Fox Studios is a pest rabbit control company commissioned to assist with a wild rabbit population growth problem on a small island. 1000 rabbits were introduced onto an island, but it was soon realised that the population growth of the rabbits would cause environmental problems on the island. Fox Studios was commissioned to assist with this problem by introducing foxes who help contain the rabbit numbers. The rabbit population growth rate is 10% each month, (This is the net of the natural birth rate and death rate). Each fox eats 10 rabbits per month. (The foxes are also fed by the company as part of their cost).

(a) Show that +1 = 1.1 10 where is the rabbit population at month t and F is the number of foxes employed, is a suitable difference equation to use.

(b) Verify that = (1000 100 ) 1.1 + 100 is the solution to the difference equation.

(c) Fox Studios is faced with the following optimisation problem:

i. The contract is for 5 years (60 months)

ii. Each fox costs $300 000 (special training, transportation, feeding and medical expenses over the 5 years)

iii. The company is liable to pay a fine of $1 per rabbit for any excess over 15 000 in every month during the 5 year life. Solve for the number of foxes that should be sent to minimise the cost to the company.

Hints: i. There are 11 cases to consider F=0, 1, 2, ... 10

ii. is an increasing function in t with a calculable time of first breach say T*.

iii. An EXCEL spread sheet may be used, but theoretical calculations should be performed.