Assume the carrying capacity of the Earth is 10 billion. Use the 1960 annual growth rate of 2.1% and population of 3 billion to predict the base growth rate and current growth rate with a logistic model. Assume a current world population of 7.8 billion. How does the predicted growth rate compare to the actual growth rate of about 1.1% per​ year?

To solve this problem, we can use the **logistic growth model**, which is often employed in population dynamics when considering a carrying capacity. The formula for the logistic growth rate is given by:

\[
\frac{dP}{dt} = rP \left( 1 - \frac{P}{K} \right)
\]

Where:
- \( P \) is the population at time \( t \),
- \( r \) is the intrinsic (base) growth rate,
- \( K \) is the carrying capacity,
- \( \frac{dP}{dt} \) is the rate of change of the population.

### Step 1: Determine the intrinsic growth rate \( r \) using the data from 1960.
In 1960, the population was 3 billion (\( P_0 = 3 \, \text{billion} \)) and the growth rate was 2.1% per year. The carrying capacity of the Earth is \( K = 10 \, \text{billion} \). So we can use the logistic growth formula at the time:

\[
\frac{dP}{dt} = 0.021P = rP_0 \left( 1 - \frac{P_0}{K} \right)
\]

Substitute the known values:

\[
0.021 \cdot 3 = r \cdot 3 \cdot \left( 1 - \frac{3}{10} \right)
\]

Simplify:

\[
0.063 = 3r \cdot \left( 1 - 0.3 \right)
\]
\[
0.063 = 3r \cdot 0.7
\]
\[
r = \frac{0.063}{2.1} = 0.03
\]

So, the intrinsic (base) growth rate \( r \) is **3%**.

### Step 2: Calculate the predicted current growth rate using the logistic model.
Now, we use the logistic growth model to predict the growth rate for the current population (\( P = 7.8 \, \text{billion} \)).

The logistic growth formula is:

\[
\frac{dP}{dt} = rP \left( 1 - \frac{P}{K} \right)
\]

Substitute \( r = 0.03 \), \( P = 7.8 \, \text{billion} \), and \( K = 10 \, \text{billion} \):

\[
\frac{dP}{dt} = 0.03 \cdot 7.8 \cdot \left( 1 - \frac{7.8}{10} \right)
\]

Simplify:

\[
\frac{dP}{dt} = 0.03 \cdot 7.8 \cdot \left( 1 - 0.78 \right)
\]
\[
\frac{dP}{dt} = 0.03 \cdot 7.8 \cdot 0.22
\]
\[
\frac{dP}{dt} = 0.05148 \, \text{billion per year}
\]

The predicted growth rate as a percentage of the current population is:

\[
\text{Growth rate} = \frac{0.05148}{7.8} \times 100 = 0.66\%
\]

### Step 3: Compare to the actual growth rate.
The predicted growth rate based on the logistic model is **0.66% per year**, while the actual current growth rate is about **1.1% per year**. 

Thus, the logistic model underestimates the current growth rate by a margin of **0.44%**.

### Summary:
- The base (intrinsic) growth rate is **3%**.
- The predicted current growth rate is **0.66%**.
- The actual current growth rate is **1.1%**, which is higher than the predicted rate.

### Relative Questions:
1. How would the model change if the carrying capacity were higher than 10 billion?
2. What impact would a lower base growth rate have on the current population predictions?
3. How would the growth model differ for populations with no environmental constraints?
4. Could immigration or emigration significantly alter the predictions made by the logistic model?
5. What happens to the population growth rate when the population approaches the carrying capacity?

**Tip:** Logistic growth models are useful for populations under resource limitations, but real-world factors like technology, migration, and policy changes can influence actual growth rates.

Assume the carrying capacity of the Earth is 10 billion. Use the 1960 annual growth rate of 2.1% and population of 3 billion to predict the base growth rate and current growth rate with a logistic model. Assume a current world population of 7.8 billion. How does the predicted growth rate compare to the actual growth rate of about 1.1% per year?

Explore how to use the logistic growth model to predict population growth. Starting with the 1960 population of 3 billion and a growth rate of 2.1%, this problem calculates the base growth rate and compares the predicted current growth rate for a population of 7.8 billion to the actual growth rate of 1.1%. Suitable for undergraduate-level studies.

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