Suppose that each pair of a genetically engineered species of rabbits left on an

Question

Suppose that each pair of a genetically engineered species of rabbits left on an island produces two new pairs of rabbits at the age of 1 month and six new pairs of rabbits at the age of 2 months and every month afterward. None of the rabbits ever die or leave the island.

Find a recurrence relation and initial conditions for the recurrence relation.

6 points (Rosen 8.2.45) Suppose that each pair of a genetically engineered species of rabbits left on an island produces two new pairs of rabbits at the age of 1 month and six new pairs of rabbits at the age of 2 months and every month afterward. None of the rabbits ever die or leave the island. (a) Find a recurrence relation for the number of pairs of rabbits on the island n months after one newborn pair is left on the island. Justify your answer. (b) at are the initial conditions for the recurrence relation in part (a)?

Explanation / Answer

Given: each pair produces 2 new pairs at the age of 1 month.
                    and 6 new pairs from next month on.
So, if a new born pair is left on the island,
at the end of 1 month: the pair produces, 2 new pairs. So, 1 + 0*6 + (1-0)*2 = total 3 pairs are there.
at the end of 2 month: pair 1 produces, 6 pairs, and the other 2 pairs produces 2 paris each.
                   So, total pairs is: 3 + 1*6 + (3-1)*2. = 13.
at the end of 3 month: 13 + 3*6 + (13-3)*2 = 51.
at the end of 4 month: 51 + 13*6 + (51-13)*2 = 205.
...
And this recurrence continues...                     
P(n) = P(n-1) + P(n-2)*6 + (P(n-1) - P(n-2)) * 2
   = P(n-1) + 6P(n-2) + 2P(n-1) -2P(n-2)
   = 3P(n-1) + 4P(n-2).
And the initial condition is: P(0) = 1.
and P(1) = 3.

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