TEST 1
Mean = 83,
Geometric mean = 82,
Harmonic mean = 81,
Median = 85.
I will add tonight the grades
to Blackboard (and also add
key on Tu/We)
To get the test back you
need to see me in my office
DSL 150-T.
Population size and Conservation
Determining whether a population is growing or shrinking
Predicting future population size
Non-genetic risks of small populations
I am in my office: Tu 10-12,
3-5:30
Population Viability Analysis (PVA)
Definitions
PVA = Use of quantitative methods to evaluate
and predict the likely future status of a
population
Use of quantitative methods to
evaluate and predict the likely
future status of a population
Status = likelihood that a population will be
above a minimum size
Minimum size, quasi-extinction threshold =
number below which extinction is very likely
due to genetic or demographic risks
Uses of PVA
Assessment
Assessing risk of a single population (for example Grizzly
population)
NPS Photo
Grizzly population size in
Yellowstone national park
Grizzlies are listed
as threatened
1975; less than 200
bears left in
Yellowstone
1983 Grizzly Bear
recovery area
(red)
Increase of
protection area
discussed (blue)
Uses of PVA
Assessment
Assessing risk of a single population (for example Grizzly
population)
Comparing risks between different populations
Sockeye and
Steelhead catch
1866
1991
Uses of PVA
Assessment
Assessing risk of a single population (for example
Grizzly population)
Comparing risks between different populations
Analyzing monitoring data – how many years of data
are needed to determine extinction risk? Example:
Gray Whale
Gray Whale
How many data points do we need?
5 years?
10 years?
15 years?
Gerber, Leah R., Douglas P. Demaster, and Peter M. Kareiva* 1999. Gray
Whales and the Value of Monitoring Data in
Implementing the U.S. Endangered Species Act. Conservation Biology 13:1215-1219.
Uses of PVA
Assessment
Identify best ways to manage.
Example: loggerhead turtles
Uses of PVA
Assessment
Identify best ways to manage.
Example: loggerhead turtles
Determine necessary reserve size.
Example: African elephants
Uses of PVA
Assessment
Assisting management
Identify best ways to manage.
Example: loggerhead turtles
Determine necessary reserve size.
Example: African elephants
Determine size of population to reintroduce
Example: European beaver
Uses of PVA
Assessment
Assisting management
Identify best ways to manage.
Example: loggerhead turtles
Determine necessary reserve size.
Example: African elephants
Determine size of population to reintroduce
Example: European beaver
Set limit to harvest (intentional and unintentional)
PVA indicates minimum of 2,500 km2 needed
to sustain population
Uses of PVA
Uses of PVA
Assessment
Assisting management
Identify best ways to manage.
Example: loggerhead turtles
Assessment
Assisting management
Identify best ways to manage.
Example: loggerhead turtles
Determine necessary reserve size.
Example: African elephants
Determine necessary reserve size.
Example: African elephants
Determine size of population to reintroduce
Example: European beaver
Set limit to harvest (intentional and unintentional)
Intentional Harvest and By-Catch
Habitat degradation
Determine size of population to reintroduce
Example: European beaver
Set limit to harvest (intentional and unintentional)
Intentional harvest
Habitat degradation
By-catch
How many populations do we need to protect?
The Saga of the Furbish Lousewort
Kate Furbish was a woman who, a century
ago, Discovered something growing, and she
classified it so That botanists thereafter, in their reference
volumes state, That the plant's a Furbish lousewort. See,
they named it after Kate. There were other kinds of louseworts, but
the Furbish one was rare. It was very near extinction when they
found out it was there. And as the years went by, it seemed, with
ravages of weather, The poor old Furbish lousewort simply
vanished altogether.
But then in 1976, our bicentennial year, Furbish lousewort fanciers had some good
news they could cheer. For along the Saint John River, guess what
somebody found? Two hundred fifty Furbish louseworts
growing in the ground. Now, the place where they were growing,
by the Saint John River banks, Is not a place where you or I would want to
live, no thanks.
For in that very area, there was a mightty
plan, An engineering project for the benefit of
man. The Dickey-Lincoln Dam it's called,
hydroelectric power. Energy, in other words, the issue of the
hour. Make way, make way for progress now,
man's ever-constant urge. And where those Furbish louseworts were,
the dam would just submerge. The plants can't be transplanted; they simply
wouldn't grow. Conditions for the Furbish louseworts have to
be just so. And for reasons far too deep for me to know
to explain, The only place they can survive is in that part of
Maine. So, obviously it was clear that something had to
give, And giant dams do not make way so that a
plant can live.
But hold the phone, for yes, they do. Indeed they
must in fact. There is a law, the Federal Endangered Species
Act, And any project such as this, though mighty and
exalted, If it wipes out threatened animals or plants, it
must be halted.
