rabbits2soln


Modeling and Simulation in Python

Rabbit example

Copyright 2017 Allen Downey

License: Creative Commons Attribution 4.0 International


In [1]:
%matplotlib inline

from modsim import *

Rabbit Redux

This notebook starts with a version of the rabbit population growth model and walks through some steps for extending it.

In the original model, we treat all rabbits as adults; that is, we assume that a rabbit is able to breed in the season after it is born. In this notebook, we extend the model to include both juvenile and adult rabbits.

As an example, let's assume that rabbits take 3 seasons to mature. We could model that process explicitly by counting the number of rabbits that are 1, 2, or 3 seasons old. As an alternative, we can model just two stages, juvenile and adult. In the simpler model, the maturation rate is 1/3 of the juveniles per season.

To implement this model, make these changes in the System object:

  1. Replace p0 with two initial populations: juvenile_pop0 and adult_pop0, with values 0 and 10.

  2. Add an additional variable, mature_rate, with the value 0.33.


In [2]:
system = System(t0 = 0, 
                t_end = 30,
                juvenile_pop0 = 0,
                adult_pop0 = 10,
                birth_rate = 0.9,
                mature_rate = 0.33,
                death_rate = 0.5)

system


Out[2]:
value
t0 0.00
t_end 30.00
juvenile_pop0 0.00
adult_pop0 10.00
birth_rate 0.90
mature_rate 0.33
death_rate 0.50

Now update run_simulation with the following changes:

  1. Add a second TimeSeries, named juveniles, to keep track of the juvenile population, and initialize it with juvenile_pop0.

  2. Inside the for loop, compute the number of juveniles that mature during each time step.

  3. Also inside the for loop, add a line that stores the number of juveniles in the new TimeSeries. For simplicity, let's assume that only adult rabbits die.

  4. During each time step, subtract the number of maturations from the juvenile population and add it to the adult population.


In [3]:
def run_simulation(system):
    """Runs a proportional growth model.
    
    Adds TimeSeries to `system` as `results`.
    
    system: System object with t0, t_end, p0,
            birth_rate and death_rate
    """
    juveniles = TimeSeries()
    juveniles[system.t0] = system.juvenile_pop0
    
    adults = TimeSeries()
    adults[system.t0] = system.adult_pop0
    
    for t in linrange(system.t0, system.t_end):
        maturations = system.mature_rate * juveniles[t]
        births = system.birth_rate * adults[t]
        deaths = system.death_rate * adults[t]
        
        if adults[t] > 30:
            market = adults[t] - 30
        else:
            market = 0
            
        juveniles[t+1] = juveniles[t] + births - maturations
        adults[t+1] = adults[t] + maturations - deaths - market
        
    system.adults = adults
    system.juveniles = juveniles

Test your changes in run_simulation:


In [4]:
run_simulation(system)
system.adults


Out[4]:
value
0 10.000000
1 5.000000
2 5.470000
3 6.209900
4 7.057723
5 8.021560
6 9.117031
7 10.362107
8 11.777219
9 13.385586
10 15.213601
11 17.291261
12 19.652658
13 22.336542
14 25.386953
15 28.853947
16 32.794414
17 34.478600
18 36.487431
19 37.893339
20 39.401924
21 40.546917
22 41.694992
23 42.613800
24 43.495581
25 44.226171
26 44.907656
27 45.485241
28 46.014130
29 46.469075
30 46.880673
31 47.238170

Next, update plot_results to plot both the adult and juvenile TimeSeries.


In [5]:
def plot_results(system, title=None):
    """Plot the estimates and the model.
    
    system: System object with `results`
    """
    newfig()
    plot(system.adults, 'bo-', label='adults')
    plot(system.juveniles, 'gs-', label='juveniles')
    decorate(xlabel='Season', 
             ylabel='Rabbit population',
             title=title)

And test your updated version of plot_results.


In [6]:
plot_results(system, title='Proportional growth model')


This notebook demonstrates the steps we recommend for starting your project:

  1. Start with one of the examples from the book, either by copying a notebook or pasting code into a new notebook. Get the code working before you make any changes.

  2. Make one small change, and run the code again.

  3. Repeat step 2 until you have a basic implementation of your model.

If you start with working code that you understand and make small changes, you can avoid spending a lot of time debugging.

One you have a basic model working, you can think about what metrics to measure, what parameters to sweep, and how to use the model to predict, explain, or design.

Bonus question

Suppose you only have room for 30 adult rabbits. Whenever the adult population exceeds 30, you take any excess rabbits to market (as pets for kind children, of course). Modify run_simulation to model this strategy. What effect does it have on the behavior of the system?

What happens if rabbits only take one season to mature? Change mature_rate to 1.0 and run the simulation again.


In [ ]: