From the image provided, the task involves analyzing a discrete-time dynamical system represented by the graph of the updating function $m_{n+1} = f(m_n)$. The graph intersects with the line $m_{n+1} = m_n$ at equilibrium points. Let us address the given statements by interpreting the graph:

1. (i) The solution starting at $m_0 = 0.5$ is increasing.

   • At $m_n = 0.5$, $f(m_n) > m_n$ (the blue curve is above the red line $m_{n+1} = m_n$).
   • Hence, the next iteration $m_{n+1} > m_n$, implying the solution is increasing.
   • This statement is true.

2. (ii) The solution starting at $m_0 = 1.25$ is increasing.

   • At $m_n = 1.25$, $f(m_n) > m_n$ (the blue curve is above the red line $m_{n+1} = m_n$).
   • Thus, $m_{n+1} > m_n$, and the solution is increasing.
   • This statement is true.

3. (iii) $m^* = 0$ is a stable equilibrium.

   • An equilibrium $m^*$ is stable if, for points near $m^*$, iterations of $f(m_n)$ tend to $m^*$.
   • At $m^* = 0$, the slope of $f(m_n)$ appears to satisfy $|f'(m^*)| < 1$, indicating stability.
   • This statement is true.

4. (iv) The non-zero equilibrium is unstable.

   • There is another equilibrium point (near $m^* \approx 2.5$) where $f(m_n) = m_n$.
   • Observing the slope of $f(m_n)$ at this point, it appears that $|f'(m^*)| > 1$, suggesting instability.
   • This statement is true.

Final Assessment:

All four statements (i), (ii), (iii), and (iv) are true.

Would you like a detailed explanation of how stability and dynamics are analyzed graphically or mathematically? Here are some related questions you may find useful:

1. How can the slope of the updating function at equilibrium points determine stability?
2. What are the mathematical conditions for equilibrium in a discrete dynamical system?
3. How can a cobweb diagram help visualize stability and convergence?
4. What happens if $f'(m^*) = 1$ or $f'(m^*) = -1$ at equilibrium?
5. Can we classify equilibrium points into types other than stable and unstable?

Tip: To analyze stability, always consider the magnitude of the derivative $|f'(m^*)|$ at equilibrium points. If $|f'(m^*)| < 1$, the equilibrium is stable; if $|f'(m^*)| > 1$, it is unstable.