Example 3

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One area of concern in the field of biology is an invasive species. This occurs when a new species, whether plant, animal, or other living organism, is introduced into a region. The new species does not have any natural predators and can start to grow out of control. This can impact the naturally occurring organisms in that region.

One of the best examples of an invasive species is the introduction of the domestic rabbit to Australia in 1859. Not thinking that letting a few rabbits into the wild could do much harm, the animals were released. Within a decade, millions of rabbits could be found in the same region.

Suppose two rabbits were bought by someone who did not know very much about rabbits. One was a female, the other was a male. One month later, the female had a litter of six babies, three of which were female. One month after that, all four females each had litters of six babies, half of which were female. This process continues on for five months before the owner starts wondering what is going on.

1. Write a sequence for the number of female rabbits born at the end of each month for five months.
   
   
   Solution

   \(\begin{array}{l}
    t_1 = {\rm{3}} \\
    t_2 = \left( {{\rm{1}} + {\rm{3}}} \right)\left( {\rm{3}} \right) = {\rm{12}} \\
    t_3 = \left( {{\rm{1}} + {\rm{3}} + {\rm{12}}} \right)\left( {\rm{3}} \right) = {\rm{48}} \\
    t_4 = \left( {{\rm{1}} + {\rm{3}} + {\rm{12}} + {\rm{48}}} \right)\left( {\rm{3}} \right) = {\rm{192}} \\
    t_5 = \left( {{\rm{1}} + {\rm{3}} + {\rm{12}} + {\rm{48}} + {\rm{192}}} \right)\left( {\rm{3}} \right) = {\rm{768}} \\
    \end{array}\)
   The sequence for the number of female rabbits born at the end of each month is \(3, 12, 48, 192, 768\).
2. Justify that this is a geometric sequence, and write the general term.
   
   
   Solution

   Determine the value for \(r \) for each pair of consecutive terms.
   

   \[r = \frac{{768}}{{192}} = \frac{{192}}{{48}} = \frac{{48}}{{12}} = \frac{{12}}{3} = 4 \]

   Because the ratio is constant, this is a geometric sequence.
   
   The general term for the geometric sequence is:
   
   \(\begin{array}{l}
    t_n = t_1 r^{n - 1}  \\
    t_n = 3\left( 4 \right)^{n - 1}  \\
    \end{array}\)
   
   where \(t_n \) is the number of female rabbits born at the end of the \(n^{th} \) month.
3. Ignoring the original female rabbit, how many female rabbits does the owner have after five months?
   
   
   Solution

   This question is asking for the total number of female rabbits, or the sum of the number of female rabbits, after five months.
   \(\begin{align}
    S_n &= \frac{{t_1 \left( {r^n - 1} \right)}}{{r - 1}} \\
    S_5 &= \frac{{3\left( {4^5 - 1} \right)}}{{4 - 1}} \\
    S_5 &= \frac{{3\thinspace 069}}{3} \\
    S_5 &= 1\thinspace 023 \\
    \end{align}\)
4. How long does it take for the female rabbit population to surpass 15 million?
   
   
   Solution

   This time, you are looking for the value of \(n\), given the value of \(S_n \).
   \(\begin{align}
    S_n &= \frac{{t_1 \left( {r^n - 1} \right)}}{{r - 1}} \\
    15\thinspace 000\thinspace 000 &= \frac{{3\left( {4^n - 1} \right)}}{{4 - 1}} \\
    15\thinspace 000\thinspace 000 &= \frac{{\cancel{3}\left( {4^n - 1} \right)}}{{\cancel{3}}} \\
    15\thinspace 000\thinspace 001 &= 4^n  \\
    \end{align}\)
   Use guess and check to find the value of \(n \) that will satisfy this equation.
   
   \( 4^{11} = 4\thinspace 194\thinspace 304 \)     too small
    \(4^{12} = 16\thinspace 777\thinspace 216\)    above the required value, but close
   
   It will take until nearly the end of 12 months for the population of female rabbits to exceed 15 million.
5. What did you assume if question c. and d.?
   
   
   Solution

   The assumption was that no female rabbits died and each one reproduced each month with a litter of exactly 3 females.