Geoscience Reference
In-Depth Information
infinity. Thus demographic variation is generally not an issue for persistence of
larger populations.
To illustrate further how demographic variation operates, consider a small
population with
N
= 100 and a second population with
N
= 10,000. Assume
both populations have identical survival rates of 0.8. With a binomial model
of the process, the probability that only 75 percent or less of the small popula-
tion survives is 0.1314 for the small population, but 3.194E - 34 for the larger
population. Thus the likelihood that up to 25 percent of the small population
is lost in 1 year is much higher than for the large population.
TEMPORAL VARIATION
A feature of all population persistence models is evident in figure 9.2. That is,
the variation of predicted population size increases with time. Some realiza-
tions of the stochastic process climb to very large population values after long
time periods, whereas other realizations drop to zero and extinction. This
result should be intuitive because as the model is projected further into the
future, certainty about the projections decreases.
However, in contrast to population size, our certainty about the extinction
probability increases as time increases to infinity. The probability of eventual
extinction is always unity if extinction is possible. This is because the only
absorbing state of the stochastic process is extinction; that is, the only popula-
tion size at which there is no chance of change is zero.
Another way to decrease persistence is to increase the stochasticity in the
model. One way would be to introduce temporal variation by making
b
and
d
random variables. Such variation would be exemplified by weather in real
populations. Some years, winters are mild and survival and reproduction are
high. Other years, winters are harsh and survival and reproduction are poor. To
incorporate this phenomenon into our simple model, suppose that the mean
birth and death rates are again 0.5, but the values of the birth rate and the
death rate at a particular time
t
are selected from a statistical distribution, say
a beta distribution. That is, each year, new values of
b
and
d
are selected from
a beta distribution.
A beta distribution is bounded by the interval 0-1 and can take on a vari-
ety of shapes. For a mean of 0.5, the distribution is symmetric about the mean,
but the amount of variation can be changed by how peaked the distribution is
(figure 9.4).
The beta distribution is described by two parameters,
a
b
> 0 and
> 0. The
