Exercise 5 (5 points) — Candidates who have NOT followed the specialization course
A biologist wishes to study the evolution of the population of an animal species in a reserve. This population is estimated at 12000 individuals in 2016. The constraints of the natural environment mean that the population cannot exceed 60000 individuals.
Part A: a first model
In a first approach, the biologist estimates that the population grows by $5\%$ per year. The annual evolution of the population is thus modelled by a sequence $(v_n)$ where $v_n$ represents the number of individuals, expressed in thousands, in $2016 + n$. We thus have $v_0 = 12$.
- Determine the nature of the sequence $(v_n)$ and give the expression of $v_n$ as a function of $n$.
- Does this model meet the constraints of the natural environment?
Part B: a second model
The biologist then models the annual evolution of the population by a sequence $(u_n)$ defined by $u_0 = 12$ and, for every natural integer $n$, $$u_{n+1} = -\frac{1.1}{605}u_n^2 + 1.1\, u_n.$$
- We consider the function $g$ defined on $\mathbb{R}$ by $$g(x) = -\frac{1.1}{605}x^2 + 1.1\, x$$ a. Justify that $g$ is increasing on $[0;60]$. b. Solve in $\mathbb{R}$ the equation $g(x) = x$.
- We note that $u_{n+1} = g(u_n)$. a. Calculate the value rounded to $10^{-3}$ of $u_1$. Interpret. b. Prove by induction that, for every natural integer $n$, $0 \leqslant u_n \leqslant 55$. c. Prove that the sequence $(u_n)$ is increasing. d. Deduce the convergence of the sequence $(u_n)$. e. It is admitted that the limit $\ell$ of the sequence $(u_n)$ satisfies $g(\ell) = \ell$. Deduce its value and interpret it in the context of the exercise.
- The biologist wishes to determine the number of years after which the population will exceed 50000 individuals with this second model. He uses the following algorithm:
| \multirow{2}{*}{Variables} | $n$ a natural integer |
| \cline{2-2} | $u$ a real number |
| Processing | $n$ takes the value 0 |
| $u$ takes the value 12 |
| While $\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots$ |
| $\quad u$ takes the value $\ldots\ldots\ldots\ldots\ldots$ |
| $n$ takes the value $\ldots\ldots\ldots\ldots\ldots$ | |
| End While |
| Output | Display $\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots$ |
Copy and complete this algorithm so that it displays as output the smallest integer $r$ such that $u_r \geqslant 50$.