16.ps (Rosen 8.2.45) Suppose that each pair of a genetically engineered species

Question

16.ps (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 R where Rn is the number of pairs of rabbits on the island n months after one newborn pair is left on the island. Justify your answer. (b) What are the initial conditions for the recurrence relation in part (a)?

Explanation / Answer

Hi,
a. The first pair of rabbits will give birth to 2 new pairs at 1st month end, therefore from second month onwards there are 3 pairs, out of which the older pair produces 6 pairs at the second month end and from here on, each pair produces 6 pairs at the second month end, therefore the values follow as
R0=2, R1=6,R2=26
R1=2+(2*2)
R2=3(6)+4(2)
therefore
Rn=3*Rn-1 +4 * Rn-2 for all n>=2
b. the initial conditions are R0=2 and R1=6

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