Let's analyze and solve the given problem step-by-step:

The logistic discrete-time dynamical system is defined as:

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

Part (a): Statements

1. The per capita production of this system is a linear function:

   • The per capita production refers to the growth rate per individual, which is proportional to $1 - x_t$. This term is linear in $x_t$, so this statement is true.

2. The derivative of the updating function is a decreasing function:

   • The updating function is $f(x_t) = \frac{39}{10} x_t (1 - x_t)$. Its derivative is: $f'(x_t) = \frac{39}{10} (1 - 2x_t)$
     • $f'(x_t)$ is a linear function of $x_t$, not necessarily a decreasing function for all $x_t$. Therefore, this statement is false.

Answer: 1, 2

Part (b): Non-zero equilibrium

To find the equilibrium, solve: $x_{t+1} = x_t = x \quad \text{(equilibrium point)}$ $x = \frac{39}{10} x (1 - x)$ Simplify: $x \left(1 - \frac{39}{10} (1 - x)\right) = 0$ Either $x = 0$ or: $1 - \frac{39}{10} + \frac{39}{10}x = 0$ $\frac{39}{10}x = \frac{29}{10}$ $x = \frac{29}{39}$

Non-zero equilibrium: $\mathbf{x = \frac{29}{39}}$.

Part (c): Stability

To assess stability, calculate $f'(x)$ at the equilibria: $f'(x) = \frac{39}{10} (1 - 2x)$ At $x = 0$: $f'(0) = \frac{39}{10} (1 - 2(0)) = \frac{39}{10} > 1$ $x = 0$ is unstable.

At $x = \frac{29}{39}$: $f'\left(\frac{29}{39}\right) = \frac{39}{10} \left(1 - 2 \cdot \frac{29}{39}\right)$ $f'\left(\frac{29}{39}\right) = \frac{39}{10} \left(1 - \frac{58}{39}\right)$ $f'\left(\frac{29}{39}\right) = \frac{39}{10} \cdot \left(-\frac{19}{39}\right)$ $f'\left(\frac{29}{39}\right) = -\frac{741}{390} \approx -1.9$ $-1.9$ lies outside the range $(-1, 1)$, so $x = \frac{29}{39}$ is also unstable.

Let me know if you would like further clarification or steps! Here are some related questions to expand your understanding:

1. How does the updating function change if the coefficient $\frac{39}{10}$ is modified?
2. What happens to stability when the derivative $f'(x)$ is adjusted to fall within the interval $(-1, 1)$?
3. How is the equilibrium $x = 0$ interpreted biologically in logistic growth models?
4. Can there be a scenario where both equilibria are stable in logistic systems?
5. What are the implications of stability analysis in predicting long-term population behavior?

Tip: To verify stability, always check the magnitude of the derivative of the updating function at equilibrium points. Stability occurs when $|f'(x)| < 1$.