Mathematics Specialty - EXERCISE II (20 points)
The questions in Part I can be treated independently. In this exercise, $K$ and $a$ are strictly positive real constants.
Part I - Preliminary Studies
Consider the differential equation $\left( E _ { 1 } \right) : z ^ { \prime } ( t ) + z ( t ) = \frac { 1 } { K }$, where $z$ is a function defined and differentiable on $[ 0 ; + \infty [$. II-1- Give the general solution of $( E _ { 1 } )$ on the interval $[ 0 ; + \infty [$. Consider the function $f$ defined for every positive real $t$ by: $f ( t ) = \frac { 10 } { 1 + a e ^ { - t } }$. II-2- Complete the table of variations of $f$ on the interval $[ 0 ; + \infty [$. Specify the value of $f$ at $0$ and the limit of $f$ at $+ \infty$. II-3- Determine, as a function of $a$, the set of solutions of the equation $f ( t ) = 5$.
Part II - Evolution of a Marmot Population
Let $y _ { 0 }$ be a strictly positive real number. We study the evolution of a marmot population, which initially numbers $y _ { 0 }$ thousand individuals. We admit that the size of the population, expressed in thousands of individuals, after $t$ years (with $t \geq 0$) is a function $y$ differentiable on $[ 0 ; + \infty [$, solution of the differential equation: $$\left( E _ { 2 } \right) : y ^ { \prime } ( t ) = y ( t ) \left( 1 - \frac { y ( t ) } { K } \right)$$ The constant $K$ is called the carrying capacity of the environment, expressed in thousands of individuals. We admit that there exists a unique function $y$ solution of $\left( E _ { 2 } \right)$ that satisfies $y ( 0 ) = y _ { 0 }$. We admit that this function takes strictly positive values on $[ 0 ; + \infty [$. We set $z ( t ) = \frac { 1 } { y ( t ) }$ for every positive real $t$. II-4-a- Express $z ^ { \prime } ( t )$ as a function of $y ^ { \prime } ( t )$ and $y ( t )$. II-4-b- We wish to show that $z$ is a solution of $\left( E _ { 1 } \right)$ if, and only if, $y$ is a solution of $\left( E _ { 2 } \right)$. Complete:
- Line 1 using an expression involving $z ^ { \prime } ( t )$ and $z ( t )$;
- Line 2 and Line 3 using an expression involving $y ^ { \prime } ( t )$ and $y ( t )$.
II-5-a- Deduce the general solution of $( E _ { 2 } )$. II-5-b- We admit that the unique solution $y$ of $\left( E _ { 2 } \right)$ satisfying $y ( 0 ) = y _ { 0 }$ is written in the form $y ( t ) = \frac { K } { 1 + a e ^ { - t } }$. Express $a$ as a function of $y _ { 0 }$ and $K$. In a certain valley with carrying capacity $K = 10$, the marmots have disappeared. Scientists wish to reintroduce $y _ { 0 }$ thousand marmots, with $0 < y _ { 0 } < 10$. In the remainder of the exercise, we will take $K = 10$. II-6- Justify that the value of $a$ obtained in question II-5-b- is indeed strictly positive. II-7-a- Using the result from question II-3-, give the value of $a$ such that $y ( 5 ) = 5$. II-7-b- Deduce the exact value of $y _ { 0 }$ such that $y ( 5 ) = 5$. Justify your answer. II-7-c- The calculator gives $0.0669285092$ as the result of the calculation of the value of $y _ { 0 }$ from the previous question. What is the minimum number of marmots to reintroduce so that at least $5$ thousand marmots are present after $5$ years following their reintroduction?