Geoscience Reference
In-Depth Information
Figure 9.1
Deterministic model of population growth. For values of R
³
0, the population persists
indefinitely. For values of R < 0, the population will eventually go extinct in that the number of ani-
mals will approach zero.
equals the death rate, say
b
=
d
= 0.5. That is, on average 50 percent of the
N
t
animals would give birth to a single individual and provide additions to the
population, and 50 percent of the
N
t
animals would die and be removed from
the population. Thus the population is expected to stay constant because the
number of births equals the number of deaths. A reasonable stochastic model
for this process would be a binomial distribution. For the binomial model, you
can think of flipping a coin twice for each animal. The first flip determines
whether the animal gives birth to one new addition to the population in
N
t
+1
and the second flip determines whether the animal currently a member of
N
t
remains in the population for another time interval, to be a member of
N
t
+1
,
or dies. If we start with
N
0
= 100, what is the probability that the population
will persist until
t
= 100? Three examples are shown in figure 9.2.
You might be tempted to say the probability is 1 that the population will
persist until
t
= 100 because the expected value of
R
is 0 given that the birth
rate equals the death rate—that is, E(
R
) = 0, so that E(
N
t
+1
) = E(
N
t
). You
would be wrong! Implementation of this model on a computer shows that the
probability of persistence is 98.0 percent; that is, 2.0 percent of the time the
