Data Analysis Module – Cheap Research Essays

ecology

Data Analysis Module
DA Figure 1 illustrates stochastic variation in births
and deaths in a population of killer whales (orcas) ( Orci-nus orca) that reside off the coast of British Columbia,
Canada. These data are reproduced in the Excel spread-sheet that accompanies this module as well. You can see
that although the number of reproductive females in the
population varied between 16 and 28 individuals, the
number of young born each year varied between 0 and
8. Annual deaths varied between 0 and 7.
Step 1: Calculate the average rate of population
change and its standard deviation.
Using the numbers of births and deaths and the initial
population size of 73 individuals in 1974, calculate the
changes in population size in the Excel spreadsheet
through 2001.
The average exponential rate of population growth (r)
during year i is calculated from the number of individuals
at the beginning and at the end of the year as
r
i  ln


N
N
i
i
1


Remember that the exponential growth rate (r) is equal
to the natural logarithm of the geometric growth rate 
(see page 225).
• Did the population increase or decrease over this
period? Was the average exponential growth rate positive
or negative?
To explore how stochastic environmental variation
affects the size of a population and the probability of its
extinction, we need to develop a model based on random
changes in the factors that determine changes in popula -tion size. Such models often include an upper limit to the
size of the population to incorporate the effect of density
dependence. However, in the simplest case, the expo-nential rate of increase of a population is independent of
its size (density-independent) and has a mean of r and a
standard deviation of S. Thus, on average, the population
grows at an exponential rate of r, but during some peri -ods the growth rate is above this level, and during other
periods it is below this level. If a population experiences
many periods of below-average growth, it runs the risk
of extinction. The critical parameter that influences the
average time to extinction under random environmental
variation is the variance in r.
The variance is the square of the standard deviation,
or S
2
. In the special case in which population size is, on
average, balanced ( r  0; that is, births equal deaths under
Stochastic Extinction with Variable
Population Growth Rates
The random nature of births, the number and sex of off-spring, and particularly deaths can lead to variation in
population size even in a constant environment. Such
stochastic variation generally is not a problem for large
populations because these chance events average out
over many individuals. However, small populations can
suffer from random variations in births and deaths, which
can lead to random variation in population size and even
to extinction.
The kakapo ( Strigops habroptilus ) is a large, flightless
parrot found only in New Zealand. Because it is flight-less, it is vulnerable to introduced predators such as
cats, opossums, and weasels. By 1976, only 14 kakapos
were known to be alive on New Zealand’s South Island;
sadly, all of them were males. If males and females were
hatched with equal frequency on average, what is the
probability that 14 individuals would include no females?
If each individual is considered a trial, and being male is
considered a success with probability (p)  0.5, then the
probability that n trials will all be successes is ( p
n
). If you
think this is unlikely, why might the New Zealand Wildlife
Service have failed to locate any female kakapos? Fortu-nately, additional kakapos were later discovered on Stew-art Island, at the southern tip of South Island, and

kakapos
were then introduced onto two small islands from which
all predators had been removed. (For more on this fas-cinating bird, see http://en.wikipedia.org/wiki/Kakapo;
http://animaldiversity.ummz.umich.edu/site/accounts/
information/Strigops_habroptila.html.)
When the environment varies, all individuals in a pop -ulation can be affected in the same way, and dramatic
changes, even in large populations, can result. All members
of a population feel a prolonged drought or a cold snap.
Conversely, a period of exceptionally favorable conditions
may increase the fecundity of all individuals or increase
their probability of survival. The kakapo, for example,
breeds primarily in years when the rimu tree, an endemic
conifer of the podocarp group, produces good fruit crops.
Variations in environmental conditions may be random
and essentially unpredictable —what is referred to as sto-chastic environmental variation—or they may occur with
some regularity. Understanding the connection between
changes in the environment and changes in population
size can suggest interventions, such as supplemental feed -ing during critical periods of limited food supply, that can
reduce the chance of population decline and extinction.
Data a nalysis MoD ule
page 2 Data Analysis Module
average conditions), the average time to extinction ( T ) of
a population of size N is
T(N )  
S
2
2
ln(1  S
2
N )  1
Step 2: Estimate the average time until extinction
for a population of killer whales.
Assume that for our killer whale population, the average
growth rate (r) is 0.
• Starting with the population size in 2001, what is your
estimate of the time to extinction?
• How does T(N ) change with the size of the initial popu -lation and with the variance in the rate of change in popu -lation size?
• Fill in the expected times to extinction for the range of
population sizes (N ) and standard deviations of the popu -lation growth rate ( S ) in DA Table 1.
Assuming that the time units are years, these values
suggest that small populations in particular have relatively
short life expectancies.
• What would T(N ) be for the killer whale population at
its largest and smallest sizes? If a population grows just by
chance, does this mean that its prospects for long-term
persistence improve? Assume that the sample standard
deviation of r in the spreadsheet accurately estimates the
underlying value of S.
The implication of this model for conservation is
clearly that populations should be managed to maintain
as large a population size as possible and to prevent strong
depression in the population growth rate during periods of
poor environmental conditions. The latter strategy might
Age and sex structure of population
Births and deaths
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2 000
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2 000
2001
10
15
20
25
30
35
40
45
Deaths
Number
Births
(a)
(b)
KEY
Calves and juveniles
Adult males
Reproductive females Births
Postreproductive females
DA Figure 1 Stochastic variation in
births and deaths of a population of killer
whales. Data from Taylor and Plater, 2001.
Initial population size
S 10 100 1,000 10,000
0.05
0.1
0.2
0.5
Da table 1 Calculating time to extinction
page 3 Data Analysis Module
involve supplemental feeding or predator and pathogen
control programs at critical times.
The model described here lacks density dependence.
Normally, ecologists believe that the growth potential of
populations reduced to low numbers is greatly increased
during normal conditions, which should allow them to
increase and draw back from the brink of extinction. This
is a fundamental message of the logistic equation and one
of the most basic foundations of ecology.
• If this were always the case, why should we be worried
about small populations? Under what conditions might
you expect a population not to increase when reduced to
low numbers? This certainly has been the case for many
endangered species that have become extinct or now tee-ter on the brink of extinction. Do some populations simply
not “have what it takes” to maintain healthy numbers?
Let’s consider the effects of adding normal density
dependence to models incorporating stochastic environ-mental variation the intrinsic exponential growth rate, r

.
According to the logistic equation, the average growth rate
of populations reduced below their usual carrying capac-ity always exceeds r  0, and such populations tend to
recover quickly. But a long series of unfavorable periods
might still be enough to drive population size below 1 indi-vidual and cause extinction. The important parameters for
predicting the average time to extinction in models with
density dependence are, first, the product of the average
value of r

and K, and second, the ratio of the standard
deviation of r

to its mean; in other words, S/r

. The equa -tions for time to extinction under density dependence are
messy, but as you would expect, the addition of negative
density dependence greatly increases the expected time
to extinction. For example, when N  100, r

 0.1, and
S  0.22; T(100) is equal to nearly 26,000 time units,
rather than the value of about 81 (see DA Table 1) in the
absence of density dependence.
Literature Cited
Taylor, M., and B. Plater. 2001. Population viability analysis for the
southern resident population of the killer whale ( Orcinus orca).
The Center for Biological Diversity, Tucson, Arizona (http://
www.biologicaldiversity.org/swcbd/species/orca/pva.pdf).