{
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{
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"
}
},
"cell_type": "markdown",
"metadata": {},
"source": [
"## Logistic Growth\n",
"In the previous notebook, we explored linear and exponential growth. In both cases, growth goes on forever -- a situation that doesn't typically happen for example since bacteria run out of food. So lets walk through such an example:\n",
"\n",
" * the food initially available is $C_0$\n",
" * division of a bacterium requires $x$ amount of food. Hence there can at most by $N = C_0/x$ new bacteria at the end\n",
" * the food remaining after time $t$ is $C(t) = C_0 - x\\times(n(t)-n_0)$.\n",
" * lets assume the rate of division decreases proportionally with the available food $\\frac{C(t)}{C_0\\tau}$\n",
" \n",
"![image.png](attachment:de09c9e7-282b-4d43-a7d4-68bc6e794baa.png)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"With these assumptions and definitions, we find a difference equation\n",
"\n",
"$$\n",
"\\begin{split}\n",
"n(t+\\Delta t) & = n(t) + \\Delta t\\times n(t)\\times \\frac{C(t)}{C_0\\tau} \\\\\n",
"&= n(t) + \\Delta t\\times \\frac{n(t)}{\\tau}\\times \\left(1- \\frac{x(n(t) - n_0)}{C_0}\\right) \\\\\n",
"&= n(t) + \\Delta t\\times \\frac{n(t)}{\\tau}\\times \\left(1- \\frac{n(t) - n_0}{N}\\right)\n",
"\\end{split}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Rearranging this into a differential equation in the usual way results in \n",
"\n",
"$$\n",
"\\lim_{\\Delta t \\to 0} \\frac{n(t+\\Delta t) - n(t)}{\\Delta t} = \\frac{dn(t)}{dt} = \\frac{n(t)}{\\tau}\\times \\left(1- \\frac{n(t) - n_0}{N}\\right)\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"This can be further simplified by realizing that whenever it matters, the $n(t)\\gg n_0$ so that we can simply drop $n_0$ from the right hand side to obtain the standard logistic differential equation:\n",
"\n",
"\n",
"$$\n",
"\\frac{dn(t)}{dt} = \\frac{n(t)}{\\tau}\\left(1- \\frac{n(t)}{N}\\right)\n",
"$$\n",
"\n",
" Here $N$ is often called carrying capacity. "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Before we start solving this equation, lets look at the case $n(t)\\ll N$!\n",
"\n",
"In this case, the equation simplifies\n",
"\n",
"$$\n",
"\\frac{dn(t)}{dt} = \\frac{n(t)}{\\tau}\\left(1- n(t)/N\\right) \\approx \\frac{n(t)}{\\tau}\n",
"$$\n",
"\n",
"This is simply exponential growth like we have seen before, but we expect this approximation only to be valid while\n",
"\n",
"$$\n",
"n(t)\\approx n_0 e^{t/\\tau} \\ll N\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"# define function that return derivative\n",
"def dndt(n, tau, N):\n",
" return n/tau*(1-n/N)"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"tau = 30 # division time of 30 minutes\n",
"N = 100\n",
"n_0 = 1\n",
"n = [n_0]\n",
"t = [0]\n",
"Delta_t = 0.1\n",
"tmax = 10*tau\n",
"for i in range(int(tmax//Delta_t)): # number of steps necessary is tmax divided by step size = tmax/Delta_t\n",
" n.append(n[i] + Delta_t * dndt(n[i],tau,N))\n",
" t.append(t[i] + Delta_t)"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
""
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
"image/png": 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\n",
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