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Loggerhead Sea Turtle Conservation Ben Marshall and Alex Varner December 16, 2009 Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation Table: Stage Based Life Table for Loggerhead Sea Turtles Stage Class Size(cm) Ages(yr)

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SLIDE 1

Loggerhead Sea Turtle Conservation

Ben Marshall and Alex Varner December 16, 2009

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 2

Table: Stage Based Life Table for Loggerhead Sea Turtles

Stage Class Size(cm) Ages(yr) Survival Fec. 1 eggs/hatchlings < 10 < 1 .6747 2

  • sm. juveniles

10.1-58 1-7 .7857 3

  • lg. juveniles

58.1-80 8-15 .6758 4 subadults 80.1-87 16-21 .7425 5 novice breeders > 87 22 .8091 127 6 1st-yr remigrants > 87 23 .8091 4 7 mature breeders > 87 24-54 .8091 80

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 3

Survival and Fecundity

◮ Pi

◮ The probability of survival while in a stage i Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 4

Survival and Fecundity

◮ Pi

◮ The probability of survival while in a stage i

◮ Fi

◮ Birth rates for each stage Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 5

Stage 1 Population

◮

n(t+1)

1

= P1n(t)

1 + F2n(t) 2 · · · + F7n(t) 7

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 6

Stage 1 Population

◮

n(t+1)

1

= P1n(t)

1 + F2n(t) 2 · · · + F7n(t) 7

◮ Survivors and hatchlings

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 7

Matrix Model

◮ n(t+1) 1

= P1n(t)

1 + F2n(t) 2 · · · + F7n(t) 7

         P1 F2 F3 F4 F5 F6 F7                     n(t)

1

n(t)

2

n(t)

3

n(t)

4

n(t)

5

n(t)

6

n(t)

7

            =           n(t+1)

1

         

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 8

Survival and Fecundity

◮ Pi

◮ The probability of survival while remaining in

same stage

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 9

Survival and Fecundity

◮ Pi

◮ The probability of survival while remaining in

same stage

◮ Fi

◮ Birth rates for each stage Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 10

Survival and Fecundity

◮ Pi

◮ The probability of survival while remaining in

same stage

◮ Fi

◮ Birth rates for each stage

◮ Gi

◮ Probability of advancing to the next

stage

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 11

Stage 2 Population

◮

n(t+1)

2

= G1n(t)

1 + P2n(t) 2

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 12

Stage 2 Population

◮

n(t+1)

2

= G1n(t)

1 + P2n(t) 2

◮ New stage members and surviving stage

members

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 13

Matrix Model cont.

◮ n(t+1) 2

= G1n(t)

1 + P2n(t) 2

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 14

Matrix Model cont.

◮ n(t+1) 2

= G1n(t)

1 + P2n(t) 2 ◮

          P1 F2 F3 F4 F5 F6 F7 G1 P2                        n(t)

1

n(t)

2

n(t)

3

n(t)

4

n(t)

5

n(t)

6

n(t)

7

             =            n(t+1)

1

n(t+1)

2

          

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 15

Final Matrix Model

◮

          P1 F2 F3 F4 F5 F6 F7 G1 P2 G2 P3 G3 P4 G4 P5 G5 P6 G6 P7                        n(t)

1

n(t)

2

n(t)

3

n(t)

4

n(t)

5

n(t)

6

n(t)

7

             =              n(t+1)

1

n(t+1)

2

n(t+1)

3

n(t+1)

4

n(t+1)

5

n(t+1)

6

n(t+1)

7

            

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 16

Calculating Fi

◮ Fi is easy.

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 17

Calculating Fi

◮ Fi is easy.

Table: Stage Based Life Table for Loggerhead Sea Turtles

◮

Stage Class Size(cm) Ages(yr) Survival Fec. 1 eggs/hatchlings < 10 < 1 .6747 2

  • sm. juveniles

10.1-58 1-7 .7857 3

  • lg. juveniles

58.1-80 8-15 .6758 4 subadults 80.1-87 16-21 .7425 5 novice breeders > 87 22 .8091 127 6 1st-yr remigrants > 87 23 .8091 4 7 mature breeders > 87 24-54 .8091 80

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 18

Calculating Fi

◮ Fi is easy.

Table: Stage Based Life Table for Loggerhead Sea Turtles

◮

Stage Class Size(cm) Ages(yr) Survival Fec. 1 eggs/hatchlings < 10 < 1 .6747 2

  • sm. juveniles

10.1-58 1-7 .7857 3

  • lg. juveniles

58.1-80 8-15 .6758 4 subadults 80.1-87 16-21 .7425 5 novice breeders > 87 22 .8091 127 6 1st-yr remigrants > 87 23 .8091 4 7 mature breeders > 87 24-54 .8091 80

◮ Found in far right column.

