Structured PVA 1 Vital rates (Processes that contribute to change - - PowerPoint PPT Presentation

Structured PVA

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Birth and death rates Growth rate Fecundity

Vital rates

(Processes that contribute to change in population size)

Vital rates often depend on age and size

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Survival rate depends on age

Hydra

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Plant fecundity depends on size

Ln(number of seeds) Plant size

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Types of PVA’s

Count based: simple -- all individuals are the same (age, size, etc.) Structured (demographic): different vital rates for different classes of individuals

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Structured (demographic) models

Age-structured - use data on each age group

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Structured (demographic) models

Age-structured - use data on each age group Stage structured - used data on size or stage groups

Adults Juveniles Tadpoles 25 50 75 100

Individuals

> 40 cm 20 < x < 40 cm < 20 cm 12.5 25.0 37.5 50.0

Individuals

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Building a stage structured model

Understand your species Decide how many stages to include

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Building a stage structured model (for loggerhead sea turtles)

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Building a stage structured model (for loggerhead sea turtles)

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nesting on beaches mating near shore foraging

• pen ocean

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How many stages to include?

Biological Intuition - stages should differ in vital rates from

• ther stages

What the data will allow - balance accuracy of more stages with amount of available data

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For turtle PVA we use 5 stages

Hatchlings (and eggs): first year Small juveniles: 1-7 years Large juveniles: 8-15 years Subadults 16-21 years (mostly non-breeding) Mature adults 22-55 years, breeding

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Nestlings Small juveniles

Life-cycle diagram

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Nestlings Small juveniles Large juveniles Subadults Adults

Life-cycle diagram

Stage Transition rate

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Nestlings Small juveniles Large juveniles Subadults Adults

Life-cycle diagram

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Building a stage structured model

Understand your species Decide how many stages to include Gather data

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Nestlings Small juveniles Large juveniles Subadults Mature adults Marked in year 1 1000 1000 1000 1000 1000 Recaptured in same class 703 657 682 809 Recaptured in next larger class 675 47 19 61

• Eggs/female/year

4.665 61.896

Turtle data

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Building a stage structured model

Understand your species Decide how many stages to include Gather data Calculate transition rates Fractions surviving but not growing Fractions surviving and growing Number of female offspring per year and female

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Nestlings Small juveniles Large juveniles Subadults Adults

Life-cycle diagram

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Nestlings Small juveniles Large juveniles Subadults Mature adults Marked in year 1 1000 1000 1000 1000 1000 Recaptured in same class 703 657 682 809 Recaptured in next larger class 675 47 19 61

• Eggs/female/year

4.665 61.896

Turtle data

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Nestlings Small juveniles Large juveniles Subadults Adults

Life-cycle diagram

0.675

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Nestlings Small juveniles Large juveniles Subadults Adults

Life-cycle diagram

0.675

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Nestlings Small juveniles Large juveniles Subadults Mature adults Marked in year 1 1000 1000 1000 1000 1000 Recaptured in same class 703 657 682 809 Recaptured in next larger class 675 47 19 61

• Eggs/female/year

4.665 61.896

Turtle data

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Nestlings Small juveniles Large juveniles Subadults Adults

Life-cycle diagram

0.703

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Nestlings Small juveniles Large juveniles Subadults Adults

Life-cycle diagram

0.703 0.675 0.703 0.047 0.657 0.019 0.682 0.061 0.809 4.665 61.896

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Building a stage structured model

Understand your species Decide how many stages to include Gather data Calculate transition rates Make model

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Population (Projection) matrix

The projection matrix is the summary of all transition probabilities (all vital rates)

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Fi

Number of new turtles (size class 1) produces by an average individual of size i per year

Si

Fraction of size i turtles surviving and STAYING in the same size class per year

Gi

Fraction of size i turtles surviving and GROWING to size class i+1 per year

Population (Projection) matrix

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A generic projection matrix

      S1 F2 F3 F4 F5 G1 S2 G2 S3 G3 S4 G4 S5      

Size this year 1 2 3 4 5 1 2 3 4 5 Size next year

Fi new Si surviving Gi advancing

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Fi

Number of new turtles (size class 1) produces by an average individual of size i per year

Si

Fraction of size i turtles surviving and STAYING in the same size class per year

Gi

Fraction of size i turtles surviving and GROWING to size class i+1 per year

Population (Projection) matrix

Note that since S and G are fractions surviving. They are between 0 and 1.

