Consider a species of elk that can be split into 4 age groupings: those aged 0-1 years, those aged 1-2 years, and those aged 2-3 years, and those aged 3-4 years. The population is observed once a year. Given that the Leslie matrix is equal to $$L = \left[\begin{array}{cccc} 0 &0.9 &1.5 &1\cr 0.9 &0 &0 &0\cr 0 &0.6 &0 &0\cr 0 &0 &0.2 &0 \end{array}\right]$$ and the initial population distribution is $42$ of the first age group, $19$ of the second age group, $11$ of the third group, and $4$ of the oldest age group, answer the following questions.

The initial population vector is $\mathbf{x}_0 =$ $\left[\Rule{0pt}{4.8em}{0pt}\right.$$\left]\Rule{0pt}{4.8em}{0pt}\right.$.

How many elk aged 1-2 years are there expected to be after 10 years?

How many elk aged 0-1 years are there expected to be after 20 years?

How many elk are there expected to be after 30 years?

Calculate the dominant eigenvalue of the Leslie matrix good to 3 decimal places. $\lambda_1\ =\ $

What is the long-term growth rate of this population of elk as a percent? growth rate = (The growth rate is the percentage of growth over/under 100%.)

Are the elk thriving, static, or going extinct? ? going extinct static thriving