Class 7: System dynamics models: Growth and decay¶

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Load packages¶

In [1]:

%matplotlib inline

import matplotlib.pyplot as plt
import numpy as np
import pandas as pd

Growth model for bacterial colony¶

Iteratively solve the difference equation using numerical methods

$\begin{equation} P(t)=P(t-\Delta{}t)-rP(t-\Delta{}t)\Delta{}t \end{equation}$

Assume the following constants:

population = 100
r = 0.1
delta_t = 0.001
time_of_simulation = 50.5

Class-led solution¶

How do I handle this in Python?

Define our constants:

In [2]:

population = 100
r = 0.1
delta_t = 0.001
time_of_simulation = 50.5

Define a list to store our simulation history.

In [3]:

trace = [[0, population]]

Iteratively solve the difference equation using a for loop and store the [time_step, population] pair in trace:

In [4]:

for time_step in range(1, int(time_of_simulation / delta_t) + 1):
    population = population + r * population * delta_t
    trace.append([time_step, population])

Convert our results into a DataFrame.

How can we extract out the first column and the second column of trace?

Remember, these are not numpy arrays!

We use list comprehensions, for example:

[time_step[0] for time_step in trace]

This is equivalent to the following:

column_one = []
for time_step in trace:
    column_one.append(time_step[0])

With list comprehensions, we can feed each column into pd.DataFrame as follows:

In [5]:

bacteria_df = pd.DataFrame({
    "time_step": [time_step[0] for time_step in trace],
    "population": [time_step[1] for time_step in trace]
})

In [6]:

bacteria_df.head()

Out[6]:

	time_step	population
0	0	100.000000
1	1	100.010000
2	2	100.020001
3	3	100.030003
4	4	100.040006

time_step is just the iteration index for each step of the simulation. To convert back to "real" time, multiply time_step by delta_t. With our Pandas data frame, this is simple to do:

In [7]:

bacteria_df["time"] = bacteria_df["time_step"] * delta_t

Finally, we should visualize the results of our simulation. Here's how to do it in matplotlib:

In [8]:

fig, ax = plt.subplots()  # Define an "axis" object for more control over plot

ax.plot(bacteria_df["time_step"], bacteria_df["population"]);

# Don't forget to label your plot!
ax.set_xlabel("time")
ax.set_ylabel("population");