The purpose of this part is to study a model of population evolution. In 2022, this population has 3000 individuals. We denote $P _ { n }$ the population size in thousands in the year $2022 + n$. Thus $P _ { 0 } = 3$. According to a model inspired by the Verhulst model, a Belgian mathematician of the XIX${}^{\mathrm{th}}$ century, we consider that, for all natural integer $n$ :

$$P _ { n + 1 } - P _ { n } = P _ { n } \left( 1 - b \times P _ { n } \right) , \text { where } b \text { is a strictly positive real number. }$$

The real number $b$ is a damping factor that allows us to account for the limited nature of the resources in the environment in which these individuals evolve.

\begin{enumerate}
  \item In this question $b = 0$.\\
a. Justify that the sequence $\left( P _ { n } \right)$ is a geometric sequence and specify its common ratio.\\
b. Determine the limit of $P _ { n }$.
  \item In this question $b = 0.2$.\\
a. For all natural integer $n$, we set $v _ { n } = 0.1 \times P _ { n }$.\\
Calculate $v _ { 0 }$ then show that, for all natural integer $n , v _ { n + 1 } = 2 v _ { n } \left( 1 - v _ { n } \right)$.\\
b. In this model, justify that the population will stabilize around a value that you will specify.
\end{enumerate}