( 2013 Kristin McCully & Myra Finkelstein) Lead Poisoning of Laysan Albatrosses at Midway Atoll: From Ecological Model to Conservation Action

( 2013 Kristin McCully & Myra Finkelstein)

Lead Poisoning of Laysan Albatrosses at Midway Atoll:

From Ecological Model to Conservation Action

Objectives• Learn how to use structured population

models to assess and protect populations

• Appreciate the importance and uses of mathematical models in ecology and conservation

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Outline• Introduction to case study:

Laysan albatrosses at Midway Atoll• Introduction to mathematical models• Structured population models– Assumptions– Model structure

• Mathematical model: Laysan albatrosses at Midway Atoll

• Importance of models to ecology and conservation

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Case Study: Lead Poisoning of Laysan

Albatrosses at Midway Atoll

Research by Myra Finkelstein

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Threats to Albatross Populations

• Longline fishing bycatch• Introduced mammals• Marine debris• Lead poisoning

Brenda Zaun/USFWS

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Where is Midway Atoll?

Papahānaumokuākea Marine National Monument

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Midway Atoll

Keith Castellano

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Wisdom: World’s Oldest- Known Wild Bird

Pete Leary/USFWSCheck her out on Facebook: https://www.facebook.com/wisdomthealbatross

https://www.facebook.com/wisdomthealbatross

https://www.facebook.com/wisdomthealbatross

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“Droopwing” syndrome

Peripheral Neuropathy

National Institutes of Health

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ProblemLead poisoning of Laysan chicks near buildings- Sileo and Fefer (1987) J Wildlife Disease- Sileo et al. (1990) J Wildlife Disease- Work et al (1996) Env Sci & Toxicology

Military base → many sources of leadNeed evidence to link source with poisoning

Away from bldgsn =15

Near bldgsn = 21

Lead

(μg/

dL w

hole

blo

od)

0

200

400

600

800

1000

1200

1400

1600

> 80% of chicks near bldgs. >100 ug/dL

Droopwing or other symptoms

Step 1: Quantify Lead Exposure

Finkelstein et al. (2003) Environmental Science & Technology

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• Fingerprint source using isotope analysisIsotopes = variants of chemical element with

different numbers of neutrons

• Lead exists as four isotopes:204Pb, 206Pb, 207Pb, 208Pb

• Lead isotope ratios in organism reflects lead isotope ratios in exposure source

Step 2: Identify Source of Lead

3 isotopes of hydrogen:

Image: Dirk Hünniger (Wikimedia)

14→ nest soil is not source of poisoning


Near Buildings: r 2 = 0.363p = 0.152

Step 2: Identify Source of LeadCh

ick

Bloo

d 20

7 Pb/

206 P

b

Nest Soil 207Pb/206Pb

Away from Buildings: r 2 = 0.224p = 0.074

15→ lead-based paint is source of poisoning


r 2 = 0.853p = 0.003

Step 2: Identify Source of LeadCh

ick

Bloo

d 20

7 Pb/

206 P

b

Nest Paint Chips 207Pb/206Pb

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Step 3: Show Population Consequences

Mathematical Model

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Discussion: What is a Model?• Simplified representation of object or process• Requires assumptions

USGS

Essentially, all models are wrong, but some are useful.

- George Box, 1987

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Mathematical Population Models

N

time

Exponential Growth Logistic Growth

N

timeMalthus (1798) Verhulst (1838)

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Steps to a Mathematical Model1. Ask research questions2. Make assumptions3. Develop conceptual model4. Formulate mathematical model5. Assign values to parameters6. Use model to answer questions

Based on: Soetaert & Herman 2008 A Practical Guide to Ecological Modeling (Fig. 1.7)

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1. Ask Research Questions• Is the population growing? • Which stage(s) and demographic processes

should managers focus monitoring and conservation effort on?

• How might threats or management actions impact the population?

