Structured PVA

Final Essay: possible topics

Historical essay: for example history of protection of Everglades

Concern: Run-off of oil-products from streets/roads

Management plan: how to manage the Wakulla river

Protection of an endangered species

Each essay needs at least 5 citations from the peer-reviewed literature (no websites!). The essay will use these citations to show facts, etc. In the reference list, these 5 used papers need to have a short summary of the paper (about half a page).

Example titles from last semester

Coral reef resilience and susceptibility due to human interference

The ripple effect: the consequences of biological control

Overfishing: without immediate reform the problems of yesterday will be here to stay

Grizzly bear population management and Grizzly bear-human conflict

Conservation efforts towards proper medical waste disposal

Endangered species protection and HIV research

Each essay needs at least 5 citations from the peer-reviewed literature (no websites!). The essay will use these citations to show facts, etc. In the reference list, these 5 used papers need to have a short summary of the paper (about half a page).

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

Survival rate depends on age

Hydra

Plant fecundity depends on size

Ln(n

um

ber

of se

eds)

Plant size

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

Structured (demographic) models

Age-structured - use data on each age group

Structured (demographic) models

Age-structured - use data on each age group

Stage structured - used data on size or stage groups

Tadpoles

Juveniles

Adults

0 25 50 75 100

Individuals

< 20 cm

20 < x < 40 cm

> 40 cm

0 12.5 25.0 37.5 50.0

Individuals

Building a stage structured model

Understand your species

Decide how many stages to include

Building a stage structured model(for loggerhead sea turtles)

Building a stage structured model(for loggerhead sea turtles)

nesting on beaches

matingnear shore

foraging

open ocean

How many stages to include?

Biological Intuition - stages should differ in vital rates from other stages

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

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

Nestlings

Small

juveniles

Life-cycle diagram

Nestlings

Small

juveniles

Large

juveniles

Subadults

Adults

Life-cycle diagram

StageTransition rate

Nestlings

Small

juveniles

Large

juveniles

Subadults

Adults

Life-cycle diagram

Building a stage structured model

Understand your species

Decide how many stages to include

Gather data

NestlingsSmall

juvenilesLarge

juvenilesSubadults

Mature adults

Marked in year 1 1000 1000 1000 1000 1000

Recaptured in same class

0 703 657 682 809

Recaptured in next larger class

675 47 19 61 -

Eggs/female/year 0 0 0 4.665 61.896

Turtle data

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 femaleNestlings

Small

juveniles

Large

juveniles

Subadults

Adults

Life-cycle diagram

NestlingsSmall

juvenilesLarge

juvenilesSubadults

Mature adults

Marked in year 1 1000 1000 1000 1000 1000

Recaptured in same class

0 703 657 682 809

Recaptured in next larger class

675 47 19 61 -

Eggs/female/year 0 0 0 4.665 61.896

Turtle data

Nestlings

Small

juveniles

Large

juveniles

Subadults

Adults

Life-cycle diagram

0.675

Nestlings

Small

juveniles

Large

juveniles

Subadults

Adults

Life-cycle diagram

0.675

NestlingsSmall

juvenilesLarge

juvenilesSubadults

Mature adults

Marked in year 1 1000 1000 1000 1000 1000

Recaptured in same class

0 703 657 682 809

Recaptured in next larger class

675 47 19 61 -

Eggs/female/year 0 0 0 4.665 61.896

Turtle data

Nestlings

Small

juveniles

Large

juveniles

Subadults

Adults

Life-cycle diagram

0.703

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

Building a stage structured model

Understand your species

Decide how many stages to include

Gather data

Calculate transition rates

Make model

Population (Projection) matrix

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

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) matrixA generic projection matrix

!

"

"

"

"

#

S1 F2 F3 F4 F5

G1 S2 0 0 0

0 G2 S3 0 0

0 0 G3 S4 0

0 0 0 G4 S5

$

%

%

%

%

&

Size this year1 2 3 4 5

1

2

3

4

5

Size next year

Fi

new

Si

surviving

Giadvancing

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.

Projection matrix for loggerhead sea turtles

Size this year1 2 3 4 5

1

2

3

4

5

Size next year

!

"

"

"

"

#

0 0 0 4.665 61.896

0.675 0.703 0 0 0

0 0.047 0.657 0 0

0 0 0.019 0.682 0

0 0 0 0.061 0.8091

$

%

%

%

%

&

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

recall count based method

Nt = !Nt!1

Structured model

Nt = PNt!1

Stage distribution vector

a column showing the number (or density) of individuals in each stage

!

"

"

"

"

#

23.85

64.78

10.33

0.73

0.31

$

%

%

%

%

&

NestlingsSmall juvenilesLarge juvenilesSubadultsAdults

100.00 Total density

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

other stages

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

other stages

Stage distribution will converge to the stable stage distribution over time

Nt = PNt!1

!

"

"

"

"

#

?

$

%

%

%

%

&

=

!

"

"

"

"

#

0 0 0 4.665 61.896

0.675 0.703 0 0 0

0 0.047 0.657 0 0

0 0 0.019 0.682 0

0 0 0 0.061 0.8091

$

%

%

%

%

&

!

