Only 5000 humans can live on Saturn. Only 2000 humans live on Saturn right now. If the human population grows exponentially, with an intrinsic rate of increase of humans of 0.0168/year, how many people will be added to the population in the next year? (Don’t round to a whole number)
The correct answer and explanation is:
To calculate how many people will be added to the population in the next year under logistic growth, we use the logistic growth model, since there’s a maximum population size (carrying capacity).
Formula for Logistic Growth (Population Increase):
ΔP=r⋅P⋅(1−PK)\Delta P = r \cdot P \cdot \left(1 – \frac{P}{K}\right)
Where:
- ΔP\Delta P = number of people added in the next year
- r=0.0168r = 0.0168 per year (intrinsic rate of increase)
- P=2000P = 2000 (current population)
- K=5000K = 5000 (carrying capacity)
Plug in the values:
ΔP=0.0168⋅2000⋅(1−20005000)\Delta P = 0.0168 \cdot 2000 \cdot \left(1 – \frac{2000}{5000}\right) =0.0168⋅2000⋅(1−0.4)= 0.0168 \cdot 2000 \cdot \left(1 – 0.4\right) =0.0168⋅2000⋅0.6= 0.0168 \cdot 2000 \cdot 0.6 =0.0168⋅1200= 0.0168 \cdot 1200 =20.16= 20.16
✅ Correct Answer: 20.16 people will be added in the next year.
🔍 Explanation (300 words):
In population ecology, exponential growth happens when a population increases by a constant proportion over time. However, in real-life situations like human habitation on a planet with limited resources—such as Saturn in this case—growth follows a logistic pattern due to environmental limits. This upper limit is called the carrying capacity (denoted KK), which is the maximum population the environment can support indefinitely.
The intrinsic rate of increase rr represents the growth potential of the population under ideal conditions. But because Saturn can only support up to 5,000 humans, the actual growth is tempered as the population approaches this limit.
We apply the logistic growth model: ΔP=r⋅P⋅(1−PK)\Delta P = r \cdot P \cdot \left(1 – \frac{P}{K}\right)
This formula accounts for how close the population is to the carrying capacity. The term (1−PK)\left(1 – \frac{P}{K}\right) is known as the environmental resistance—it reduces the growth rate as the population nears the limit.
In this problem:
- The current population is 2,000.
- The carrying capacity is 5,000.
- The growth rate is 0.0168/year.
The calculation yields an increase of 20.16 people in the next year, indicating a moderate growth rate that is still under the influence of environmental limits. As the population gets closer to 5,000, the growth rate will slow even further.
This model helps scientists and planners estimate population changes and prepare for future needs in constrained environments like space colonies.