To solve this logistic growth problem, we use the logistic growth model equation:

$p_{n+1} = r \cdot p_n \cdot (1 - p_n)$

where:

• $r$ is the growth rate ($r = 0.8$),
• $p_0$ is the initial population ($p_0 = 0.7$),
• $p_n$ is the population at generation $n$,
• $p_{n+1}$ is the population at the next generation.

We will compute $p_1$ through $p_{10}$ step by step using this recurrence relation.

Step-by-Step Calculation

Given $p_0 = 0.7$:

1. $p_1 = 0.8 \cdot 0.7 \cdot (1 - 0.7)$
2. $p_2 = 0.8 \cdot p_1 \cdot (1 - p_1)$
3. Repeat until $p_{10}$.

Let’s calculate numerically: Logistic growth model parameters r = 0.8 # Growth rate p0 = 0.7 # Initial population n_steps = 10 # Number of iterations to calculate

List to store population values for p_0 to p_10

population_values = [p0]

Iteratively compute population values

for i in range(n_steps): pn = population_values[-1] pn_next = r * pn * (1 - pn) # Logistic growth formula population_values.append(pn_next)

population_valuesHere are the computed values of $p_n$ from $p_0$ to $p_{10}$:

p_0 & = 0.7 \\ p_1 & = 0.168 \\ p_2 & \approx 0.1118 \\ p_3 & \approx 0.07945 \\ p_4 & \approx 0.05851 \\ p_5 & \approx 0.04407 \\ p_6 & \approx 0.03370 \\ p_7 & \approx 0.02605 \\ p_8 & \approx 0.02030 \\ p_9 & \approx 0.01591 \\ p_{10} & \approx 0.01253 \\ \end{aligned}$$ The population quickly decreases with each step under this logistic growth model due to the high initial value of $$p_0 = 0.7$$, which is close to the equilibrium $$p = 0$$. --- Would you like further explanation of these results, or assistance with part (b)? Here are some related questions for you to consider: 1. What is the long-term behavior of the logistic growth model for this system? 2. How does changing the growth rate $$r$$ affect the population trend? 3. What happens if the initial population $$p_0$$ is below 0.5? 4. Can you plot the values of $$p_n$$ over time to visualize the decay? 5. How does logistic growth compare to exponential growth in this case? **Tip:** Logistic growth models are particularly useful for modeling populations constrained by resources or competition!