"

"

"

"

#

23.85

64.78

10.33

0.73

0.31

$

%

%

%

%

&

Nt P Nt!1

Use matrix algebra.....

Nt P Nt!1

!

"

"

"

"

#

22.59

61.64

9.83

0.69

0.30

$

%

%

%

%

&

=

!

"

"

"

"

#

0 0 0 4.665 61.896

0.675 0.703 0 0 0

0 0.047 0.657 0 0

0 0 0.019 0.682 0

0 0 0 0.061 0.8091

$

%

%

%

%

&

!

"

"

"

"

#

23.85

64.78

10.33

0.73

0.31

$

%

%

%

%

&

Time

#

EggsJuveniles

Large juvenilesSubadults

Adults

EggsJuveniles

Large juvenilesSubadults

Adults

Same graph as last slide, but changing

scale on y-axis

Time

#

EggsJuveniles

Large juvenilesSubadults

Adults

Stable stage distribution

Time

Freq

Nt P Nt!1

How do we know if population is growing or shrinking?

!

"

"

"

"

#

22.59

61.64

9.83

0.69

0.30

$

%

%

%

%

&

=

!

"

"

"

"

#

0 0 0 4.665 61.896

0.675 0.703 0 0 0

0 0.047 0.657 0 0

0 0 0.019 0.682 0

0 0 0 0.061 0.8091

$

%

%

%

%

&

!

"

"

"

"

#

23.85

64.78

10.33

0.73

0.31

$

%

%

%

%

&

Recall that:

! =

Nt

Nt!1

Nt P Nt!1

!

"

"

"

"

#

22.59

61.64

9.83

0.69

0.30

$

%

%

%

%

&

=

!

"

"

"

"

#

0 0 0 4.665 61.896

0.675 0.703 0 0 0

0 0.047 0.657 0 0

0 0 0.019 0.682 0

0 0 0 0.061 0.8091

$

%

%

%

%

&

!

"

"

"

"

#

23.85

64.78

10.33

0.73

0.31

$

%

%

%

%

&

95.05 100.0

95.05/100 = 0.9505 = !

Time

lambda

! again

Nt = !Nt!1

Nt = PNt!1

In a count based model

In a structured model

P is playing the same role as the count based !.

! again

Nt = !Nt!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)

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

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

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 = !X

(right) eigenvector

eigenvalues

Using the turtle model for PVA

Beaches (nestlings)

Ocean (juveniles, subadults, adults)

Sources of turtle mortality:

Predation of eggs by racoons, dogs, and lizards, among others

Hatchlings emerging at night (fish, crabs)

Hatchlings emerging at day (sea birds)

Beach lights affects hatchlings

Threats to juveniles and adults Using the turtle model for PVA

Beaches (nestlings)

Ocean (juveniles, subadults, adults)

Sources of turtle mortality:

Status: population is declining (!=0.951)

Decline of loggerhead turtle

5 10 15 20

50

60

70

80

90

Years

Tota

l den

sity

of

logg

erhea

d

Using the PVA

Can we stop this decline of loggerhead turtle populations?

What if we protect all turtles on the beach?

What element would protecting nestlings on the beach change?

Size this year1 2 3 4 5

1

2

3

4

5

Size next year

!

"

"

"

"

#

0 0 0 4.665 61.896

0.675 0.703 0 0 0

0 0.047 0.657 0 0

0 0 0.019 0.682 0

0 0 0 0.061 0.8091

$

%

%

%

%

&

What element would protecting nestlings on the beach change?

Size this year1 2 3 4 5

1

2

3

4

5

Size next year

!

"

"

"

"

#

0 0 0 4.665 61.896

0.675 0.703 0 0 0

0 0.047 0.657 0 0

0 0 0.019 0.682 0

0 0 0 0.061 0.8091

$

%

%

%

%

&

1.00

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

5 10 15 20

50

60

70

80

90

100

Decline of loggerhead turtles

Years

Tota

l den

sity

of

logg

erhea

d

Protected beach

No protection

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?

Turtle excluder device (TED)

What element change would protecting large juveniles ?

Size this year1 2 3 4 5

1

2

3

4

5

Size next year

!

"

"

"

"

#

0 0 0 4.665 61.896

0.675 0.703 0 0 0

0 0.047 0.657 0 0

0 0 0.019 0.682 0

0 0 0 0.061 0.8091

$

%

%

%

%

&

What element change would protecting large juveniles ?

Size this year1 2 3 4 5

1

2

3

4

5

Size next year

!

"

"

"

"

#

0 0 0 4.665 61.896

0.675 0.703 0 0 0

0 0.047 0.657 0 0

0 0 0.019 0.682 0

0 0 0 0.061 0.8091

$

%

%

%

%

&

25%

25%

What element change would protecting large juveniles ?

Size this year1 2 3 4 5

1

2

3

4

5

Size next year

!

"

"

"

"

#

0 0 0 4.665 61.896

0.675 0.703 0 0 0

0 0.047 0.657 0 0

0 0 0.019 0.682 0

0 0 0 0.061 0.8091

$

%

%

%

%

&

0.821

0.024

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

INCREASE of loggerhead turtles

Years

Tota

l den

sity

of

logg

erhea

d

Protected beach

No protection

5 10 15 20

60

80

100

120

140

160TED and beach