blue and the code below to model the impact of humans on the Dodo population.\n",
        "\n",
        "We numerically approximate the differential equation using Eulers method to give the difference equation\n",
        "\\begin{equation}\n",
        "P_{i}=P_{i-1}+h \\big(\\alpha_{Birth} P_{i-1}-\\alpha_{Death}P_{i-1}^2+ \\ \\color{blue}{HUMANS} \\big),\n",
        "\\end{equation}\n",
        "where $h$ is the stepsize and where $P_i$ is the numerical approximation of $p(t_i)$ at distance $t_i$, for $i=0,...,10.$\n",
        "\n",
        "HINT: use np.exp(0.1*t[i]) for $e^{0.1t_i}$."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "nGUnLfZPWoTx"
      },
      "source": [
        "for i in range (1,N+1):\n",
        "    ### UPDATED DIFFERENCE FOR DODOS TO INCLUDE HUMANS\n",
        "    P[i]=P[i-1]+h*(alpha_birth*P[i-1]-alpha_death*P[i-1]*P[i-1]-)\n",
        "    ###"
      ],
      "execution_count": 7,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 246
        },
        "id": "HUmhx3UWWoTx",
        "outputId": "c94798a7-a31c-4675-860d-b9223ec60dac"
      },
      "source": [
        "## Plotting Figure\n",
        "fig = plt.figure(figsize=(14,4))\n",
        "plt.plot(t,P,'o:',color='red',label='Population')\n",
        "plt.legend()\n",
        "plt.xlabel('Time (yrs)')\n",
        "plt.ylabel('Population (thousands)')\n",
        "#ax.legend(loc='best')\n",
        "plt.title('Dodo Population, P')\n",
        "plt.show()"
      ],
      "execution_count": 8,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 1008x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "jAh9Gkp2WoTy"
      },
      "source": [
        "## Problem 2. \n",
        "\n",
        "Write a brief description of what the solution of the model tells you about the population of Dodos.\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "JETpiXdxWoTy"
      },
      "source": [
        "# References\n",
        "\n",
        "[1] Stover, Christopher and Weisstein, Eric W. \"Population Growth.\" From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/PopulationGrowth.html\n",
        "\n",
        "[2] Wikipedia contributors. (2021, January 13). Dodo. In Wikipedia, The Free Encyclopedia. Retrieved 15:19, February 22, 2021, from https://en.wikipedia.org/w/index.php?title=Dodo&oldid=1000163941"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "_uARRxUrZ7jW"
      },
      "source": [
        ""
      ],
      "execution_count": 8,
      "outputs": []
    }
  ]
}