Modeling population growth - Variance

Question

I've been studing population growth models, but there's something I can not find that's fustrating. That's a formula for the variance in population growth. I know that other models can be aplied, but I want to start with the simplest case.

Let's assume a population is growing exponentially as defined by: $N(t) = N(0)W^{t}$

The parameters $W$(absolute fitness in my definition) tell us that the time for division (if the species reproduces binarally) is given by an exponential distribution with $\lambda = \dfrac{\ln W}{\ln 2}$

So I see a model for $N(t)$ and a random variable associated with this model. What is the pdf, expected value and variance of $N(t)$? The expected value I expect it to be given by the first equation (or the value to be the same) but what about the variance?

Answer 1

Check out Kendall (1949), section 2. He shows that the pdf is a geometric distribution. In your notation, it's $P_n(t) = N(0)W^{-t}\left(1-W^{-t}\right)^{n-1}$, which implies that the mean is indeed $E[N(t)] = N(0)W^t$ and the variance is $\text{Var}[N(t)] = N(0) W^t(W^t-1)$. (Be careful -- his definition of $\lambda$ differs from yours by a factor of $\ln 2$.)

Answer 2

If $W$ is a parameter, I'd expect (in the simplest case) that $W$ itself is a constant (i.e. it doesn't have a distribution). (By the way, I would normally use $T_{1/2}$ to denote the doubling time, $\ln 2/\ln W$, and would use $\lambda = \ln W$ for the exponential rate, i.e. $W^t= \exp(\ln W)^t = \exp(t \ln W) = \exp(\lambda t)$. I don't normally see people express rates scaled by $\ln 2$.)

In the case of a continuous-time birth-death process (the technical jargon for your model), I believe the analytical answer will be that the standard deviation of the population size at time $t$ will grow at the same exponential rate as the mean population size, and that the variance will grow twice as fast (because it has units of $\textrm{population size}^2$), i.e. $\propto \exp(2 \lambda t)$)

I don't have a derivation/reference handy, will dig around for one. This is fairly basic stochastic-process math, but stochastic-process math also gets pretty hairy-looking (to a biologist) quickly. Bartlett 1960 (Stochastic population models in ecology and epidemiology) is one place to look for basic results, although I'm sure there are many others. This recent PhD thesis gives a thorough introduction to the math of birth-death processes ... although, not to my disappointment, a canned derivation of the variance result I'm too lazy to derive for you.