QUESTION ONE

The common penguin is endangered in some areas. One way to model the population is with a transition matrix for a Markov chain that can provide a system of expression that estimates how many birds will be born, die, and move on to the next stage of life $($chick $($c$),$ juvenile $($j$),$ single adult $(s),$ and mating adult $(m).$

Given the transition matrix, $T_1=\left[\begin{array}{cccc}0& 0& 0& 0\mathrm{.}145\\ 0\mathrm{.}6& 0\mathrm{.}375& 0& 0\\ 0& 0\mathrm{.}225& 0\mathrm{.}56& 0\mathrm{.}05\\ 0& 0& 0\mathrm{.}35& 0\mathrm{.}86\end{array}\right]$ and the initial penguin population, $P_0=\left[\begin{array}{c}c\\ j\\ 5\\ m\end{array}\right]$

a$)$ Generate a system of mathematical expressions that model the state of the penguin population next year.

$[4]$

b$)$ Interpret each mathematical expression in relation to the different stages of the penguins indicated in the case study.

$10$

c$)$ Given that the initial population $P_0=\left[\begin{array}{c}41\\ 61\\ 52\\ 64\end{array}\right]$ calculate the total number of penguins next year in each category rounding the answers correct to the nearest whole number.

$[6]$