## Elk abundance thousands

to survive, but rather anticipate that by chance sometimes a larger fraction will survive, sometimes a smaller fraction. We consider this process in more exact mathematical detail in Chapter 17, when we discuss population viability analysis. In wildlife management we need to disentangle demographic from environmental sources of stochasticity (S^ther et al. 2000; Bj0rnstad and Grenfell 2001).

We should include in our population models the variability in growth rates due to environmental and demographic stochasticity. We do this by simulating natural stochastic variation and adding this variation to the exponential growth rate rt predicted by population density. We first need to calculate the residual variation in growth from the data in Fig. 8.15:

where 0.518 is the intercept (rmax) of the regression line drawn through the observed values of rt versus Nt, and -0.00004404 is the slope. We calculate the deviation between each observation of r and the value predicted by the regression line at that population density, square each deviation to standardize positive versus negative values, sum the squared deviations, and divide by the sample size (16 in this case) to estimate the mean-squared deviation. This is the residual variability, denoted by a2. For the Northern Yellowstone elk, a2 = 0.0361.

Once equipped with an estimate of the residual variation based on the observed data, we draw values of the random variable efrom a bell-shaped (i.e. normal) probability distribution with the same magnitude of residual variation at. In mathcad, this normal probability distribution is the function called rnorm, which also requires the user to input the required number of random values, and mean and standard deviation of the normal distribution from which these values will be drawn. For the elk example:

Fig. 8.16 Simulated dynamics of elk, based on the Yellowstone National Park population. (Data from Coughenour and Singer 1996.)