Environmental Engineering Reference
In-Depth Information
5.3 Simple
algebraic models of
population viability
analysis
If the fundamental net per capita rate of increase
is constant and greater than 1
(or equivalent ly, if the intrinsic rate of population increase r is constant and greater
than zero) a population will grow indefi nitely. Conversely, if
λ
is constant and less
than 1 (or r is less than 0) the population will shrink to extinction. When
λ
equals
1 (or r equals 0) the population size remains unchanged. In fact, of course, envi-
ronmental variation causes survival and reproduction to vary from year to year, and
thus population growth rate also varies, and usually in an unpredictable manner.
Thus, we must view population growth rates as variable, each population having a
growth rate with a mean and a variance about the mean. Sometimes managers are
faced with populations at risk for which little detailed demographic information is
available. At best, they may only have a run of several years of population density
estimates. However, such census data can be used to roughly calculate the mean
and variance of population growth rate and, given a range of simplifying assump-
tions, to derive an estimate of the probability of extinction over a particular time
period.
At its simplest, the likely persistence time of a population, T , can be expected to
increase with population size N (as you saw in Figure 5.3), to increase with average
population growth rate ( r or
λ
), and to decrease with an increase in variance in
growth rate V resulting from temporal variation in environmental conditions.
λ
5.3.1 The case of
Fender's blue
butterfl y
Schultz and Hammond (2003) performed an analysis based on this approach for the
rare Fender's blue butterfl y ( Icaricia icarioides fenderi ) in the USA, using population
data from surveys lasting at least 8 years in 12 different local populations (Figure
5.4). The butterfl y lives exclusively in the few surviving patches of prairie grassland
in Oregon that support plants used as food by its caterpillars - Kincaid's lupine
( Lupinus sulphureus kincaidii ) and spur lupine ( Lupinus arbustus ). They followed a
procedure described by Morris et al. (1999) to estimate the mean and variance of
growth rate for each population, and then to determine the probability that each
population would persist for 100 years. The key assumptions of the method were:
(i) that the data represent a complete count of individuals or a constant fraction of
the population; (ii) that variability between years is true environmental variability
and not infl uenced by observer error; (iii) that there are no extreme catastrophes or
bonanza years in the data set; (iv) that population growth rate is independent of
density; and (v) that current environmental conditions will persist for at least a
century. The probability of population persistence for 100 years varied from less
than 0.01 (i.e. lower than a 1% chance of survival) at Gopher Valley to 0.92 (92%
chance of survival) at Butterfl y Meadows. You can see in Figure 5.4 that Gopher
Valley has a small population that follows a particularly erratic pattern (high vari-
ance in population growth rate) while the population at Butterfl y Meadows is both
large and relatively constant through time. Schultz and Hammond estimated that
given the average observed variation in population growth rate ( V
0.79) and an
initial population size of 300 butterfl ies, a minimum average population growth rate
λ
=
of 1.83 is needed for a 95% probability that a population will survive 100 years.
Regrettably, none of the populations look set to achieve this. To provide for a 95%
probability of persistence of a minimum of one population, they estimate that three
independent populations will need to be managed, or restored, each to have a popu-
lation growth rate
λ
of at least 1.55.
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