b) A farmer decides to breed fish in a special pond built for the purpose breeding and selling them later. (5) Suppose he buys 1000 kg of a species of fish. He is told by experts that this species grows according to a logistic population model with $$ a=\frac{1}{10} $$. If the population doubles in 12 months, calculate the value of b. Then calculate the limiting population size in kilograms.

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To solve this problem, we'll use the **logistic population model**, which is appropriate for populations with limited resources. The logistic model is given by:

$$
\frac{dP}{dt} = aP\left(1 - \frac{P}{b}\right)
$$

Where:
- $$ P(t) $$ is the population at time $$ t $$,
- $$ a $$ is the intrinsic growth rate,
- $$ b $$ is the carrying capacity (limiting population size).

### Given:
- Initial population, $$ P_0 = 1000 $$ kg
- Growth rate, $$ a = \frac{1}{10} $$ per month
- Population doubles in 12 months, so $$ P(12) = 2000 $$ kg

### Step 1: Determine the Carrying Capacity ($$ b $$)

The logistic equation can be solved to find $$ P(t) $$:

$$
P(t) = \frac{b}{1 + C e^{-at}}
$$

Where $$ C $$ is a constant determined by the initial condition.

**Applying the Initial Condition $$ P(0) = 1000 $$ kg:**

$$
1000 = \frac{b}{1 + C}
$$
$$
1 + C = \frac{b}{1000}
$$
$$
C = \frac{b}{1000} - 1
$$

**Applying the Condition at $$ t = 12 $$ months, $$ P(12) = 2000 $$ kg:**

$$
2000 = \frac{b}{1 + \left(\frac{b}{1000} - 1\right)e^{-a \cdot 12}}
$$

Plugging in the values:

$$
2000 = \frac{b}{1 + \left(\frac{b}{1000} - 1\right)e^{-1.2}}
$$

Calculating $$ e^{-1.2} \approx 0.301194 $$:

$$
2000 = \frac{b}{1 + \left(\frac{b}{1000} - 1\right) \times 0.301194}
$$

Solving for $$ b $$:

$$
2000 \left[1 + \left(\frac{b}{1000} - 1\right) \times 0.301194\right] = b
$$
$$
2000 \times 0.698806 + 2000 \times 0.301194 \times \frac{b}{1000} = b
$$
$$
1397.612 + 0.602388b = b
$$
$$
1397.612 = b - 0.602388b
$$
$$
1397.612 = 0.397612b
$$
$$
b \approx \frac{1397.612}{0.397612} \approx 3515 \text{ kg}
$$

### Step 2: Determine the Limiting Population Size

The **limiting population size** is the carrying capacity $$ b $$ of the environment. From the calculation above:

$$
\text{Limiting Population Size} = b \approx 3515 \text{ kg}
$$

### **Final Answer:**
- **Value of $$ b $$:** Approximately **3515 kg**
- **Limiting population size:** **3515 kg**
The carrying capacity $$ b $$ is approximately 3515 kg, and the limiting population size is 3515 kg.

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