Answer Happy

Difference equations are often used to model population dynamics for species that reproduce only at particular time intervals. A scientist studied a beetle which in the laboratory live for a maximum of 3 years. • Only half the beetles born will survive their first year. • Only one third of the beetles that survive the first year, survive their second year. • Every beetle that enters the third year produces exactly 6 beetles as they die (at the end of the third year). Consider the following matrix: A= 0 1/2 0 6 0 0 1/30
(a) Explain why the matrix A is a reasonable model for the beetle growth described in this problem. That is, if Ek is a vector with three components representing the number of newly born beetles (x{k)), 1 year old beetles (x%), and 2 year old beetles(x*)), in year k then: (k) Tk+1 = Ačk. (b) If we start off with 6 newborn beetles, how many beetles will be in each of the next 3 years? (Do you see a pattern, can you predict how many there will be for all future years?) Hint: Diagonalization, while possible, is not the best way to solve this problem. Just go ahead and plug in solutions. If you do feel strongly committed to diagonalization, you might find it helpful to remember that: 13 =1 = I1 = 1, 12,3 = (1/2)(-1 V-3).