uses of pv a - department of scientific computingpbeerli/bsc3052/restricted/slides/02-19-p… ·...

Mean = 83, Geometric mean = 82,Harmonic mean = 81,

Median = 85.

TEST 1

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.

I am in my office: Tu 10-12, 3-5:30

Population size and Conservation

Determining whether a population is growing or shrinking

Predicting future population size

Non-genetic risks of small populations

Population Viability Analysis (PVA)

Use of quantitative methods to evaluate and predict the likely future status of a population

Definitions

PVA = 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



Determine necessary reserve size.

Example: African elephants

PVA indicates minimum of 2,500 km2 needed

to sustain population

Uses of PVA

Assessment

Assisting management





Determine size of population to reintroduce Example: European beaver

Uses of PVA

Assessment







Set limit to harvest (intentional and unintentional)

Uses of PVA

Assessment








Intentional Harvest and By-Catch

Habitat degradation

Uses of PVAAssessment








Intentional harvest

Habitat degradation

By-catch

How many populations do we need to protect?

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

The Saga of the Furbish Lousewort

Kate Furbish was a woman who, a centuryago, Discovered something growing, and sheclassified it so That botanists thereafter, in their referencevolumes state, That the plant's a Furbish lousewort. See,they named it after Kate. There were other kinds of louseworts, butthe Furbish one was rare. It was very near extinction when theyfound out it was there. And as the years went by, it seemed, withravages of weather, The poor old Furbish lousewort simplyvanished altogether.But then in 1976, our bicentennial year, Furbish lousewort fanciers had some goodnews they could cheer. For along the Saint John River, guess whatsomebody found? Two hundred fifty Furbish lousewortsgrowing 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 tolive, no thanks.For in that very area, there was a mighttyplan, An engineering project for the benefit ofman. The Dickey-Lincoln Dam it's called,hydroelectric power. Energy, in other words, the issue of thehour. 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 simplywouldn't grow. Conditions for the Furbish louseworts have tobe just so. And for reasons far too deep for me to know to explain, The only place they can survive is in that part ofMaine. So, obviously it was clear that something had togive, And giant dams do not make way so that aplant can live.But hold the phone, for yes, they do. Indeed theymust in fact. There is a law, the Federal Endangered SpeciesAct, And any project such as this, though mighty andexalted, If it wipes out threatened animals or plants, itmust be halted.And since the Furbish lousewort is endangeredas can be, They had to call the dam off, couln't build it,don't you see. For to flood that louseworth haven, where theFurbishes were at, Would be to take away their only extant habitat.And the only way to save the day, to end thisawful stall, Would be to find some other louseworts,anywhere at all. And sure enough, as luck would have it, strangethough it may seem, They found some other Furbish lousewortsgrowing just downstream. Four tiny little colonies, one with just a singleplant.

Types of PVA

Count based PVA: simple -- uses census data (head counts)

Structured PVA: uses demographic models (age structure)

PVM: Count based model

Nt = !Nt!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 mean!

Nt = !Nt!1

Population is shrinkingPopulation is stablePopulation is growing

! < 1

! = 1

! > 1

Predicting future population sizes

N1 = !N0

N2 = !N1 = !(!N0) = !2N1

Nt = !tN0

Example

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

Nt = !tN0

Measuring ! from data

! =

Nt

Nt!1

If we known the population size in two generations we are able to calculate the growth rate

Incorporating stochasticity into model

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

! !

Cyclical example

Stochasticity

With no variability of ! (= 1.2) With variability of ! (!good = 1.3, !bad = 1.1)

N10= 929

N10= 897

Influence of small ! is larger

Cyclical example

Stochasticity

N0 = 150

! =

!

1.3 with p = 0.5

1.1 with p = 0.5

N10 = 150 ! 1.1 ! 1.1 ! 1.3 ! 1.3 ! 1.1 ! 1.1 ! 1.3 ! 1.1 ! 1.3 ! 1.3 = 897

N10 = 150 ! 1.1 ! 1.1 ! 1.3 ! 1.3 ! 1.1 ! 1.1 ! 1.1 ! 1.1 ! 1.1 ! 1.3 = 642

Stochasticity

N0 = 150

! =

!

