untitled

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 math-

ematical detail in Chapter 17, when we discuss population viability analysis. In wildlife

management we need to disentangle demographic from environmental sources of

stochasticity (Sæther et al. 2000; Bjørnstad 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 rtpre-

dicted 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 rtversus Nt, and −0.00004404 is the slope. We calculate the deviation between

each observation of rand the value predicted by the regression line at that popula-

tion 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 σ^2. For the

Northern Yellowstone elk, σ^2 =0.0361.

Once equipped with an estimate of the residual variation based on the observed

data, we draw values of the random variable εfrom a bell-shaped (i.e. normal) prob-

ability distribution with the same magnitude of residual variation σt. 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:

ε=rnorm(51, μ, σ)

σ

[ (.. )]

=

−−

=

∑

rNtt

t

0 518 0 00004404

16

2

0

15

124 Chapter 8

1.0

0.5

0

• 0.5

–1.0

0 5 10 15 20

Elk abundance (thousands)

rt

Fig. 8.15Exponential
growth rates for
Northern Yellowstone
elk between 1968 and
1989 in relation to
population density at
the beginning of each
yearly interval. (Data
from Coughenour and
Singer 1996.)