And since the Furbish lousewort is endangered
as can be, They had to call the dam off, couln't build it,
don't you see. For to flood that louseworth haven, where the
Furbishes were at, Would be to take away their only extant habitat.
And the only way to save the day, to end this
awful stall, Would be to find some other louseworts,
anywhere at all. And sure enough, as luck would have it, strange
though it may seem, They found some other Furbish louseworts
growing just downstream. Four tiny little colonies, one with just a single
plant.
Types of PVA
Count based PVA: simple -- uses census data (head counts)
Structured PVA: uses demographic models (age structure)
Dickey-Lincoln Dam was too laden with
ecological and economic problems to ever be
built, and the Furbish lousewort has held its
own along the ice-scoured banks of the Saint
John. In 1989 the U.S. Fish and Wildlife Service
reported finding 6,889 flowering stems--far
more than the 250 or so that were thought to
exist earlier. Pedicularis furbishiae, a species
with close relatives in Asia but nowhere else in
North America, is still endangered, however.
The current threats are new dam proposals,
logging, and real-estate development
PVM: Count based model
Nt = λNt−1
population size at time t
population size at time t-1
‘lambda’ = growth rate
includes birth and death
does not include gene flow (movement among different
populations)
What does
λ
mean
Nt = λNt−1
N1 = λN0
N2 = λN1 = λ(λN0 ) = λ2 N1
λ < 1 Population is shrinking
λ = 1 Population is stable
λ > 1 Population is growing
Example
Predicting
future population sizes
Nt = λt N0
Nt = λt N0
For some insect, !=1.2 and N0 = 150
How many insects will we have in 10 years?
N10! =! 150*1.2*1.2*1.2*1.2*1.2*1.2*1.2*1.2*1.2*1.2
!!!!!!! =! 150 * 1.210
!!!!!!! =! 150 * 6.19
!!!!!!! =!! 929
Incorporating stochasticity into
model
Cyclical example
Measuring ! from data
If we known the population size in two generations we are
able to calculate the growth rate
λ=
Nt
Nt−1
Stochasticity
Cyclical example
For some insect, !good = 1.3, !
bad = 1.1 and N0 = 150
Assume good/bad years alternate.
How many insects will we have in 10 years?
N10 = 150*1.1*1.3*1.1*1.3*1.1*1.3*1.1*1.3*1.1*1.3
!!!!!! = 897
With no variability of ! (= 1.2)
With variability of !
(!good = 1.3, !bad = 1.1)
N10= 929
N10= 897
Influence of small ! is larger
Stochasticity
N0 = 150
!
1.3
λ=
1.1
with p = 0.5
with p = 0.5
N10 = 150 × 1.1 × 1.1 × 1.3 × 1.3 × 1.1 × 1.1 × 1.1 × 1.1 × 1.1 × 1.3 = 642
N10 = 150 × 1.1 × 1.1 × 1.3 × 1.3 × 1.1 × 1.1 × 1.3 × 1.1 × 1.3 × 1.3 = 897
Stochasticity
N0 = 150
!
1.3
λ=
1.1
with p = 0.5
with p = 0.5
N10 = 150 × 1.1 × 1.1 × 1.3 × 1.3 × 1.1 × 1.1 × 1.1 × 1.1 × 1.1 × 1.3 = 642
N10 = 150 × 1.1 × 1.1 × 1.3 × 1.3 × 1.1 × 1.1 × 1.3 × 1.1 × 1.3 × 1.3 = 897
Stochasticity
Population Density (Ln)
Mean ! = 1
Models without a stochastic component produce ONE
population size
Models with a stochastic component produce a distribution
of possible population sizes
TIME
Distribution of population sizes
Population size
Population size frequency
400
350
Measuring ! from data
If we known the population size in two generations we are
able to calculate the growth rate
300
250
200
λ=
150
100
50
0
0 1 2 3 4 5 6 7 8 9 10
Nt
Nt−1
Estimation of growth rate with
more than two time points
Nt = λt N0
λtG = λt λt−1 λt−2 ...λ0
λG = (λt λt−1 λt−2 ...λ0 )1/t
λ=
Estimation of average growth rate
µ=
(ln(λt ) + ln(λt−1 ) + ln(λt−2 )...ln(λ0 )
t
Nt
Nt−1
" is the average over all ln(!)
µ = ln(λG )
(ln(λt ) + ln(λt−1 ) + ln(λt−2 )...ln(λ0 )
=
t


> 0 then λG > 1and population is mostly growing
µ ∼ 0 then λG ∼ 1 and population size is constant


< 0 then λG < 1 and population is mostly shrinking
Example: Grizzly bears in
Yellowstone park
Grizzly population size in
Yellowstone national park
NPS Photo
Grizzly population size in
Yellowstone national park
100
Growth rate !