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Projection matrix for loggerhead sea turtles

Size this year 1 2 3 4 5 1 2 3 4 5 Size next year

      4.665 61.896 0.675 0.703 0.047 0.657 0.019 0.682 0.061 0.8091      

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Nestlings Small juveniles Large juveniles Subadults Adults

Life-cycle diagram

0.703 0.675 0.703 0.047 0.657 0.019 0.682 0.061 0.809 4.665 61.896

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recall count based method

Nt = λNt−1

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Structured model

Nt = PNt−1

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Stage distribution vector

a column showing the number (or density)

• f individuals in each stage

      23.85 64.78 10.33 0.73 0.31       Nestlings Small juveniles Large juveniles Subadults Adults

100.00 Total density

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Stable stage (or age or size) distribution

distribution of individuals among stages that won’t change over time (if population size changes at a constant rate) Example: 100% of individuals in stage 1 is not stable – the next year there will be individuals in

• ther stages

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Stable stage (or age or size) distribution

distribution of individuals among stages that won’t change over time (if population size changes at a constant rate) Example: 100% of individuals in stage 1 is not stable – the next year there will be individuals in

• ther stages

Stage distribution will converge to the stable stage distribution over time

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Nt = PNt−1

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      ?       =       4.665 61.896 0.675 0.703 0.047 0.657 0.019 0.682 0.061 0.8091             23.85 64.78 10.33 0.73 0.31      

Nt P Nt−1

Use matrix algebra.....

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Nt P Nt−1

      22.59 61.64 9.83 0.69 0.30       =       4.665 61.896 0.675 0.703 0.047 0.657 0.019 0.682 0.061 0.8091             23.85 64.78 10.33 0.73 0.31      

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Time # Eggs Juveniles Large juveniles Subadults Adults

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Eggs Juveniles Large juveniles Subadults Adults Same graph as last slide, but changing scale on y-axis Time #

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Eggs Juveniles Large juveniles Subadults Adults Stable stage distribution Time Freq

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Nt P Nt−1

How do we know if population is growing or shrinking?

      22.59 61.64 9.83 0.69 0.30       =       4.665 61.896 0.675 0.703 0.047 0.657 0.019 0.682 0.061 0.8091             23.85 64.78 10.33 0.73 0.31      

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Recall that:

λ = Nt Nt−1

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Nt P Nt−1

      22.59 61.64 9.83 0.69 0.30       =       4.665 61.896 0.675 0.703 0.047 0.657 0.019 0.682 0.061 0.8091             23.85 64.78 10.33 0.73 0.31      

95.05 100.0 95.05/100 = 0.9505 =

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Time lambda

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again

Nt = λNt−1 Nt = PNt−1

In a count based model In a structured model P is playing the same role as the count based .

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again

Nt = λNt−1 Nt = PNt−1

In a count based model In a structured model P is playing the same role as the count based . The information in P can be summarized by a matrix (dominant eigenvalue)

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In structured models, change in N is still called but can be measured in two ways

Summarize the information P as a single number, the dominant eigenvalue .

Nt/Nt−1

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In structured models, change in N is still called but can be

Summarize the information P as a single number, the dominant eigenvalue .

Nt/Nt−1

This only will be constant if the population is at the stable stage distribution, variable until then

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In structured models, change in N is still called but can be

Summarize the information P as a single number, the dominant eigenvalue .

Nt/Nt−1

This only will be constant if the population is at the stable stage distribution, variable until then This will be constant as long as P doesn’t change

AX = λX

(right) eigenvector eigenvalues

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Using the turtle model for PVA

Beaches (nestlings) Ocean (juveniles, subadults, adults)

Sources of turtle mortality:

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Predation of eggs by racoons, dogs, and lizards, among others Hatchlings emerging at night (fish, crabs) Hatchlings emerging at day (sea birds)

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Beach lights affects hatchlings

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Threats to juveniles and adults

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Using the turtle model for PVA

Beaches (nestlings) Ocean (juveniles, subadults, adults)

Sources of turtle mortality: Status: population is declining (=0.951)

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Decline of loggerhead turtle

5 10 15 20 50 60 70 80 90

Years Total density of loggerhead

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Using the PVA

Can we stop this decline of loggerhead turtle populations? What if we protect all turtles on the beach?

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What element would protecting nestlings

• n the beach change?

Size this year 1 2 3 4 5 1 2 3 4 5 Size next year

      4.665 61.896 0.675 0.703 0.047 0.657 0.019 0.682 0.061 0.8091      

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What element would protecting nestlings

• n the beach change?

