Population growth models help biologists predict how the number of individuals in a population changes over time. These models are used to study bacteria in a lab, invasive species, endangered animals, and human impacts on ecosystems. Exponential growth describes a population increasing rapidly when resources are abundant.
Logistic growth adds the real-world limit that environments can only support a certain population size.
Key Facts
- Exponential growth model: dN/dt = rN.
- Logistic growth model: dN/dt = rN(1 - N/K).
- N is population size, t is time, r is the per capita growth rate, and K is carrying capacity.
- Exponential growth produces a J-shaped curve when r is positive and resources are unlimited.
- Logistic growth produces an S-shaped curve because growth slows as N approaches K.
- In the logistic model, population growth is fastest when N = K/2.
Vocabulary
- Population
- A population is a group of individuals of the same species living in the same area at the same time.
- Exponential Growth
- Exponential growth is population increase at a rate proportional to the current population size.
- Logistic Growth
- Logistic growth is population increase that slows as the population approaches the environment's carrying capacity.
- Carrying Capacity
- Carrying capacity is the maximum population size that an environment can support over time with available resources.
- Limiting Factor
- A limiting factor is any resource or condition that restricts population growth, such as food, space, disease, or predators.
Common Mistakes to Avoid
- Treating exponential growth as realistic forever is wrong because real populations eventually face limits such as food, space, waste buildup, and disease.
- Confusing r with total growth is wrong because r is the per capita growth rate, while dN/dt is the total change in population size per unit time.
- Assuming carrying capacity is always fixed is wrong because K can change when resources, climate, habitat quality, or human impacts change.
- Thinking logistic growth stops only when every individual dies or reproduces equally is wrong because growth slows due to population-level limits, not because each organism behaves the same way.
Practice Questions
- 1 A bacterial population has N = 500 and r = 0.40 per hour. Using dN/dt = rN, what is the instantaneous growth rate in bacteria per hour?
- 2 A deer population has N = 200, r = 0.30 per year, and K = 1000. Using dN/dt = rN(1 - N/K), calculate the growth rate in deer per year.
- 3 A population first grows rapidly, then its growth rate decreases and the curve levels off near a stable size. Explain which growth model fits this pattern and what biological factors could cause the leveling off.