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 × × (1 – ÷ 100)= 19.52% Next % = 2 × × (1 – ÷ 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=

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= “sensitively dependent on initial conditions” or not?