How long would it take two rats to fill up Lake Superior with rats?

Background Story:  

It is a fairly common saying: breeding like rats. Or, multiplying like rats. One day after hearing this saying, I wondered, "How fast would two rats take to breed enough to fill up a swimming pool with rats?" I decided to expand this into the fermi problem:How long would two adult brown rats take to breed enough to fill up Lake Superior, if they had an unlimited supply of food and water?(I also assumed that there was only one breeding pair of rats at a time.) My reason for choosing brown rats is because they are the most widespread species of rats. 

Ask Yourself This:  

How long is a brown rat's life span?

How long does it take a brown rat to mature enough to breed?

How long is a brown rat's breeding cycle (how long between litters)

How large is an average brown rat litter?

What is the volume of an average brown rat?

What is the volume of Lake Superior?

Helpful Hints:  

a.  

Rat data

Data from http://animaldiversity.ummz.umich.edu/accounts/Rattus_norvegicus/

Gestation period: about 23 days

Males usually reach sexual maturity at 3-4 months

Females usually reach sexual maturity at 3-4 months

Around 18 hours after giving birth, the female mates again

Typical lifespan: 2-3 years (30 months)

Number of rats per litter: about 8

Data from:http://www.arkive.org/brown-rat/rattus-norvegicus/ 

Head/body length of a brown rat: 20-28 cm

Tail length:17-23 cm

Data from:http://www.epa.gov/greatlakes/factsheet.html 

Lake Superior info

Volume: About 23,000 cubic km 

b.  

Length between  brown rat litters: 23.75 days

23 days = 552 hrs   552 hrs + 18 hrs = 570 hrs

570 hrs/ 24 hrs= 23.75 days

Volume of a brown rat: 2646 cubic cm

I estimated based on a diagram that the ratio of height to body length is about 3.5:8, and that the height and width were about the same. Since the  body length is about 24 cm, and 24 cm is 8 x 3, I multiplied 3.5(height in the ratio of height to body length) by 3 to get 10.5 cm, which is the height of an average brown rat. Then, I multiplied length and height to get 252 square cm for the area. I multiplied that by the width to get 2646 cubic cm.

Construct a Formula:  

First, I’ll have to divide the volume of Lake Superior by the volume of each rat, to find out how many rats we need to fill it up. Next I’ll divide the rat’s life span(minus 3.5 months) by the length between litters, to find the # of times 2 rats can mate in their lifetimes. Then I’ll have to multiply that # by the # of rats per litter, to find how many babies those rats make in their lifetimes. Then I’ll divide the number of rats it takes to fill up Lake Superior by that number, to find how many rat life spans it will take to fill up Lake Superior. 

Since each rat lifespan is 2.5 years, I’ll multiply that # by 2.5. Along the way, I’ll also have to exchange the units.

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Sorry if this is a little faint.

Messy Math:  

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Again, sorry if this is a little faint. Also, I kind of had a hard time doing the last part in fractions.

Answer:  

If two brown rats had an unlimited amount of food and water, it would take them about 2.15x10^23 years to fill up Lake Superior with rats.