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 19

Stage Specific Survival pi

◮ Each stage has specific survival rate, found in

table

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 20

Stage Specific Survival pi

◮ Each stage has specific survival rate, found in

table

Table: Stage Based Life Table for Loggerhead Sea Turtles

◮

Stage Class Size(cm) Ages(yr) Survival Fec. 1 eggs/hatchlings < 10 < 1 .6747 2

  • sm. juveniles

10.1-58 1-7 .7857 3

  • lg. juveniles

58.1-80 8-15 .6758 4 subadults 80.1-87 16-21 .7425 5 novice breeders > 87 22 .8091 127 6 1st-yr remigrants > 87 23 .8091 4 7 mature breeders > 87 24-54 .8091 80

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 21

Calculating Gi

Table: Population and Survival

◮

Pop. Years M 1 Mpi 2 Mp2

i

3 Mp3

i

4 Mp4

i

5 Mp5

i

6 Mp6

i

7

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 22

Calculating Gi

Table: Population and Survival

◮

Pop. Years M 1 Mpi 2 Mp2

i

3 Mp3

i

4 Mp4

i

5 Mp5

i

6 Mp6

i

7

◮ Eligible:

Mp5

i

M + Mpi + Mp2

i + Mp3 i + Mp4 i + Mp5 i

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 23

Calculating Gi cont.

◮ Proportion:

  • p5

i

1 + pi + p2

i + p3 i + p4 i + p5 i

  • Ben Marshall and Alex Varner

Loggerhead Sea Turtle Conservation

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SLIDE 24

Calculating Gi cont.

◮ Proportion:

  • p5

i

1 + pi + p2

i + p3 i + p4 i + p5 i

  • ◮ Hit this again with pi

◮ Yields the percentage that actually advance. Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 25

Calculating Gi cont.

◮ Proportion:

  • p5

i

1 + pi + p2

i + p3 i + p4 i + p5 i

  • ◮ Hit this again with pi

◮ Yields the percentage that actually advance.

◮

Gi =

  • p6

i

1 + pi + p2

i + p3 i + p4 i + p5 i

  • Ben Marshall and Alex Varner

Loggerhead Sea Turtle Conservation

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SLIDE 26

Generalizing Gi

◮

Gi =

  • p5

i

1 + pi + p2

i + p3 i + p4 i + p5 i

  • pi

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 27

Generalizing Gi

◮

Gi =

  • p5

i

1 + pi + p2

i + p3 i + p4 i + p5 i

  • pi

◮ Gi generalized: di represents stage duration.

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 28

Generalizing Gi

◮

Gi =

  • p5

i

1 + pi + p2

i + p3 i + p4 i + p5 i

  • pi

◮ Gi generalized: di represents stage duration. ◮

Gi =

  • pdi−1

i

1 + pi + p2

i + · · · + pdi−1 i

  • pi

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 29

Generalizing Gi

◮

Gi =

  • p5

i

1 + pi + p2

i + p3 i + p4 i + p5 i

  • pi

◮ Gi generalized: di represents stage duration. ◮

Gi =

  • pdi−1

i

1 + pi + p2

i + · · · + pdi−1 i

  • pi

◮

Gi = (1 − pi)pdi

i

1 − pdi

i

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 30

Calculating Pi

Table: Population and Survival

◮

Pop. Years M Mpi 1 Mp2

i

2 Mp3

i

3 Mp4

i

4 Mp5

i

5 Mp6

i

6

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 31

Calculating Pi

Table: Population and Survival

◮

Pop. Years M Mpi 1 Mp2

i

2 Mp3

i

3 Mp4

i

4 Mp5

i

5 Mp6

i

6

◮ Mp5 i are eligible to advance

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 32

Calculating Pi

Table: Population and Survival

◮

Pop. Years M Mpi 1 Mp2

i

2 Mp3

i

3 Mp4

i

4 Mp5

i

5 Mp6

i

6

◮ Mp5 i are eligible to advance ◮ then M + Mpi + Mp2

i + Mp3 i + Mp4 i are ineligible.

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 33

Calculating Pi cont.

◮ M + Mpi + Mp2 i + Mp3 i + Mp4 i are ineligible to advance

◮ Only option for survival is remain in class. Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 34

Calculating Pi cont.

◮ M + Mpi + Mp2 i + Mp3 i + Mp4 i are ineligible to advance

◮ Only option for survival is remain in class.

◮ Proportion:

1 + pi + p2

i + p3 i + p4 i

1 + pi + p2

i + p3 i + p4 i + p5 i

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 35

Calculating Pi cont.

◮ M + Mpi + Mp2 i + Mp3 i + Mp4 i are ineligible to advance

◮ Only option for survival is remain in class.

◮ Proportion:

1 + pi + p2

i + p3 i + p4 i

1 + pi + p2

i + p3 i + p4 i + p5 i

◮ Hit this with pi

◮ Percentage that actually survive and remain. Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 36

Generalizing Pi

◮

Pi =

  • 1 + pi + p2

i + p3 i + p4 i

1 + pi + p2

i + p3 i + p4 i + p5 i

  • pi

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 37

Generalizing Pi

◮

Pi =

  • 1 + pi + p2

i + p3 i + p4 i

1 + pi + p2

i + p3 i + p4 i + p5 i

  • pi

◮ Pi generalized: di represents stage duration.

