A biologist is interested in the evolution of the population of an animal species on an island in the Pacific. At the beginning of 2020, this population had 600 individuals. We consider that the species will be threatened with extinction on this island if its population becomes less than or equal to 20 individuals. The biologist models the number of individuals by the sequence $(u_n)$ defined by:
$$\begin{cases} u_{0} & = 0.6 \\ u_{n+1} & = 0.75 u_{n} \left( 1 - 0.15 u_{n} \right) \end{cases}$$
where for every natural integer $n$, $u_{n}$ denotes the number of individuals, in thousands, at the beginning of the year $2020 + n$.
- Estimate, according to this model, the number of individuals present on the island at the beginning of 2021 and then at the beginning of 2022.
Let $f$ be the function defined on the interval $[ 0 ; 1 ]$ by
$$f ( x ) = 0.75 x ( 1 - 0.15 x )$$
\setcounter{enumi}{1} - Show that the function $f$ is increasing on the interval $[ 0 ; 1 ]$ and draw up its variation table.
- Solve in the interval $[ 0 ; 1 ]$ the equation $f ( x ) = x$.
We note for the rest of the exercise that, for every natural integer $n$, $u_{n+1} = f \left( u_{n} \right)$.
4. a. Prove by induction that for every natural integer $n$, $0 \leqslant u_{n+1} \leqslant u_{n} \leqslant 1$. b. Deduce that the sequence $\left( u_{n} \right)$ is convergent. c. Determine the limit $\ell$ of the sequence $(u_{n})$.
5. The biologist has the intuition that the species will sooner or later be threatened with extinction. a. Justify that, according to this model, the biologist is correct. b. The biologist has programmed in Python language the function menace() below:
\begin{verbatim} def menace() : u = 0.6 n = 0 while u>0.02 : u=0.75*u*(1-0.15*u) n = n+1 return n \end{verbatim}
Give the numerical value returned when the function menace() is called. Interpret this result in the context of the exercise.