MathBench > Population Dynamics

Exponential Growth and Decay

A quick review of exponential models

  1. This model works any time that something (amount of money, population size, concentration, etc) grows or shrinks by the same PERCENT each timestep.
  2. Timesteps can be days, years, nanoseconds, whatever you want as long as you are consistent.
  3. Over time, populations that are growing exponentially will grow faster and faster -- the graph will "curve" upward.
  4. Over time, populations that are shrinking exponentially will shrink MORE SLOWLY (that is, first they decline precipitously, then they start to level out, and they never quite get to zero) -- the graph will curve downward and level out.

If you don’t believe me about the never-getting-to-zero part, think about what happens when two very polite people share a brownie.  In the first minute, they eat half of it.  That’s a shrinkage rate of 50%.  In the next minute, they eat half of what was left.  Now we’re down to a quarter.  In the third minute they eat half again, leaving an eighth.  Since they are both polite and no one wants to finish it off, each minute they take smaller and smaller bites.  Soon there is only a thirty-second left, then a sixty-fourth… But of course they will never get to zero this way (unless they use calculus, but that’s a different story).

5. You find an exponential multiplier by adding the rate of growth (expressed as a decimal number) to 1, i.e., for 5% growth the multiplier is 1.05, for -5% growth the multiplier is 0.95.

6. You find the final population size by multiplying the original population by the multiplier RAISED TO THE POWER of the number of timesteps (number of timesteps is the exponent).