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2. Make Assumptions• Constant environment (Deterministic)• Unlimited resources

Constant birth & death rates• No immigration or emigration• No time lags• All individuals are identical

N

time

Exponential Growth

From Gotelli (2001) A Primer of Ecology

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Constant birth & death rates• No immigration or emigration• No time lags• All individuals are identical• All individuals in each stage are identical


vital rates

→ structured population model

Complex Life Histories

Transition Matrix

Structured Population Model

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n1 P1 F2 F3 F4 F5 F6 n1

n2 G1 P2 0 0 0 0 n2

n3 =

0 G2 P3 0 0 0 n3

n4 0 0 G3 P4 0 0 n4

n5 0 0 0 G4 P5 0 n5

n6 0 0 0 0 G5 P6 n6

tt+1

• Age • Size • Sex • Developmental stageStructured by

Transition Matrix

Structured Population Model

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n1 P1 F2 F3 F4 F5 F6 n1

n2 G1 P2 0 0 0 0 n2

n3 =

0 G2 P3 0 0 0 n3

n4 0 0 G3 P4 0 0 n4

n5 0 0 0 G4 P5 0 n5

n6 0 0 0 0 G5 P6 n6

t

1 2 3 4 5 6

1

2

=3

4

5

6t+1

1 2 3 4 5 6

1 P1

2 P2

3 P3

4 P4

5 P5

6 P6

1 2 3 4 5 6

1

2 G1

3 G2

4 G3

5 G4

6 G5

1 2 3 4 5 6

1 F2 F3 F4 F5 F6

2

3

4

5

6

0 0 0 0

=0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

P = Persistence G = Growth F = FecundityFROM

TO

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PJ

GL GJ

PA

P = Persistence

Simpler Example: Lionfish (Pterois volitans)

NOAA SE Fisheries Science Center

Todd Gardner

L J A

Larvae Juveniles Adults

FAG = Growth

F = Fecundity

Morris et al. (2011) Biological Invasions

Time step: 1 month

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Larvae Juveniles Adults

Larvae

Juveniles

Adults

ToStage

From Stage

GL

GJ

PL

PA

FA

Simpler Example: Lionfish (Pterois volitans)

0

0

0

0

PL

GL GJ

PA

P = Persistence L J AFAG = Growth

F = Fecundity

Time step: 1 month

Seeds

Immature Plants

Mature Plants

F3

G1P1

P2

P3G2

F = Fecundity G = Growth P = PersistenceEx2: Life Cycle Diagram → Transition Matrix

Ex2: Life Cycle Diagram → Transition Matrix

Stage-Based ModelF = Fecundity G = Growth P = Persistence

Seeds Immature Plants

Mature Plants

Seeds

Immature Plants

MaturePlants

ToStage

From Stage

P1

G1

0

F

0

P3

0

P2

G2

Seeds

Immature Plants

Mature Plants

F

G1P1

P2

P3G2

Ex3: Transition Matrix → Life Cycle Diagram

0 0 F3 F4

G1 P2 0 00 G2 P3 00 0 G3 P4

1 2 3 4P2 P3 P4

G3G2G1

F4F3

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1. Determine Research Questions

• How does lead poisoning of chicks affect population growth?

• Will lead remediation help protect the population?

Finkelstein et al. 2009 Animal Conservation

35


Constant birth & death rates• All individuals are identical• No immigration or emigration• No time lags• All individuals in each stage are identical


vital rates

→ structured population model

Complex Life Histories

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3. Develop Conceptual ModelVideo: Albatross Life History

3. Develop Conceptual Model

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E 5421 3 6 7 8

Pete Leary/USFWS

SubadultImmatureJuvenileEggs

Chicks

AdultsGina

Ruttle

Time step: 1 year

4. Formulate Math Model

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E 5421 3 6 7 8

F

Gone Gj Gj Gj Gi Gi Gsa Pa

Gsa

G = GrowthP = PersistenceF = Fecundity


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F

E 1 2 3 4 5 6 7 8E

1

2

3

4

5

6

7

8

E 5421 3 6 7 8Gone Gj Gj Gj Gi Gi Gsa Pa

Gsa


40

F

E 1 2 3 4 5 6 7 8E F

1 Gone

2 Gj

3 Gj

4 Gj

5 Gi

6 Gi

7 Gsa

8 Gsa Pa


Gsa


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F

E 1 2 3 4 5 6 7 8E 0 0 0 0 0 0 0 0 F

1 Gone 0 0 0 0 0 0 0 0

2 0 Gj 0 0 0 0 0 0 0

3 0 0 Gj 0 0 0 0 0 0

4 0 0 0 Gj 0 0 0 0 0

5 0 0 0 0 Gi 0 0 0 0

6 0 0 0 0 0 Gi 0 0 0

7 0 0 0 0 0 0 Gsa 0 0

8 0 0 0 0 0 0 0 Gsa Pa


Gsa

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5. Assign Values to Parameters

Pete Leary/USFWS

Andrew Derocher (IUCN)

