The logistic model is used to predict population growth by considering the limits imposed by a carrying capacity. The logistic growth equation is given by:

$\frac{dP}{dt} = r P \left( 1 - \frac{P}{K} \right)$

Where:

• $P$ is the population at time $t$,
• $r$ is the base growth rate (intrinsic growth rate when population is much smaller than the carrying capacity $K$),
• $K$ is the carrying capacity of the environment,
• $\frac{dP}{dt}$ is the rate of change of the population.

Step 1: Solve for the base growth rate $r$

Given:

• Carrying capacity $K = 18$ billion,
• Initial population $P_0 = 3$ billion in 1960,
• Growth rate in 1960 $r_0 = 2.1\% = 0.021$.

We can substitute these values into the logistic growth equation and solve for the base growth rate $r$ using the fact that $P = 3$ billion and the growth rate $r_0 = 0.021$ in 1960:

$r_0 = r \left( 1 - \frac{P_0}{K} \right)$

Substituting the values:

$0.021 = r \left( 1 - \frac{3}{18} \right)$

$0.021 = r \left( 1 - 0.1667 \right)$

$0.021 = r \times 0.8333$

Solving for $r$:

$r = \frac{0.021}{0.8333} \approx 0.0252$

So, the base growth rate $r$ is approximately 0.0252 or 2.52%.

Step 2: Predict the current growth rate

Now, we can predict the current growth rate using the current population $P = 7.8$ billion:

$r_{\text{current}} = r \left( 1 - \frac{P}{K} \right)$

Substitute the values $r = 0.0252$, $P = 7.8$ billion, and $K = 18$ billion:

$r_{\text{current}} = 0.0252 \left( 1 - \frac{7.8}{18} \right)$

$r_{\text{current}} = 0.0252 \left( 1 - 0.4333 \right)$

$r_{\text{current}} = 0.0252 \times 0.5667$

$r_{\text{current}} \approx 0.0143 \text{ or } 1.43\%$

Step 3: Compare with actual growth rate

The predicted current growth rate using the logistic model is 1.43%. The actual current growth rate is approximately 1.1%. The predicted rate is slightly higher than the actual rate, but they are close, indicating that the logistic model provides a reasonable approximation, though not perfect. The difference might be due to additional factors not captured by the model, such as economic, social, and political influences that could slow down growth more than the simple logistic model predicts.

Summary:

• Base growth rate $r$: 2.52%
• Predicted current growth rate: 1.43%
• Actual current growth rate: 1.1%

Would you like further details or have any questions?

Expanding questions:

1. How does a logistic growth model differ from an exponential growth model in population predictions?
2. What other factors could explain the difference between the predicted and actual growth rates?
3. How would a change in the carrying capacity of the Earth affect the growth rate prediction?
4. What assumptions are made in the logistic growth model, and how do they impact its accuracy?
5. How would the model's predictions change if the initial population in 1960 was different?

Tip:

The logistic model assumes a constant carrying capacity and does not account for technological advances or policies that might affect growth rates, so its predictions may become less accurate over time.