untitled

Now consider a subset of this population restricted to a reserve of 200 km^2. The

density of 0.01/ km^2 translates to a population size of two individuals. These two

obviously cannot increase by 32% to 2.64 individuals as the large-population estimate

would imply. They can only increase to 3, or remain at 2, or decline to 1 or even to

0. Table 17.1 gives the probabilities of those outcomes.
   Table 17.1 shows that the most likely outcome is 3 animals and a rate of increase
   of r=0.41. But even though the population is “trying” to increase, the actual rate
   of increase may by chance vary between a low of minus infinity to a high of r=0.41.
   Hence the demographic behavior of a small population is determined by the luck
   and misfortune of individuals. It is a lottery. That of a large population is ruled by
   the law of averages. We say that the outcome for a small population is stochastic and
   for a large population deterministic.
   The extent to which actual ris likely to vary from its deterministic value in a
   constant environment is measured by Var(r) =Var(r) 1 /N, where Var(r) 1 is the com-
   ponent of variance in rattributable to the demographic behavior of an average indi-
   vidual. For a population with a relatively low rm, as in our hypothetical example,
   Var(r) 1 will be in the region of 0.5. We adopt that value for purposes of illustration.
   Var(r) declines progressively as population size Nrises. Variance of rat any popu-
   lation size Ncan be estimated for this “population” as Var(r) =0.5/N.
   Table 17.2 shows that at a population size of N=50 the effects of small numbers
   and hence necessarily unstable age distribution can result in a rate of increase varying
   (at 95% confidence) between r=0.48 (i.e. 0.28 +0.202) and r=0.08 (i.e. 0.28 −0.202).
   At N=10 the possible outcomes vary between a high rate of increase and a steep
   decline. In this example the deterministic rate of increase becomes a safe guide to the
   actual rate of increase only after the population has attained a size of several hundred.
   Although the details are special the message is general: populations containing fewer
   than about 30 individuals can quite easily be walked to extinction by the random

290 Chapter 17

Probability of outcome

Nt+ 1 What happened Symbolic Numerical r

0 Both die (1 −p)^2 0.01 −∞

1 One dies 2 p(1−p) 0.18 −0.69

2 Both live, no offspring p^2 (1 −m) 0.0405 0.0

3 Both live, one offspring p^2 m 0.7695______ 0.41

1.0000

Table 17.1Probabilities
for the population
outcome over a year of
a population comprising
two individuals, one of
each sex. The chance of
an individual surviving
the year is p=0.9 and
the chance of the female
producing an offspring
at the end of that year is
m=0.95.

95% confidence

N Expectedr Var(r) SE(r) limits of r

10 0.28 0.05 0.224 ±0.500

50 0.28 0.01 0.100 ±0.202

100 0.28 0.005 0.071 ±0.139

500 0.28 0.001 0.032 ±0.063

1000 0.28 0.0005 0.022 ±0.043

Table 17.2Deviation
from expected rate of
increase resulting from
stochastic variation.
The influence of one
individual on variance
in ris taken as
Var (r) 1 =0.5.