Can a pair of rats multiply to 200 within a year?

Question

A story on the BBC News website makes the following claim:

Brown rats are one of the planet's most serious mammalian pests, ruining crops and harbouring disease. Expert jumpers, climbers and swimmers, a single pair can multiply to 200 within a year.

Can rats multiply this quickly?

Answer 1

According to this site rats:

• reach sexual maturity at 4 months (for females, who are the limiting factor here)
• can give birth seven times a year
• have litters of about 8

Starting with two sexually mature rats (and assuming averages of all the above throughout), the original pair will produce 7 litters of 8 rats in a year - 56 rats. The first litter in the second generation become fertile at 4 months after birth (5 months from the start of the year), and so the four females in that litter will have around 7/12*7 = 4 litters each in the year - 128 rats. 128 + 56 = 184. That's so close to the 200 that we don't need to worry about the remaining 6 litters or further generations. Rats are clearly capable of becoming at least 200 within a year.

Answer 2

Using the statistics cited in DJClayworth's answer, this Python program calculates that a male/female pair of mature rats could grow (under ideal conditions) to a population of about 250 in one year.

import matplotlib.pyplot as plt

maturity_age = 120     # days
litter_frequency = 52  # 7 litters per year implies a new litter is born every 52 (=365/7) days
# starting population consists of a mature male and female pair
rats = [{'gender':'male', 'age':litter_frequency*2}, 
        {'gender':'female', 'age':litter_frequency*2}]
pop = []
for day in range(365):
    new_rats = []
    for i, rat in enumerate(rats):
        if ((rat['age'] >= maturity_age) 
            and (rat['gender'] == 'female') 
            and (rat['age'] % litter_frequency == 0)):
            # new litter consists of 4 males, 4 females
            new_rats.extend([
                {'gender': gender, 'age': 0} for gender in ['male', 'female'] 
                for i in range(4)])
            print('Day {}: rat #{} ({} days old) gives birth to a new litter. (Pop size = {})'.format(day, i, rat['age'], len(rats)+len(new_rats)))
        rat['age'] += 1
    rats.extend(new_rats)
    pop.append(len(rats))

plt.plot(pop)
plt.xlabel('days')
plt.ylabel('population')
plt.show()

yields

Day 52: rat #1 (156 days old) gives birth to a new litter. (Pop size = 10)
Day 104: rat #1 (208 days old) gives birth to a new litter. (Pop size = 18)
Day 156: rat #1 (260 days old) gives birth to a new litter. (Pop size = 26)
Day 208: rat #1 (312 days old) gives birth to a new litter. (Pop size = 34)
Day 209: rat #6 (156 days old) gives birth to a new litter. (Pop size = 42)
Day 209: rat #7 (156 days old) gives birth to a new litter. (Pop size = 50)
Day 209: rat #8 (156 days old) gives birth to a new litter. (Pop size = 58)
Day 209: rat #9 (156 days old) gives birth to a new litter. (Pop size = 66)
Day 260: rat #1 (364 days old) gives birth to a new litter. (Pop size = 74)
Day 261: rat #6 (208 days old) gives birth to a new litter. (Pop size = 82)
Day 261: rat #7 (208 days old) gives birth to a new litter. (Pop size = 90)
Day 261: rat #8 (208 days old) gives birth to a new litter. (Pop size = 98)
Day 261: rat #9 (208 days old) gives birth to a new litter. (Pop size = 106)
Day 261: rat #14 (156 days old) gives birth to a new litter. (Pop size = 114)
Day 261: rat #15 (156 days old) gives birth to a new litter. (Pop size = 122)
Day 261: rat #16 (156 days old) gives birth to a new litter. (Pop size = 130)
Day 261: rat #17 (156 days old) gives birth to a new litter. (Pop size = 138)
Day 312: rat #1 (416 days old) gives birth to a new litter. (Pop size = 146)
Day 313: rat #6 (260 days old) gives birth to a new litter. (Pop size = 154)
Day 313: rat #7 (260 days old) gives birth to a new litter. (Pop size = 162)
Day 313: rat #8 (260 days old) gives birth to a new litter. (Pop size = 170)
Day 313: rat #9 (260 days old) gives birth to a new litter. (Pop size = 178)
Day 313: rat #14 (208 days old) gives birth to a new litter. (Pop size = 186)
Day 313: rat #15 (208 days old) gives birth to a new litter. (Pop size = 194)
Day 313: rat #16 (208 days old) gives birth to a new litter. (Pop size = 202)
Day 313: rat #17 (208 days old) gives birth to a new litter. (Pop size = 210)
Day 313: rat #22 (156 days old) gives birth to a new litter. (Pop size = 218)
Day 313: rat #23 (156 days old) gives birth to a new litter. (Pop size = 226)
Day 313: rat #24 (156 days old) gives birth to a new litter. (Pop size = 234)
Day 313: rat #25 (156 days old) gives birth to a new litter. (Pop size = 242)
Day 364: rat #1 (468 days old) gives birth to a new litter. (Pop size = 250)

To run the script on a machine (with Python and matplotlib installed), save the code to a file called script.py, then run from the command-line:

python script.py

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Note that DJClayworth's answer estimates that the second generation of rats give birth to 4 litters. Assuming the first litter is born on day 52, the simulation above shows females from the second generation may only have 3 litters each. So the accounting is a bit different, though the conclusion (that a single pair can give rise to >200 rats) is the same.