04_Logisitic Population Differential Equations.ipynb

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      "name": "04_Logisitic Population Differential Equations.ipynb",
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        "<a href=\"https://colab.research.google.com/github/john-s-butler-dit/CaseStudy_PredatorPrey/blob/master/04_Logisitic%20Population%20Differential%20Equations.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
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      "source": [
        "## Logistic Population Differential Equations"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "9tMdcbBWWl0O"
      },
      "source": [
        "\n",
        "In this notebook we will model non-linear logistic population growth [1]"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "mSSbP6n3Wl0O"
      },
      "source": [
        "## Library\n",
        "import numpy as np\n",
        "import math \n",
        "import matplotlib.pyplot as plt"
      ],
      "execution_count": 1,
      "outputs": []
    },
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        "id": "Hf8BMVZwWl0P"
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      "source": [
        "## Logistic Population Equation\n",
        "The logistic growth differential equation is a non-linear first order differential equation\n",
        "\\begin{equation}\n",
        "\\frac{d p}{dt}=\\alpha_{Birth} p-\\alpha_{Death} p^2\n",
        "\\end{equation}\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",
        "As the equation is non-linear we must numverically approximate the solution.\n",
        "In 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 $p(0)=3$, over the time period $0\\leq t \\leq 20$."
      ]
    },
    {
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      "source": [
        "alpha_Birth=0.3\n",
        "alpha_Death=0.01"
      ],
      "execution_count": 7,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "bvohyQ-BWl0P"
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      "source": [
        "## Discrete Domain \n",
        "To numerically approximate the solution we discretise the continuous domain using a step size, $h=1$, \n",
        "which gives\n",
        "\\begin{equation}\n",
        "t_i=0+i h,\n",
        "\\end{equation}\n",
        "for $i=0,...20$.\n",
        "The figure below illustrate the discrete domain."
      ]
    },
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      "source": [
        "h=1\n",
        "a=0\n",
        "b=20\n",
        "t=np.arange(a,b+h/2,h)\n",
        "fig = plt.figure(1,figsize=(5,4))\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 Domain')\n",
        "plt.show()"
      ],
      "execution_count": 13,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
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\n",
            "text/plain": [
              "<Figure size 360x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "pqMl2cClWl0R"
      },
      "source": [
        "N=int((b-a)/h) # Number of Steps\n",
        "P=np.zeros(N+1) # Numerical Solution P\n",
        "P[0]=3\n"
      ],
      "execution_count": 14,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "LwujljB9Wl0R"
      },
      "source": [
        "We numerically approximate the differential equation using Eulers method to give the difference equation\n",
        "\\begin{equation}\n",
        "\\frac{P_{i+1}-P_i}{h}=\\alpha_{Birth}P_i-\\alpha_{Death}P_i^2\n",
        "\\end{equation}\n",
        "re-arranging\n",
        "\\begin{equation}\n",
        "\\color{red}{P_{i+1}}=\\color{green}{P_i}+h\\big(\\alpha_{Birth}\\color{green}{P_i}-\\alpha_{Death}\\color{green}{P_i^2}\\big)\n",
        "\\end{equation}\n",
        "\n",
        "where $h$ is the stepsize and where $P_i$ is the numerical approximation of $p(t_i)$ at time $t_i$, for $i=0,...,10$\n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "51anXmYqWl0S"
      },
      "source": [
        "for i in range (0,N):\n",
        "    P[i+1]=P[i]+h*(alpha_Birth*P[i]-alpha_Death*P[i]*P[i])\n",
        "\n"
      ],
      "execution_count": 15,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "RsaqDSO8Wl0S"
      },
      "source": [
        "## Results\n",
        "The plot bellow shows the Numerical solution $P_i$ of the Logistic Population as a function of time $t$."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "I-EcDL94Wl0S",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 281
        },
        "outputId": "c0444cb5-1124-492d-e291-754cd7285630"
      },
      "source": [
        "## Plotting Figure\n",
        "fig = plt.figure(1,figsize=(4,4))\n",
        "\n",
        "ax = fig.add_subplot(1,1,1)\n",
        "\n",
        "ax.plot(t,P,'o:',color='red',label='Population')\n",
        "ax.legend()\n",
        "\n",
        "ax.set_xlabel('Time (yrs)')\n",
        "ax.set_ylabel('Population (billions)')\n",
        "#ax.legend(loc='best')\n",
        "ax.set_title('Numerical Solution, P')\n",
        "\n",
        "\n",
        "\n",
        "plt.tight_layout()\n",
        "plt.subplots_adjust(top=0.85)\n",
        "plt.show()\n"
      ],
      "execution_count": 16,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 288x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "oSybAbgxWl0S"
      },
      "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]  Vito Volterra.  Fluctuations in the abundance of a species considered mathematically. Nature,118:558–560,1926.\n",
        "\n",
        "[3] Alfred J Lotka.   Analytical note on certain rhythmic  relations inorganic  systems.Proceedings of the National Academy of Sciences,6(7):410–415,1920."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Z0qm9M2yWl0T"
      },
      "source": [
        ""
      ],
      "execution_count": 6,
      "outputs": []
    }
  ]
}