Geoscience Reference
In-Depth Information
George (1992, 1993) support the general statements presented earlier in this
chapter. As with all ecological generalities, exceptions quickly appear.
j
Sources of Variation Affecting Population Persistence
The persistence of a population depends on stochasticity, or variation (Dennis
et al. 1991). Sources of variation, and their magnitude, determine the proba-
bility of extinction, given the population growth mechanisms specific to the
species. The total variance of a series of population measurements is a function
of process variation (stochasticity in the population growth process) and sam-
pling variation (stochasticity in measuring the size of the population). Process
variation is a result of demographic, temporal and spatial (environmental), and
individual (phenotypic and genotypic) variation. In this section, I define these
sources of variation more precisely and develop a simple mathematical model
to illustrate these various sources of stochasticity, thus demonstrating how sto-
chasticity affects persistence.
NO VARIATION
Consider a population with no variation, one that qualifies for the simple,
density-independent growth model
N
t
+1
=
N
t
(1 +
R
), where
N
t
is the popula-
tion size at time
t
and
R
is the finite rate of change in the population. This
model is deterministic, and hence, so is the population.
R
³
0 guarantees that
the population will persist, in contrast to
R
< 0, which guarantees that the pop-
ulation will go extinct (albeit in an infinite amount of time because a fraction
of an animal is allowed in this model).
R
can be considered to be a function of
birth and death rates, so that
R
=
b
-
d
defines the rate of change in the popu-
lation as a function of birth rate (
b
) and death rate (
d
). When the birth rate
exceeds or equals the death rate, the population will persist with probability 1
in this deterministic model. These examples are illustrated in figure 9.1.
STOCHASTIC VARIATION
Let us extend this naive model by making it stochastic. I will change the
parameter
R
to be a function of two random variables. At each time
t
, I deter-
mine stochastically the number of animals to be added to the population by
births and then the number to be removed by deaths. Suppose the birth rate
