Let’s imagine a large empty island – Australia before we Europeans invaded it, maybe – and an invasion of just two rabbits. Assume there is enough food for rabbits and no predators. Rabbits make more rabbits, surprisingly rapidly. A little research says that a doe (female) is pregnant for 28-31 days and is then, amazingly ready to be made pregnant all over again, so we will assume that our basic counting unit is the ‘month’, and that each doe is made pregnant pretty immediately. if we need to stretch a month to 35 days, we can do that later. the gestation period defines our time measure.

Further let us assume that the litter size is 4 [L=4] and that the male:female split is even, so each litter produces L/2=2 females. A little more research says these baby rabbits are ready to make babies after six months, so we’ll say at seven months the new females become pregnant, and we do not much care by which male. If it makes you happier, assume the original rabbit was pregnant by a different male than the one she arrived with.

At month 0 we have one male, one female and she is pregnant

At month 1 we have 1 adult male, 1 adult female, 2 baby males and 2 baby females, total 6

At month 2: 1 adult male, 1 adult female, 2 more baby males and 2 more baby females, total 10

At month 3: 1 adult male, 1 adult female, 2 more baby males and 2 more baby females,  total 14

At month 4: 1 adult male, 1 adult female,2 more baby males and 2 more baby females, total 18

At month 5: 1 adult male, 1 adult female, 2 more baby males and 2 more baby females, total 22

At month 6: 1 adult male, 1 adult female, 2 more baby males and 2 more baby females, total 26

in month 7 the oldest of the baby girls can become pregnant and so the pattern changes from here onwards. the count at month 7 is: 3 adult males, 3 adult females, 2 more baby males and 2 more baby females, total 30

Month 8 all three females birth a litter of four and litter Two becomes adult. Count is: 5 adult males, 5 adult females, 6 more baby males and 6 more baby females, total 42

Month 9: 7 adult females, 20 babies (last month’s 5 adults females * litter size), total 62

Month 10: 9 adult females, 7x4=28 babies, total 62+28=90

Month 11: 11 adult females, 9x4=36 babies, total 90+36=126

Month 12: 13 adult females, 11x4=44 babies, total 126+44=170

Month 13: 15 adult females, 13x4=52 babies, total 170+52=222

Month 14: 21 adult females, because the month 8 babies are now adult, 15x4=60 babies, total 222+60=282

Month 15: 31 adult females (21+ half of month 9’s 20 new adults) 21x4=84 babies, total 282+84=366

Month 16: 45 adult females (31+ half of month 10’s 28 new adults) 31x4=124 babies, total 366+124=490

Month 17: 63 adult females (45+ half  of 36 new adults) 45x4=180 babies, total 490+180=670

Month 18: 63 + 44/2 = 85 adult females, 63x4=252 babies, total 670+252=922

Month 19: 85 + 52/2 = 111 adult females, 85x4=340 babies, total 922+340=1262

Month 20: 111 + 60/2 = 141 adult females, 111 x4=444 babies, total 1262+444=1706

Month 21: 141 + 84/2 = 183  adult females, 141 x4=564 babies, total 1706+564=2270

Month 22: 183 + 124/2 = 245 adult females, 183 x4=732 babies, total 2270+732=3002

Month 23: 245 + 180/2 = 335 adult females, 245 x4=980 babies, total 3002+980=3982

Month 24: 335 + 252/2 = 461 adult females, 335 x4=1340 babies, total 3982+1340=5322

Month 25: 461 + 340/2 = 631 adult females, 461 x4=1844 babies, total 5322+1844=7166

Month 26: 631 + 444/2 = 853 adult females, 631 x4=2524 babies, total 7166+2524=9690

So after two years we have 10 thousand rabbits. At 34 months we have more than a million. if we change the breeding cycle (birth to the next birth by the same mother) to 35 days, so 11 cycles a year instead of 12, we still have a million in three years.

Write a spreadsheet to show this pattern. 

If you didn’t do it straight away, put the litter size as a control figure (mine is in the first row, in D1) and set the m/f split as 50% (mine is in E1) and do the multiplying with a fixed reference.

In my model, I have these formulas:

Month N          Rabbits (N) = Rabbits (N-1) + Babies (N)

                        Babies (N) = Biddable females (N-1) * Litter          where Litter is in D1 and set at 4

                        Biddable females (N) = Biddable females (N-1) + New Females (N-6)

                        New Females (N) = Babies (N) * M’F split              where M/F split is set at 50%

A smart user of Excel will Define Names much as I have done, but probably would use shorter names.

If you then graph Month (x) and Rabbits (y) you have a spectacular graph whose curve is called an exponential.

If you change Litter from 4 (my reading said it depends on the breed whether you have 4 to 14 in a litter) and if you modified your counting of breeding cycles to recognise that a year might be 12 or down to 10 per year, you have an idea of the spectacular growth of rabbits while the food lasts.

I modified my graph to reflect a doe wearing out at two years and all rabbits dying at three (or being eaten). It doesn’t slow the growth by much. What makes a difference is when there is something that reduces the population significantly (such as humans eating them). Which would be the next model to explore.

Think of this as the beginning of a GCSE (Y9-11) modelling exercise.

© David Scoins 2017