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
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