A growing population adds more each day
- The idea
- Predict
- See it
- Why
- Check
The idea
“A steady growth factor means the same gain every day, so I add that gain or multiply the factor by the days.”
After one period, adding and multiplying give the same number. They split apart once the new individuals start having offspring of their own.
The short version
The mistake: A steady growth factor means the same gain every day, so I add that gain or multiply the factor by the days.
What's actually true: In exponential growth, each period multiplies the current population by the same factor, so the gain gets bigger every period.
- With unlimited resources, a population grows exponentially. Each period it's multiplied by the same factor. At 1.5, every 10 individuals become 15.
- The new individuals reproduce too, so each gain is the same fraction of a bigger population. From 1,000 at a factor of 1.5, the gains are 500, then 750, then 1,125.
- To go n periods ahead, multiply by the factor n times: N = N0 × factor^n. Three periods at 1.5 is 1.5 × 1.5 × 1.5, or 3.375. Doing 1.5 × 3 gives 4.5, which is wrong.
- In dN/dt = r_max N, the population's rate of increase is proportional to its size, N. That's why exponential growth graphs as a J-shaped curve.
On the exam: Multiply by the growth factor once per period. The gain grows every time.