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

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

Where:

• $r = 0.9$ (growth parameter),
• $p_0 = 0.3$ (initial population),
• $n$ denotes the step.

Given $p_0 = 0.3$, let's calculate $p_1, p_2, \ldots, p_{10}$ using the recurrence formula. We'll round each result to four decimal places.

Step-by-step Calculations:

1. Initial Population: $p_0 = 0.3$.
2. Calculate $p_1$: $p_1 = r \cdot p_0 \cdot (1 - p_0) = 0.9 \cdot 0.3 \cdot (1 - 0.3) = 0.9 \cdot 0.3 \cdot 0.7 = 0.189$
3. Use the same formula iteratively for $p_2$ through $p_{10}$.

Let me compute these values step by step. Parameters r = 0.9 # growth parameter p_0 = 0.3 # initial population

Iterative calculation of p_n for n = 1 to 10

populations = [p_0] # list to store p_n values for n in range(1, 11): p_next = r * populations[-1] * (1 - populations[-1]) populations.append(round(p_next, 4)) # round to 4 decimal places

populationsThe values of $p_1$ through $p_{10}$ are:

p_1 &= 0.189 \\ p_2 &= 0.138 \\ p_3 &= 0.1071 \\ p_4 &= 0.0861 \\ p_5 &= 0.0708 \\ p_6 &= 0.0592 \\ p_7 &= 0.0501 \\ p_8 &= 0.0428 \\ p_9 &= 0.0369 \\ p_{10} &= 0.0320 \end{aligned}$$ Let me know if you need help with part (b), further explanation, or any other details. ### Expanding Questions: 1. How does the logistic growth model compare to exponential growth? 2. What would happen to the population if the growth parameter $$r$$ were increased? 3. How does the value of $$p_n$$ change as $$n \to \infty$$? 4. What is the significance of $$1 - p_n$$ in the logistic growth formula? 5. Can you deduce a general behavior or trend in $$p_n$$ values from this problem? ### Tip: In a logistic model, the population tends to stabilize (approach a steady state) as $$n$$ increases, depending on the value of $$r$$.