1.3 with p = 0.5

1.1 with p = 0.5

N10 = 150 ! 1.1 ! 1.1 ! 1.3 ! 1.3 ! 1.1 ! 1.1 ! 1.3 ! 1.1 ! 1.3 ! 1.3 = 897

N10 = 150 ! 1.1 ! 1.1 ! 1.3 ! 1.3 ! 1.1 ! 1.1 ! 1.1 ! 1.1 ! 1.1 ! 1.3 = 642

TIME

Pop

ula

tion

Den

sity

(L

n)

Mean ! = 1

Stochasticity

Models without a stochastic component produce ONE population size

Models with a stochastic component produce a distribution of possible population sizes

0

50

100

150

200

250

300

350

400

0 1 2 3 4 5 6 7 8 9 10

Popula

tion s

ize

Population size frequency

Distribution of population sizes Measuring ! from data

! =

Nt

Nt!1

If we known the population size in two generations we are able to calculate the growth rate

Estimation of growth rate with more than two time points

!tG = !t!t!1!t!2...!0

!G = (!t!t!1!t!2...!0)1/t

=(ln(!t) + ln(!t−1) + ln(!t−2)...ln(!0)

t

µ = ln(!G)

! =

Nt

Nt!1

Nt = !tN0

Estimation of average growth rate

µ =(ln(!t) + ln(!t!1) + ln(!t!2)...ln(!0)

t

µ

!

"

#

"

$

> 0 then !G > 1and population is mostly growing

! 0 then !G ! 1 and population size is constant

< 0 then !G < 1 and population is mostly shrinking

" is the average over all ln(!)

NPS Photo

Example: Grizzly bears in Yellowstone park



1960 1970 1980 1990

20

40

60

80

100

Fem

ale

griz

zly

bea

rs

Census year


1960 1970 1980 1990 2000

0.75

1

1.25

Gro

wth

rat

e !

Census year

!t =

Nt

Nt!1

Change of growth rate over time


Ln(G

row

th r

ate !)

Census year

ln(!t) = ln(Nt

Nt!1

)

1960 1970 1980 1990 2000

!0.25

0

0.25

0.5

Change of ln(growth rate) over time

Average growth curve

Ln(G

row

th r

ate !)

Census year

ln(!t) = ln(Nt

Nt!1

)

1960 1970 1980 1990 2000

!0.25

0

0.25

0.5

average growth

Change of ln(growth rate) over time


!̂2 =1

T ! 1

T!

t=1

(ln(Nt

Nt!1

) ! µ̂)2

!2

= 0.0130509µ = 0.0213403

µ̂ =1

T

T!

t=1

ln(Nt

Nt!1

)

The average growth rate is positive, so Grizzlies survive in Yellowstone park, right?

Confidence intervals

!0.4 !0.2 0 0.2 0.4

0.5

1

1.5

2

2.5

3

3.5

!0.4 !0.2 0 0.2 0.4

0.5

1

1.5

2

2.5

3

3.5

µ̂

µ

!2

= 0.0130509

µ̂ + !µ̂ ! !

µ̂ = 0.0213403

Freq.

95% of all values

2.5% 2.5% µ̂ ! 2!

µ̂ + 2!

! = 0.114

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 quasi-

extinction 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

!2

= 0.04

µ = !0.01

µ = !0.01

!2

= 0.12

Years from now

Extinct

ion p

robab

ility

20 40 60 80 100 120

0.1

0.2

0.3

0.4

0.5

0.6

Nthreshold = 1

Ncensus = 10

Cumulative risk of extinction

20 40 60 80 100 120

0.1

0.2

0.3

0.4

0.5

0.6

!2

= 0.04

µ = !0.01

µ = !0.01

!2

= 0.12

!2

= 0.12

µ = 0.01

µ = 0.01

!2

= 0.04

Years from now

Extinct

ion p

robab

ility

Nthreshold = 1

Ncensus = 10

Extinction risk for Grizzlies

25 50 75 100 125 150 175 200

!25

!20

!15

!10

!5

0[Log10 ]

Extinct

ion P

robab

ility

Years from now

25 50 75 100 125 150 175 200

0.0000001

0.00001

0.001

0.01

[Blow-up of top part of graph]

Confidence interval}

Using Extinction time estimates for conservation planning

460

400

60

30

Adult birds: NORTH CAROLINA

Adult birds: FLORIDA

1980

1980 1990

1990

Year

Year

1

0.5

0.1

100 300Year

Extinction probability

When to use count-based PVMs ?

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]

Key assumptions of count based PVAs

The parameters " and #2 are constant over time.

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 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

!t!1 ! !tNO

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