\section*{Exercise 4 (5 points)}
\section*{Candidates who followed the specialization course}

We denote $u_n$ as the number of voles and $v_n$ as the number of foxes on July $1^{\text{st}}$ of the year $2012 + n$.

\section*{Part A - A simple model}
We model the evolution of populations using the following relations:
$$\left\{\begin{array}{l} u_{n+1} = 1{,}1\, u_n - 2000\, v_n \\ v_{n+1} = 2 \times 10^{-5}\, u_n + 0{,}6\, v_n \end{array}\right. \quad \text{for all integers } n \geqslant 0, \text{ with } u_0 = 2000000 \text{ and } v_0 = 120.$$

\begin{enumerate}
  \item a. We consider the column matrix $U_n = \binom{u_n}{v_n}$ for all integers $n \geqslant 0$.

Determine the matrix $A$ such that $U_{n+1} = A \times U_n$ for all integers $n$ and give the matrix $U_0$.\\
b. Calculate the number of voles and foxes estimated using this model on July $1^{\text{st}}$ 2018.
  \item Let the matrices $P = \left(\begin{array}{cc} 20000 & 5000 \\ 1 & 1 \end{array}\right)$, $D = \left(\begin{array}{cc} 1 & 0 \\ 0 & 0{,}7 \end{array}\right)$ and $P^{-1} = \dfrac{1}{15000}\left(\begin{array}{cc} 1 & -5000 \\ -1 & 20000 \end{array}\right)$.

We admit that $P^{-1}$ is the inverse matrix of matrix $P$ and that $A = P \times D \times P^{-1}$.\\
a. Show that for all natural integers $n$, $U_n = P \times D^n \times P^{-1} \times U_0$.\\
b. Give without justification the expression of matrix $D^n$ as a function of $n$.\\
c. We admit that, for all natural integers $n$:
$$\left\{\begin{array}{lcl} u_n & = & \dfrac{2{,}8 \times 10^7 + 2 \times 10^6 \times 0{,}7^n}{15} \\[6pt] v_n & = & \dfrac{1400 + 400 \times 0{,}7^n}{15} \end{array}\right.$$
Describe the evolution of the two populations.
\end{enumerate}

\section*{Part B - A model more in line with reality}
We construct another model using the following relations:
$$\left\{\begin{array}{l} u_{n+1} = 1{,}1\, u_n - 0{,}001\, u_n \times v_n \\ v_{n+1} = 2 \times 10^{-7}\, u_n \times v_n + 0{,}6\, v_n \end{array}\right. \quad \text{for all integers } n \geqslant 0, \text{ with } u_0 = 2000000 \text{ and } v_0 = 120.$$

\begin{enumerate}
  \item What formulas must be written in cells B4 and C4 and copied downwards to fill columns B and C?
  \item With the second model, from what year do we observe the phenomenon described (decrease in foxes and increase in voles)?
\end{enumerate}

\section*{Part C}
In this part we use the model from Part B.\\
Is it possible to give $u_0$ and $v_0$ values such that the two populations remain stable from one year to the next, that is, such that for all natural integers $n$ we have $u_{n+1} = u_n$ and $v_{n+1} = v_n$? (We then speak of a stable state.)