Consider a population of insects that consists of juveniles (1 year and under) and adults. Each year,
20%
of juveniles reproduce and
70%
of adults reproduce.
70%
of juveniles survive to adulthood the next year and
20%
of adults survive the year. The transition matrix for this population is then given by
A=[ .2
.7
​
.7
.2
​
]
. (a) Find the eigenvalues of
A
. What is the dominant eigenvalue
λ 1
​
(largest absolute value)? (b) Find an eigenvector corresponding to the dominant eigenvalue. (c) Divide your eigenvector by the sum of its entries to find an eigenvector
v 1
​
whose entries sum to one that gives the long term probability distribution. (d) Describe what will happen to the insect population long term based on your longterm growth rate
λ 1
​
and corresponding eigenvector
v 1
​