NOAA SEFSC

NOAA


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E 5421 3 6 7 8

F

Gone Gj Gj Gj Gi Gi Gsa Pa

Gsa

G = GrowthP = PersistenceF = Fecundity

0.482 0.84 0.84 0.84 0.906 0.906 0.9450.916

0.945

= 0.366


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F

E 1 2 3 4 5 6 7 8E 0 0 0 0 0 0 0 0 F

1 Gone 0 0 0 0 0 0 0 0

2 0 Gj 0 0 0 0 0 0 0

3 0 0 Gj 0 0 0 0 0 0

4 0 0 0 Gj 0 0 0 0 0

5 0 0 0 0 Gi 0 0 0 0

6 0 0 0 0 0 Gi 0 0 0

7 0 0 0 0 0 0 Gsa 0 0

8 0 0 0 0 0 0 0 Gsa Pa


Gsa


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F

E 1 2 3 4 5 6 7 8E 0 0 0 0 0 0 0 0 0.3661 0.482 0 0 0 0 0 0 0 02 0 0.840 0 0 0 0 0 0 03 0 0 0.840 0 0 0 0 0 04 0 0 0 0.840 0 0 0 0 05 0 0 0 0 0.906 0 0 0 06 0 0 0 0 0 0.906 0 0 07 0 0 0 0 0 0 0.945 0 08 0 0 0 0 0 0 0 0.945 0.916


Gsa

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6. Use Model to Answer QuestionsA. Is the population growing?

B. Which vital rate(s) should managers focus monitoring and conservation effort on?

C. Which stage(s) should managers focus monitoring and conservation effort on?

D. Which stage(s) contain most of the population?

E. How might management actions impact population?

Transition Matrix

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n1 P1 F2 F3 F4 F5 F6 n1

n2 G1 P2 0 0 0 0 n2

n3 =

0 G2 P3 0 0 0 n3

n4 0 0 G3 P4 0 0 n4

n5 0 0 0 G4 P5 0 n5

n6 0 0 0 0 G5 P6 n6

tt+1

A. Is the population growing?

Math: multiplication of transition matrix & initial population vectorPopTools: Matrix Tools → Matrix Projection

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Popu

latio

n Si

zeA. Is the population growing?

Time

50


Population Growth Rate (λ) • λ = Nt+1 / Nt

• If λ = 1 , Nt+1 = Nt → population size

• If λ > 1, Nt+1 > Nt → population size

• If λ < 1, Nt+1 < Nt → population size

Math: dominant eigenvalue of transition matrixPopTools: = DomEig(TM)

is constant

is increasing

is decreasing

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Time (Months)

# Li

onfis

h

Morris et al. (2011) Biological Invasions

λ = 1.134


0 4 8 12 16 20 24 28 32 36 40 44 480

250,000

500,000

750,000

1,000,000

1,250,000

1,500,000

1,750,000

Time (years)

Num

ber o

f Ind

ivid

uals

in P

opu-

latio

n

A. Is the population growing? [After many years with these vital rates]

With initial population vector based on stable stage distributionWith initial population vector not based on stable stage distribution

λ = 0.995

Elasticity= Effect that a change in each one of the vital

rates has on population growth rate λ • Used to determine which vital rate changes

the population growth rate λ most


Math: , where <> denotes the scalar productPopTools: Matrix Tools → Elasticity


Elasticity:

Corresponds to Pa

0 0 0 0 0 0 0 0 0.048720.04872 0 0 0 0 0 0 0 0

0 0.04872 0 0 0 0 0 0 00 0 0.04872 0 0 0 0 0 00 0 0 0.04872 0 0 0 0 00 0 0 0 0.04872 0 0 0 00 0 0 0 0 0.04872 0 0 00 0 0 0 0 0 0.04872 0 00 0 0 0 0 0 0 0.04872 0.56151

00.10.20.30.40.50.6

Elas

ticity

Vital Rate

Elasticity

Gone 0.04872Gj 0.04872Gi 0.04872Gsa 0.04872Ga 0.56152F 0.04872


Gone Gj Gi Gsa Ga F

Reproductive Value= expected number of offspring that remain to be born

to each individual of a stage

• Essentially which stages are most valuable for future population growth

• Math: left eigenvector of transition matrix• PopTools: Matrix Tools → Age Distribution

v(x) =

# of offspring produced by individuals of age x or older (discounted by likelihood of surviving to reproduce)

# of individuals of age x


Egg 1 2 3 4 5 6 7 80.000.020.040.060.080.100.120.140.160.18

Stage

Repr

oduc

tive

Valu

e

Stage Reproductive Value

Egg 0.035

1 0.0718

2 0.0851

3 0.1009

4 0.1195

5 0.1314

6 0.1443

7 0.1520

8 0.1601


Reproductive Value• Humans?