Size this year 1 2 3 4 5 1 2 3 4 5 Size next year

      4.665 61.896 0.675 0.703 0.047 0.657 0.019 0.682 0.061 0.8091      

1.00

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Using the turtle model for PVA

What if we protect turtles on the beach Change nestling survival to 100% (so G1=1) and turns to =0.974

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5 10 15 20 50 60 70 80 90 100

Decline of loggerhead turtles

Years Total density of loggerhead Protected beach No protection

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Using the turtle model for PVA

What if we protect turtles on the beach? Change nestling survival to 100% (so G1=1) and turns to =0.974 What happens if we protect larger turtles in the ocean?

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Turtle excluder device (TED)

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What element change would protecting large juveniles ?

Size this year 1 2 3 4 5 1 2 3 4 5 Size next year

      4.665 61.896 0.675 0.703 0.047 0.657 0.019 0.682 0.061 0.8091      

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What element change would protecting large juveniles ?

Size this year 1 2 3 4 5 1 2 3 4 5 Size next year

      4.665 61.896 0.675 0.703 0.047 0.657 0.019 0.682 0.061 0.8091      

25% 25%

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What element change would protecting large juveniles ?

Size this year 1 2 3 4 5 1 2 3 4 5 Size next year

      4.665 61.896 0.675 0.703 0.047 0.657 0.019 0.682 0.061 0.8091      

0.821 0.024

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Using the turtle model for PVA

What if we protect turtles on the beach? Change nestling survival to 100% (so G1=1) and the growth rate =0.974 What happens if we protect larger turtles in the ocean? Change mortality of large juvenile mortality by 25% and the growth rate =1.006

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INCREASE of loggerhead turtles

Years Total density of loggerhead Protected beach No protection

5 10 15 20 60 80 100 120 140 160

TED and beach

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Quantifying relative effects of protecting different stages

ELASTICITY = proportional change in as a results

• f proportional change in a particular stage (ri). It is a

proportional measure of sensitivity to change.

Eri =

λnew−λoriginal λoriginal ri,new−ri,original ri,original

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Quantifying relative effects of protecting different stages

ELASTICITY = proportional change in as a results

• f proportional change in a particular stage (ri).

Eri =

λnew−λoriginal λoriginal ri,new−ri,original ri,original

=

∆λ λoriginal ∆ri ri,original

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Size this year 1 2 3 4 5 1 2 3 4 5 Size next year

      4.665 61.896 0.675 0.703 0.047 0.657 0.019 0.682 0.061 0.8091      

1.00

Elasticity

0.675 NEW OLD

EG1 =

0.023 0.952 1−0.675 0.675

= 0.049

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Size this year 1 2 3 4 5 1 2 3 4 5 Size next year

      4.665 61.896 0.675 0.703 0.047 0.657 0.019 0.682 0.061 0.8091      

0.812

Elasticity

0.657 NEW OLD

ES3 =

0.041 0.952 0.821−0.657 0.657

= 0.171

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Effect of management

Improvement Elasticity Effect Hatchling survival 0.05 small Large juvenile survival (stay in stage class) 0.17 large Large juvenile survival (grow to subadults) 0.05 small

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Conservation outcome of this PVA

1987 TED required offshore May-August 1988-89 Court challenges, delay 1990 TEDs implemented seasonally 1993 Year-round TEDs offshore 1994 Year-round TEDs offshore and inshore

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Assumptions of structured PVA

(as discussed)

All individuals in a stage class are the same No density dependence does not change Only one population

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Density dependent survival

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Density dependent fecundity

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Density and growth rate

Nt = λNt−1

Time Number of individuals

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Density and Growth

Density with independent discrete growth Independent continuous growth

Nt = λNt−1 dN dt = rN

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Density and Growth

Density with independent discrete growth Independent continuous growth Density-dependent continuous growth

Nt = λNt−1 dN dt = rN dN dt = rN(1 − N K )

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Carrying capacity K

Carrying capacity K is the maximum stable population size

dN dt = rN(1 − N K )

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Density and carrying capacity

20 40 60 80 100 20 40 60 80 100

Time Population size Carrying capacity K

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Allee effect

Any mechanism that can lead to a positive relationship between individual fitness and numbers or densities of conspecifics

Antipredator aggression: Colonial Bluegill sunfish males spend less time chasing predators than solitary males Predator swamping: Cicadas, Mast-seeding in plants Modification of the environment: hemlock Social facilitation of reproduction: helpers in monkeys, birds.

Examples:

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Assumptions and remedies for PVAs

Individuals are not the same !"use structured model No density dependence !"add density dependence never change !"add stochasticity to model Only one population !"use meta-population model Stochasticity will add variability, for example environmental variability induces changes in vital rates

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