Overview
This lecture explains how a rabbit population grows exponentially with a constant growth rate, then introduces the concept of environmental limits and logistic growth.
Exponential Growth Model
- Starting population: 1,000 rabbits.
- Monthly growth rate: 10%.
- After each month, population is multiplied by 1.1 (equivalent to a 10% increase).
- Population after n months: ( P(n) = 1000 × 1.1 to the nth power
- Example: After 120 months (10 years), population is ( 1{,}000 \times (1.1)^{120} \approx 93,000,000 ).
- Exponential growth graph has a J- or hockey-stick shape.
Environmental Limits & Logistic Growth
- Exponential growth assumes unlimited food, space, and no predators or competition.
- In reality, resources are limited and populations cannot grow indefinitely.
- Maximum population an environment can sustain is called "carrying capacity."
- As population nears carrying capacity, growth rate slows and population levels off.
- This S-shaped curve is modeled by logistic growth.
Key Terms & Definitions
- Exponential Growth — Population increases by a constant percentage each time period, growth rate appears in the exponent.
- Carrying Capacity — The maximum population size an environment can sustainably support.
- Logistic Growth — Population growth that slows as it approaches carrying capacity, forming an S-shaped curve.
- Population Growth Formula — ( P(n) = P_0 \times r^n ), where ( P_0 ) is the starting population and ( r ) is the growth factor per period.
Action Items / Next Steps
- Review the exponential growth and logistic growth curves.
- Practice calculating population growth using the exponential formula for different values of n.
- Learn more about logistic functions if interested.