Two rats, one male and one female, scampered on board a ship anchored at a local

Question

Two rats, one male and one female, scampered on board a ship anchored at a local dock. The ship set sail across the ocean. It reached a deserted island in late December. The rats abandoned the ship to make their home on the island. Given these ideal conditions, you can estimate the number of offspeing produced from this pair of rats in one year. To do so, make these four assumptions The original female gives birth to eight young on January 1. She produces another litter of eight rats every 40 days thereafter as long as she lives. Each female rat born on the island will produce her first litter of eight young 120 days after her birth She will produce a new litter of eight rats every 40 days thereafter Every litter has four males and four females The rats have no natural enemies on the island and plenty of food. This means that no rats die in the first year. How many rats will live on the island by the following January 1. including the original pair?

Explanation / Answer

If we look at the population every 40 days, there is the first birth (on Jan 1) and then 9 more births. There are 4 kinds of rats - newborns, rats 40 days old, rats 80 days old, and adults (breeders). We can make a list like this to show the populations. After the births on January 1: (8,0,0,2) -- 8:newborns, 0:40-day-olds, 0:80-day-olds, and 2:breeders. If the list looks like this: (w,x,y,z) -- w:newborns, x:40-day-olds, etc., then at the next period, there will be: (4z, w, x, y+z) rats, right? Each z makes 4 babies (2 parents make 8, so each makes 4), everybody advances to the next age group, except the 80-day olds become breeders, and the breeders remain breeders. Let's call Jan 1 the 0th breeding, and there are 9 more, each after 40 days: 0: (8,0,0,2) 1: (8,8,0,2) 2: (8,8,8,2) 3: (40,8,8,10) 4: (72,40,8,18) 5: (104,72,40,26) 6: (264,104,72,66) 7: (552,264,104,138) 8: (968,552,264,242) 9: (2024,968,552,506) The grand total is 2024 + 968 + 552 + 506 = 4050 rats are there on the island by the following january 1.

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