Modelling Breeding
Rabbits
“Breeding like Rabbits!”
Suppose we have the following model for the breeding
of rabbits:
We start off with two rabbits in a field.
Each year the number of rabbits doubles.
What happens?
Is this realistic?
A more realistic model
Suppose we think of the maximum number that a field
would support as 100%.
If the rabbits don’t breed so they may go down to 70%.
If there are lots of foxes, it might go down to 40%, say.
Iteration
A much better formula than just doubling, is given by:
N = as × (1 – s ÷ 100)
N = % at the end of the year
a is a fixed number between 0 and 4.
s = % at the start of the year
If a is 2, and the starting % = 3%.
Next % = 2 × 3 × (1 – 3 ÷ 100)
= 5.82%
Next % = 2 × 5.82 × (1 – 5.82 ÷ 100)
= 10.96%
Next % = 2 × 10.96 × (1 – 10.96 ÷ 100) = 19.52%
Next % = 2 × 19.52 × (1 – 19.52 ÷ 100) = 31.42%
“Ans” button on DAL calculators
Iteration can be done very easily using DAL calculators.
I’ll give an example of generating a sequence.
4, 7, 10, 13, 16, 19, ...
What you do on the calculator is:
Enter 4 then press = button
ANS [2ndF =] +3 then every time you press =
you get the next term.
The 2ndF = just takes your last answer
Our Initial Rabbit Problem
N = as × (1 – s ÷ 100)
N = % at the end of the year
a is a fixed number between 0 and 4.
s = % at the start of the year
If a is 2, and the starting % = 3%.
We can solve this efficiently on the calculator by:
Type 3= (This stores 3 as our starting value)
Type 2 x ANS x (1 – ANS ÷ 100)
Then each time you press = you get the population at the end of
the next year.
“sensitively dependent on initial conditions”
or not?
N = as × (1 – s ÷ 100)
N = % at the end of the year
a is a fixed number between 0 and 4.
s = % at the start of the year
If a is 2.7, and the starting % = 12%.
We can solve this efficiently on the calculator by:
Type 12= (This stores 12 as our starting value)
Type 2.7 x ANS x (1 – ANS ÷ 100)
Then each time you press = you get the population at the end of
the next year.
You should get 12, 28.51, 55.03, 66.82, 59.86, ...
Now start with current % being 12.5%. i.e start: 12.5=
“sensitively dependent on initial conditions”
or not?
N = as × (1 – s ÷ 100)
N = % at the end of the year
a is a fixed number between 0 and 4.
s = % at the start of the year
If a is 3.8, and the starting % = 7%.
We can solve this efficiently on the calculator by:
Type 7= (This stores 7 as our starting value)
Type 3.8 x ANS x (1 – ANS ÷ 100)
Then each time you press = you get the population at the end of
the next year.
Record your answers
Now start with current % being 7.1%. i.e start: 7.1=