If a population of 100 individuals has an r of 0.1, what is the growth rate (dN/dt) under exponential conditions?

10 individuals per time unit. dN/dt = rN = 0.1 \times 100 = 10.

Population Growth Models

Q: What happens to the growth rate in the logistic model when N = K?

The growth rate becomes zero. When N=K, the term (K-N)/K becomes (K-K)/K = 0, so dN/dt = 0.

Compare exponential and logistic population growth models to understand how populations change over time. Visualize the J-shaped curve of exponential growth (unlimited resources) versus the S-shaped curve of logistic growth (limited by carrying capacity). Explore how density-dependent and density-independent factors regulate population size, and calculate growth rates using the equations dN/dt = rN and dN/dt = rN(K-N)/K.

EXPONENTIAL VS. LOGISTIC GROWTH

Populations change over time based on birth rates, death rates, and resource availability. **Exponential growth** () occurs when resources are unlimited, resulting in a J-shaped curve. **Logistic growth** () occurs when resources become limiting, causing the growth rate to slow as the population approaches the **carrying capacity** (K).

THE LOGISTIC EQUATION

The term represents the fraction of the environment's carrying capacity that is still available. As the population size () gets closer to , this fraction gets closer to zero, causing the growth rate to level off, resulting in an S-shaped (sigmoidal) curve.

HOW TO USE THIS VISUALIZATION

1. **Set Carrying Capacity (K)**: Adjust the maximum number of individuals the environment can support. 2. **Adjust Growth Rate (r)**: See how quickly the population reaches its limit. 3. **Add Disturbance**: Introduce a sudden drop in population and watch the recovery phase. **Try This**: Set equal to . Notice that the growth rate becomes zero. Now set higher than . What happens to the population curve and why?

AP EXAM CONNECTION

Unit: Unit 8: Ecology (Topic 8.3 & 8.4)
Learning Objective: SYI-1.G

COMMON MISCONCEPTIONS

Thinking carrying capacity (K) is a fixed number (it can change if the environment changes).
Assuming exponential growth can continue forever in nature.
Confusing population size (N) with growth rate (dN/dt).