{
"cells": [
{
"cell_type": "markdown",
"metadata": {
"cell_style": "center",
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"\n",
"\n",
"# Class 7: System dynamics models: Growth and decay\n",
"\n",
"---\n",
"\n",
"\n",
"\n",
"This notebook is licensed under a [Creative Commons Attribution-ShareAlike 4.0 International License](http://creativecommons.org/licenses/by-sa/4.0/)."
]
},
{
"cell_type": "markdown",
"metadata": {
"cell_style": "center",
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"## Load packages"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"cell_style": "center",
"slideshow": {
"cell_style": "split",
"slide_type": "-"
}
},
"outputs": [],
"source": [
"%matplotlib inline\n",
"\n",
"import matplotlib.pyplot as plt\n",
"import numpy as np\n",
"import pandas as pd"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"## Growth model for bacterial colony\n",
"\n",
"Iteratively solve the difference equation using numerical methods\n",
"\n",
"\\begin{equation}\n",
"P(t)=P(t-\\Delta{}t)-rP(t-\\Delta{}t)\\Delta{}t\n",
"\\end{equation}\n",
"\n",
"Assume the following constants:\n",
"\n",
"```python\n",
"population = 100\n",
"r = 0.1\n",
"delta_t = 0.001\n",
"time_of_simulation = 50.5\n",
"```"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"### Class-led solution\n",
"\n",
"How do I handle this in Python?"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"Define our constants:"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"outputs": [],
"source": [
"population = 100\n",
"r = 0.1\n",
"delta_t = 0.001\n",
"time_of_simulation = 50.5"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"Define a list to store our simulation history."
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"outputs": [],
"source": [
"trace = [[0, population]]"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"Iteratively solve the difference equation using a `for` loop and store the `[time_step, population]` pair in `trace`:"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"outputs": [],
"source": [
"for time_step in range(1, int(time_of_simulation / delta_t) + 1):\n",
" population = population + r * population * delta_t\n",
" trace.append([time_step, population])"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"Convert our results into a DataFrame.\n",
"\n",
"**How can we extract out the first column and the second column of `trace`?**\n",
"\n",
"Remember, these are not `numpy` arrays!"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"We use list comprehensions, for example:\n",
"\n",
"```\n",
"[time_step[0] for time_step in trace]\n",
"```\n",
"\n",
"This is equivalent to the following:\n",
"\n",
"```python\n",
"column_one = []\n",
"for time_step in trace:\n",
" column_one.append(time_step[0])\n",
"```"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"With list comprehensions, we can feed each column into `pd.DataFrame` as follows:"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [],
"source": [
"bacteria_df = pd.DataFrame({\n",
" \"time_step\": [time_step[0] for time_step in trace],\n",
" \"population\": [time_step[1] for time_step in trace]\n",
"})"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"outputs": [
{
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