Population Viability Analysis
293
George (1992, 1993) support the general statements presented earlier in this
chapter. As with all ecological generalities, exceptions quickly appear.
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 probability 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 sampling 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 stochasticity 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 population 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 ≥ 0 guarantees that
the population will persist, in contrast to R < 0, which guarantees that the population 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 population 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 determine 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
293
George (1992, 1993) support the general statements presented earlier in this
chapter. As with all ecological generalities, exceptions quickly appear.
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 probability 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 sampling 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 stochasticity 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 population 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 ≥ 0 guarantees that
the population will persist, in contrast to R < 0, which guarantees that the population 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 population 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 determine 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