Female grizzly bears
Grizzly population size in
Yellowstone national park
80
60
40
20
1960
1970
1980
1990
Census year
1.25
Change of growth rate
over time
Nt
λt =
Nt−1
1
0.75
1960
1970
1980
1990
2000
Census year
Grizzly population size in
Yellowstone national park
Change of ln(growth rate) over time
Nt
ln(λt ) = ln(
)
Nt−1
0.5
0.25
0
!0.25
1960
1970
1980
1990
Ln(Growth rate !)
Ln(Growth rate !)
Change of ln(growth rate) over time
Average growth curve
ln(λt ) = ln(
0.5
0.25
0
average growth
!0.25
2000
1960
1970
1980
Census year
T
1!
Nt
ln(
)
T t=1
Nt−1
1 !
Nt
(ln(
) − µ̂)2
T − 1 t=1
Nt−1
µ = 0.0213403
σ = 0.0130509
3.5
Freq.
3
1
µ̂ + σ
µ̂ − σ
2
1.5
The average growth rate is positive, so Grizzlies survive in
Yellowstone park, right?
σ 2 = 0.0130509
µ̂ σ = 0.114
µ̂ = 0.0213403
2.5
2
2000
Confidence intervals
T
σ̂ 2 =
1990
Census year
Grizzly population size in
Yellowstone national park
µ̂ =
Nt
)
Nt−1
95% of all values
2.5%
2.5%
µ̂ − 2σ
0.5
µ̂ + 2σ
!0.4
!0.2
0
0.2
0.4
µ
Extinction probability
One does not try to predict when the last individual is gone
but when the population size goes below a threshold, the
quasi-extinction threshold, under which the population is
critically and immediately imperiled (Ginzburg et al. 1982)
A value of 20 reproductive individuals is often used for
practical purposes.
Genetic arguments would ask for 100 or more reproductive
individuals
Relationship between the
probability of extinction and the
parameters " and #2
A population goes extinct when its size falls below the
quasi-extinction threshold.
if " is negative, eventually the population goes extinct,
independent of the variance of the growth rate #2.
if " is positive, there is still a risk to fall under the quasiextinction threshold, depending on the magnitude of the
variance of the growth rate, with high variance, the risk
is higher to go extinct than with low variance.
Extinction risk depends on the average growth rate ", the
variance of the growth rate, and time.
Cumulative risk of extinction
µ = −0.01
σ 2 = 0.12
0.6
µ = −0.01
σ 2 = 0.04
0.5
0.4
0.3
0.2
0.1
20
40
60
80
100
120
Years from now
Nthreshold = 1
Ncensus = 10
[Log10 ]
µ = −0.01
σ 2 = 0.12
0.6
µ = −0.01
σ 2 = 0.04
0.5
0.4
µ = 0.01
σ 2 = 0.12
0.3
0.2
0.1
20
40
60
80
100
120
µ = 0.01
σ 2 = 0.04
Years from now
Nthreshold = 1
Ncensus = 10
Extinction risk for Grizzlies
Extinction Probability
Extinction probability
Extinction probability
Cumulative risk of extinction
Using Extinction time estimates for
conservation planning
0
}
!5
Confidence
interval
!10
!15
Adult birds: NORTH CAROLINA
Extinction probability
1
460
400
!20
1980
!25
Year
1990
0.5
25
50
75
100 125 150 175 200
Years from now
0.01
Adult birds: FLORIDA
60
0.001
0.00001
0.1
30
0.0000001
100
25
50
75 100 125 150 175 200
[Blow-up of top part of graph]
When to use count-based PVMs ?
1980
Year
300
Year
1990
Key assumptions of count based PVAs
The parameters " and #2 are constant over time.
When only “few” data is available, we still need about 10 years
of census data to get usable extinction risk estimates.
This method is useful to compare multiple populations; to
give a relative population health (in comparison to these
other populations).
Simplicity [there are many assumption]
No density dependence: growth rate is independent of
population size. Small populations might enjoy more
resources. Pessimistic estimates of extinction risk. [but
density dependence of finding mates: Allee effect]
Demographic stochasticity is ignored for derivation of
extinction probability formula: #2 is considered to be
constant over time and independent of the population size
(Remedy is to set quasi-extinction threshold sufficiently
high)
Environmental trends are ignored.
Key assumptions of count based PVAs
The parameters " and #2 are constant over time.
No environmental autocorrelation:
NO
λt−1 ∼ λt
No catastrophes or bonanzas
The extinction probability calculations was derived by
assuming only small changes of the population size over
time: Catastrophes, such as ice storms, wild fires, droughts,
etc. will reduce numbers fast, and are not taken into
account [too positive view]; bonanzas (good years) are also
not taken into account [too pessimistic view]
No observation error, the numbers are treated as true
population sizes.
(only one population: no population structure)
Individuals are all treated all the same