I'm a microbiologist, but I'm teaching some ecology in my intro-level course, so when we got to population growth, I thought I'd use the example of a microbial population. But, I hit a strange problem that I thought maybe an ecologist could help me understand:
Imagine a population of bacteria that can divide every 30 minutes. In one hour, every cell produces four cells. Since no one really dies, the intrinsic growth rate (r) is 4. The exponential growth equation, dN/dt = rN works fine to show the growth of the population: starting with one cell, in one hour it's 4, then in two hours rN = 4*4 = 16, in three hours rN = 16*4 = 64 and so on. At 16 hours, we get to about 4 billion bacteria, which is exactly what the microbiologist expects.
But, we then tell the students that to make it easy to predict future numbers, we can do some math and get N(t) = N(0) e^rt. If N(0) = 1, t = 16 hours and r = 4 bacteria/hour/cell, this gives e^64 which is about 6x10^27. Yikes! Kinda far from 4 billion.
So...why doesn't this work? I feel like either I'm not understanding r correctly (it would have to be down around 0.6 for this to work out, I think) or maybe there's a limit on the equation and it just doesn't work for the absurdly high r of bacteria?
Thanks for the help...