Demographic vs. environmental stochasiticity

Remember that demographic stochasticity refers to the inevitable variability in actual population growth rate that occurs even if the population vital rates (expected rates of survival and reproduction) don't change from one season to the next. Environmental stochasticity refers to the variability in population growth rate that occurs because vital rates differ from one season to the next. Making an analogy with evolutionary processes may help: Demographic stochasticity is like genetic drift. It is non-directional and non-deterministic. Environmental stochasticity is like a variable selection pressure. In any one generation there is a directional change and it is deterministic. It would be very useful if we could tease apart these two stochastic influecnes on population dynamics so that we could more accurately assess their effect.

Unfortunately, the relative contributions of demographic and environmental stochasticity to variation in population growth rate cannot be directly measured. All we know is that we see $N_{t+1}$ plants or animals at time $t+1$ and $N_t$ at time $t$. Now if we had a large enough set of data, we might be able to predict pretty closely what $N_{t+1}$ should be in terms of $N_t$ and the relevant demographic parameters. The $N_{t+1}$ we observe will almost certainly be different from the one we expect. But how much of this difference is because the environment in generation $t$ was different from its long-term average and how much is because of inherent randomness associated with birth and reproduction? There's no way to tell.⁵

If, however, we assume that the number of births and deaths are approximately distributed as independent Poisson random variables, then the demographic variance is equal to $(1+R)/N$. We can use this observation to calculate the population size (labeled ``Equivalent $N$'' in Table 1) that would produce a variance in population growth rate equivalent to what is observed in a particular data set.<sup>6</sup> For example, the observed variance in population growth rate of a British heron population is 0.01438. Given that its observered growth rate, $R$, is 0.025, we can calculate the ``equivalent N'' from

$$\begin{eqnarray*} (1+R)/N &=& \hbox{Var(R)} \\ (1+0.025)/N &=& 0.01438 \\ N &=& 1.025/0.01438 \\ &=& 71 \end{eqnarray*}$$

Such calculations give us a way of assessing whether it's reasonable to think that the observed magnitude of population fluctuations we see are consistent with purely demographic variability or if it's more reasonable to think that there must be some environmental variability, too.

Table 1: Demographic versus environmental stochasticity for sevearl animal populations (from [3])

Species	$\bar r$	Variance	Equivalent $N$	Observed $N$
Great tit	0.20 (-0.045,0.44)	0.4151	3	20-95
Heron	0.025 (-0.035,0.084)	0.01438	71	274-484
Laysan finch	0.89 (0.33,1.44)	0.0802	24	5000-21000
Palila	0.52 (-0.66,1.70)	0.3625	5	2000-6400
Palila	0.0017 (-1.0,1.0)	0.3931	3	2000-6400
Grizzly bear	-0.0018 (-0.036,0.032)	0.006444	155	33-47

Notice that the equivalent $N$ is much smaller than the observed $N$ for every species except the grizzly bear. That means that in every species except grizzly bear there is substantially more variation in population growth rate than is likely to be accounted for by demographic stochasticity, i.e., it is likely that environmental stochasticity makes a large contribution to the population dynamics of the other species. This may not be much of a surprise, but the quantitative results suggest that environmental stochasticity is 10-100 times more important than deographic stochasticity.