To address Problem #9, we analyze the given logistic discrete-time dynamical system:

$x_{t+1} = \frac{39}{10} x_t (1 - x_t)$

(a) Analysis of the statements:

(i) "The per capita production of this system is a linear function."

The per capita production can be represented as:

$\text{Per capita production} = \frac{x_{t+1}}{x_t} = \frac{39}{10}(1 - x_t)$

This expression is linear in $x_t$ because $1 - x_t$ is a linear function of $x_t$. Hence, the statement is true.

(ii) "The derivative of the updating function is a decreasing function."

The updating function is given by:

$f(x) = \frac{39}{10}x(1 - x)$

To determine if the derivative is a decreasing function, we first compute the derivative:

$f'(x) = \frac{39}{10}(1 - 2x)$

This derivative is a linear function of $x$ with a negative slope ($-\frac{78}{10}$). Since $f'(x)$ decreases as $x$ increases, the statement is true.

Would you like a more detailed explanation or further clarifications?

Questions to expand on:

1. What are the equilibrium points of this logistic map, and are they stable?
2. How does the behavior of the system change for different initial values of $x_t$?
3. Can we explore the bifurcation behavior of the given logistic map?
4. How does the coefficient $\frac{39}{10}$ influence the dynamics of the system?
5. What would the updating function and its behavior look like graphically?

Tip: When working with logistic maps, analyzing equilibrium points and stability provides deep insight into the system's behavior.