Growth of Rat Populations

[yaas]

Two rats, one make and one female, scampered on board a ship that was anchored at a local dock. The ship set sail across the ocean. When it anchored at a deserted island in late December, the two rats abandoned the ship to make their home on the island.

* The number of young produced in every liters is six, and three of those six are females.
* the original female gives birth to six young on January 1 and produces another little of six every 40 days thereafter as long as she lives.
*Each female born on the island will produce her first litter 120 days after her birth and then produce a new litter every 40 days thereafter.
* The rats are on the island with no enemies and plenty of food, so no rats will dies in this first year.

Question : What will be the total number of rats by the following January 1, including the original pair?

* Thanks for the Help. And please Help with the equation and clear explanations. *

 

[Unknown008]

Two rats, one make and one female, scampered on board a ship that was anchored at a local dock. The ship set sail across the ocean. When it anchored at a deserted island in late December, the two rats abandoned the ship to make their home on the island.

* The number of young produced in every liters is six, and three of those six are females.
* the original female gives birth to six young on January 1 and produces another little of six every 40 days thereafter as long as she lives.
*Each female born on the island will produce her first litter 120 days after her birth and then produce a new litter every 40 days thereafter.
* The rats are on the island with no enemies and plenty of food, so no rats will dies in this first year.

Question : What will be the total number of rats by the following January 1, including the original pair?

* Thanks for the Help. And please Help with the equation and clear explanations. *

Day 0 - 2 arrive on island
Day 1 - 6 are born
Day 40 - 6 are born
Day 80 - 6 are born
Day 120 - 24 are born. 6 are born from the initial parents and each 6 born on day 1 give birth to 6 more each, for a total of 6+(3x6) = 24
Day 160 - 84 are born. 6 are born from the initial parents, and each 12 born on days 1 and 40 give birth to 6 more each. So we get 6+(6x6) = 42

I think it's easier to do it step by step.

So, the next amount of rats born on the next 40 days is 3n (where n is the number of rats born that are mature enough to liter. 3 because you need 2 rats to give 1 liter. Hence if there are 6 rats, they will make 3 pairs which make (6x3) = 18 rats). And we get:

Day:  0   1  40  80 120  160  200  240  280  320  360
Born: 2   6   6   6  24   42   60  132  258  438  834

To get 438, for instance you do:
(2+6+6+6+24+42+60)x3

I hope it's clear enough.

[Note that this gives the number born each time, not the total number of rats!]