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 38

Generalizing Pi

◮

Pi =

  • 1 + pi + p2

i + p3 i + p4 i

1 + pi + p2

i + p3 i + p4 i + p5 i

  • pi

◮ Pi generalized: di represents stage duration. ◮

Pi =

  • 1 + pi + p2

i + · · · + pdi−2 i

1 + pi + p2

i + · · · + pdi−1 i

  • pi

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 39

Generalizing Pi

◮

Pi =

  • 1 + pi + p2

i + p3 i + p4 i

1 + pi + p2

i + p3 i + p4 i + p5 i

  • pi

◮ Pi generalized: di represents stage duration. ◮

Pi =

  • 1 + pi + p2

i + · · · + pdi−2 i

1 + pi + p2

i + · · · + pdi−1 i

  • pi

◮

Pi =

  • 1 − pdi−1

i

1 − pdi

i

  • pi

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 40

Final Matrix

          127 4 80 .6747 .7371 .0486 .6611 .0147 .6907 .0518 .8091 .8091 .8088                       n(t)

1

n(t)

2

n(t)

3

n(t)

4

n(t)

5

n(t)

6

n(t)

7

            =             n(t+1)

1

n(t+1)

2

n(t+1)

3

n(t+1)

4

n(t+1)

5

n(t+1)

6

n(t+1)

7

           

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 41

Population After 50 Years

◮

10 20 30 40 50 10

−2

10 10

2

10

4

10

6

Time (years) Population 1st 2nd 3rd 4th 5th 6th 7th

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 42

Diagonalizing

◮ Call matrix model L

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 43

Diagonalizing

◮ Call matrix model L ◮

Lk = V ΛkV −1

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 44

Diagonalizing

◮

xk = Lk x0

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 45

Diagonalizing

◮

xk = Lk x0

◮

xk =

  • V ΛkV −1
  • x0

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 46

Diagonalizing

◮

xk = Lk x0

◮

xk =

  • V ΛkV −1
  • x0

◮ V −1

x0 = c

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 47

Diagonalizing

◮

xk = Lk x0

◮

xk =

  • V ΛkV −1
  • x0

◮ V −1

x0 = c

◮

xk = c1λk

1

v1 + · · · + cnλk

n

vn

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 48

Eigenvalues

Table: Eigenvalues

Gi = 1 λ1 λ2 λ3 λ4 λ5 λ6 λ7 Stage 1 −.100 −.100 .317 .317 .750 .750 .963 Stage 2 −.230 −.230 .320 .320 1.16 .778 .778 Stage 3 −.320 −.320 1.30 .791 .791 .336 .336 Stage 4 −.225 −.225 .320 .320 1.15 .775 .775 Stage 5 −.090 −.090 .314 .314 .753 .753 .953 Stage 6 −.095 −.095 .314 .314 .753 .753 .953

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 49

Stage 3 Eigenvalues

◮ Using λi for Stage 3 from eigenvalue table

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 50

Stage 3 Eigenvalues

◮ Using λi for Stage 3 from eigenvalue table ◮ k = 50

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 51

Stage 3 Eigenvalues

◮ Using λi for Stage 3 from eigenvalue table ◮ k = 50 ◮

x(50) = c1(−.32)(50) v1 +c2(−.32)(50) v2 +c3(1.3)(50) v3 . . .

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 52

Stage 3 Eigenvalues

◮ Using λi for Stage 3 from eigenvalue table ◮ k = 50 ◮

x(50) = c1(−.32)(50) v1 +c2(−.32)(50) v2 +c3(1.3)(50) v3 . . .

◮ When λi < 1 then λk i → 0

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 53

Stage 3 Eigenvalues

◮ Using λi for Stage 3 from eigenvalue table ◮ k = 50 ◮

x(50) = c1(−.32)(50) v1 +c2(−.32)(50) v2 +c3(1.3)(50) v3 . . .

◮ When λi < 1 then λk i → 0 ◮

x(50) ≈ c3λ50

3

v3

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 54

G3 Set to 100%

◮

10 20 30 40 50 10 10

2

10

4

10

6

10

8

10

10

Time(years) Population Ratio 1st 2nd 3rd 4th 5th 6th 7th

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

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SLIDE 55

G3 Set to 4%

◮

10 20 30 40 50 10 10

1

10

2

10

3

10

4

10

5

Time(years) Population Ratio 1st 2nd 3rd 4th 5th 6th 7th

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

slide-56
SLIDE 56

G3 Set to 5%

◮

10 20 30 40 50 10 10

1

10

2

10

3

10

4

10

5

Time(years) Population Ratio 1st 2nd 3rd 4th 5th 6th 7th

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

slide-57
SLIDE 57

Population After 50 Years

◮

10 20 30 40 50 10

−2

10 10

2

10

4

10

6

Time (years) Population 1st 2nd 3rd 4th 5th 6th 7th

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

slide-58
SLIDE 58

The Bibliography

Deborah Crouse, Larry Crowder, Hal Caswell, 1987 A Stage Based Population Model For Loggerhead Sea Turtles And Implications For Conservation Wikipedia, Oct. 1, 2009 Loggerhead Sea Turtles David Arnold, Kevin Yokoyama, The Leslie Matrix David Arnold, Kevin Yokoyama, The Leslie Matrix Part Two

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation