Logistic Population Growth Study Guide

Population Growth Models

• Exponential Growth Model
  • Represents ideal conditions where population keeps doubling.
  • Assumes unlimited resources; however, it cannot be sustained long-term.

Carrying Capacity (K)

• Definition: The maximum number of individuals an ecosystem can support.
• Influence: Varies depending on organism and environment.
  • Example: An organism may thrive in Environment A but struggle in Environment B.

Logistic Population Growth

• Equation: $\frac{dN}{dt} = rN \left(\frac{K - N}{K}\right)$
  • Components:
  • $N$: population size
  • $r$: intrinsic rate of increase
  • $K$: carrying capacity
• Behavior:
  • When $N$ is small:
  • Growth rate is close to $r$ (exponential-like growth)
  • When $N$ approaches $K$:
  • Growth rate decreases, reaching zero when $N = K$ (stabilizes at carrying capacity).

Example Calculation

• Assume:
  • Intrinsic rate of increase $r = 1$
  • Carrying capacity $K = 1500$
• Population Size:
  • Starts at 25:
  • Growth = $\frac{K - N}{K} = \frac{1500 - 25}{1500} \approx 1$ (add ~25 individuals)
  • At 100:
  • Growth slows (~100 added)
  • At 500:
  • Only ~300 added (growth diminishing)
  • At capacity 1500:
  • Growth = $0$

Graphical Representation

• S-curve (Sigmoid):
  • Reflects logistic growth.
  • Initial low growth, acceleration in mid-range, then leveling off at $K$.
  • Comparison to exponential growth (continuous increase).

Real-Life Examples

• Paramecium:
  • Show classic sigmoid growth under ideal conditions with enough food and no predators.
• Daphnia:
  • Exhibits irregular bouncing population growth with lag phases and crashes, complicating carrying capacity estimates.

Related Concepts

• Factors Affecting Population Growth:
  • Include predators, seasonal changes, food availability.
• Ecological Applications:
  • Models are used to predict the recovery of endangered populations (e.g., white rhinos).