(based on Daly and Wilson 1988 Homicide)

• To maximize sustainable harvest yield, harvest stages with reproductive value

• To maximize impact of restoration, transplant individuals with reproductive value

• Natural selection will act most strongly on stages with reproductive value

low

high

high

RV

Age

Gotelli 2008 A Primer of Ecology, Caswell 1980 Ecology

D. Which stage(s) contain most of the population?

0 3 6 9 12 15 18 21 24 27 300

200,000

400,000

600,000

800,000

1,000,000

1,200,000

Stage E

Stage 1

Stage 2

Stage 3

Stage 4

Stage 5

Stage 6

Stage 7

Stage 8Time (years)

Num

ber o

f Ind

ivid

uals

in E

ach

Stag

e

PopTools: Matrix Tools → Matrix Projection

0 3 6 9 12 15 18 21 24 27 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Stage EStage 1Stage 2Stage 3Stage 4Stage 5Stage 6Stage 7Stage 8Time (years)

Prop

ortio

n of

Indi

vidu

als

in

Each

Sta

geD. Which stage(s) contain most of the population?

[After many years with these vital rates]

Stable Stage Distribution

Stable Stage Distribution= constant proportion of individuals in each stage

(derived from constant vital rates)

• Math: Right eigenvector of transition matrix• PopTools: Matrix Tools → Age Distribution

w(x) = # of individuals of age x in SSD

total # of individuals

D. Which stage(s) contain most of the population? [After many years with these vital rates]

Stage Stable Stage Distribution

Egg 0.1671 0.0812 0.0683 0.0584 0.0495 0.0446 0.0417 0.0388 0.455

D. Which stage(s) contain most of the population? [After many years with these vital rates]

Egg

1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

Stage

Stab

le S

tage

Dis

trib

ution

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• How does lead poisoning of chicks affect population growth?

• Will lead remediation help protect the population?

E. How might management actions impact population?

SIMULATION – What if . . .

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How does lead poisoning of chicks affect population growth?

E 5421 3 6 7 8

65Finkelstein et al. 2009 Animal Conservation

How does lead poisoning of chicks affect population growth?

3.5%7%14%

Lead-induced chick mortality

0%

66

starting in 21 years

Will lead remediation help protect the population?

Lead Remediation

1% Reduction in Adult Bycatch Mortality

Finkelstein et al. 2009 Animal Conservation

No Action

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Result: Conservation Action

For more info: http://www.fws.gov/midway/lpa.html

Pete Leary / USFWS USFWS

http://www.fws.gov/midway/lpa.html

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Discussion: How do we use and why do we need mathematical models in ecology and

conservation?

Objectives• Learn how scientists use structured population

models to assess and protect populations

• Appreciate the importance and uses of mathematical models in ecology and conservation

70

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Lab: Cooperative Activity Using PopTools

Dwayne Meadows, NOAA/NMFS/OPR

John Snow

Chris Muenzer

Using PopTools Add-on to MS Excelhttp://www.poptools.org/

http://www.poptools.org/

Resources• Crouse, D. T., Crowder, L. B., & Caswell, H. (1987). A stage-

based population model for loggerhead sea turtles and implications for conservation. Ecology 68(5):1412-1423. http://dx.doi.org/10.2307/1939225

• Finkelstein, M. E., Doak, D. F., Nakagawa, M., Sievert, P. R., & Klavitter, J. (2010). Assessment of demographic risk factors and management priorities: impacts on juveniles substantially affect population viability of a long-lived seabird. Animal Conservation, 13(2), 148-156. http://dx.doi.org/10.1111/j.1469-1795.2009.00311.x

• Gotelli, N. (Various) A Primer of Ecology. Chapter 3: Age-Structured Population Growth. Sinauer.

http://dx.doi.org/10.2307/1939225

http://dx.doi.org/10.2307/1939225

http://dx.doi.org.oca.ucsc.edu/10.1111/j.1469-1795.2009.00311.x

http://dx.doi.org.oca.ucsc.edu/10.1111/j.1469-1795.2009.00311.x

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Questions?

Prettier photo?

Documents

( 2013 Kristin McCully & Myra Finkelstein) Lead Poisoning of Laysan Albatrosses at Midway Atoll: From Ecological Model to Conservation Action