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) Survival Fec.

1 eggs/hatchlings < 10 < 1 .6747 02 sm. juveniles 10.1-58 1-7 .7857 03 lg. juveniles 58.1-80 8-15 .6758 04 subadults 80.1-87 16-21 .7425 05 novice breeders > 87 22 .8091 1276 1st-yr remigrants > 87 23 .8091 47 mature breeders > 87 24-54 .8091 80

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

Survival and Fecundity

I PiI The probability of survival while in a stage i

I FiI Birth rates for each stage

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

Survival and Fecundity

I PiI The probability of survival while in a stage i

I FiI Birth rates for each stage

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

Stage 1 Population

I

n(t+1)1 = P1n

(t)1 + F2n

(t)2 · · ·+ F7n

(t)7

I Survivors and hatchlings

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

Stage 1 Population

I

n(t+1)1 = P1n

(t)1 + F2n

(t)2 · · ·+ F7n

(t)7

I Survivors and hatchlings

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

Matrix Model

I 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

Survival and Fecundity

I PiI The probability of survival while remaining in

same stage

I FiI Birth rates for each stage

I GiI Probability of advancing to the next

stage

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

Survival and Fecundity

I PiI The probability of survival while remaining in

same stage

I FiI Birth rates for each stage

I GiI Probability of advancing to the next

stage

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

Survival and Fecundity

I PiI The probability of survival while remaining in

same stage

I FiI Birth rates for each stage

I GiI Probability of advancing to the next

stage

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

Stage 2 Population

I

n(t+1)2 = G1n

(t)1 + P2n

(t)2

I New stage members and surviving stagemembers

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

Stage 2 Population

I

n(t+1)2 = G1n

(t)1 + P2n

(t)2

I New stage members and surviving stagemembers

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

Matrix Model cont.

I n(t+1)2 = G1n

(t)1 + P2n

(t)2

I

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

Matrix Model cont.

I n(t+1)2 = G1n

(t)1 + P2n

(t)2

I

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

Final Matrix Model

I

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

Calculating Fi

I Fi is easy.

Table: Stage Based Life Table for Loggerhead Sea Turtles

I

Stage Class Size(cm) Ages(yr) Survival Fec.

1 eggs/hatchlings < 10 < 1 .6747 02 sm. juveniles 10.1-58 1-7 .7857 03 lg. juveniles 58.1-80 8-15 .6758 04 subadults 80.1-87 16-21 .7425 05 novice breeders > 87 22 .8091 1276 1st-yr remigrants > 87 23 .8091 47 mature breeders > 87 24-54 .8091 80

I Found in far right column.

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

Calculating Fi

I Fi is easy.

Table: Stage Based Life Table for Loggerhead Sea Turtles

I

Stage Class Size(cm) Ages(yr) Survival Fec.

1 eggs/hatchlings < 10 < 1 .6747 02 sm. juveniles 10.1-58 1-7 .7857 03 lg. juveniles 58.1-80 8-15 .6758 04 subadults 80.1-87 16-21 .7425 05 novice breeders > 87 22 .8091 1276 1st-yr remigrants > 87 23 .8091 47 mature breeders > 87 24-54 .8091 80

I Found in far right column.

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

Calculating Fi

I Fi is easy.

Table: Stage Based Life Table for Loggerhead Sea Turtles

I

Stage Class Size(cm) Ages(yr) Survival Fec.

1 eggs/hatchlings < 10 < 1 .6747 02 sm. juveniles 10.1-58 1-7 .7857 03 lg. juveniles 58.1-80 8-15 .6758 04 subadults 80.1-87 16-21 .7425 05 novice breeders > 87 22 .8091 1276 1st-yr remigrants > 87 23 .8091 47 mature breeders > 87 24-54 .8091 80

I Found in far right column.

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

Stage Specific Survival pi

I Each stage has specific survival rate, found intable

Table: Stage Based Life Table for Loggerhead Sea Turtles

I

Stage Class Size(cm) Ages(yr) Survival Fec.

1 eggs/hatchlings < 10 < 1 .6747 02 sm. juveniles 10.1-58 1-7 .7857 03 lg. juveniles 58.1-80 8-15 .6758 04 subadults 80.1-87 16-21 .7425 05 novice breeders > 87 22 .8091 1276 1st-yr remigrants > 87 23 .8091 47 mature breeders > 87 24-54 .8091 80

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

Stage Specific Survival pi

I Each stage has specific survival rate, found intable

Table: Stage Based Life Table for Loggerhead Sea Turtles

I

Stage Class Size(cm) Ages(yr) Survival Fec.

1 eggs/hatchlings < 10 < 1 .6747 02 sm. juveniles 10.1-58 1-7 .7857 03 lg. juveniles 58.1-80 8-15 .6758 04 subadults 80.1-87 16-21 .7425 05 novice breeders > 87 22 .8091 1276 1st-yr remigrants > 87 23 .8091 47 mature breeders > 87 24-54 .8091 80

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

Calculating Gi

Table: Population and Survival

I

Pop. Years

M 1Mpi 2Mp2

i 3Mp3

i 4Mp4

i 5Mp5

i 6

Mp6i 7

I Eligible:

Mp5i

M + Mpi + Mp2i + Mp3

i + Mp4i + Mp5

i

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

Calculating Gi

Table: Population and Survival

I

Pop. Years

M 1Mpi 2Mp2

i 3Mp3

i 4Mp4

i 5Mp5

i 6

Mp6i 7

I Eligible:

Mp5i

M + Mpi + Mp2i + Mp3

i + Mp4i + Mp5

i

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

Calculating Gi cont.

I Proportion:(p5

i

1 + pi + p2i + p3

i + p4i + p5

i

)

I Hit this again with piI Yields the percentage that actually advance.

I

Gi =

(p6i

1 + pi + p2i + p3

i + p4i + p5

i

)

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

Calculating Gi cont.

I Proportion:(p5

i

1 + pi + p2i + p3

i + p4i + p5

i

)I Hit this again with pi

I Yields the percentage that actually advance.

I

Gi =

(p6i

1 + pi + p2i + p3

i + p4i + p5

i

)

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

Calculating Gi cont.

I Proportion:(p5

i

1 + pi + p2i + p3

i + p4i + p5

i

)I Hit this again with pi

I Yields the percentage that actually advance.

I

Gi =

(p6i

1 + pi + p2i + p3

i + p4i + p5

i

)

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

Generalizing Gi

I

Gi =

(p5i

1 + pi + p2i + p3

i + p4i + p5

i

)pi

I Gi generalized: di represents stage duration.

I

Gi =

(pdi−1

i

1 + pi + p2i + · · ·+ pdi−1

i

)pi

I

Gi =(1− pi)p

di

i

1− pdi

i

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

Generalizing Gi

I

Gi =

(p5i

1 + pi + p2i + p3

i + p4i + p5

i

)pi

I Gi generalized: di represents stage duration.

I

Gi =

(pdi−1

i

1 + pi + p2i + · · ·+ pdi−1

i

)pi

I

Gi =(1− pi)p

di

i

1− pdi

i

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

Generalizing Gi

I

Gi =

(p5i

1 + pi + p2i + p3

i + p4i + p5

i

)pi

I Gi generalized: di represents stage duration.

I

Gi =

(pdi−1

i

1 + pi + p2i + · · ·+ pdi−1

i

)pi

I

Gi =(1− pi)p

di

i

1− pdi

i

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

Generalizing Gi

I

Gi =

(p5i

1 + pi + p2i + p3

i + p4i + p5

i

)pi

I Gi generalized: di represents stage duration.

I

Gi =

(pdi−1

i

1 + pi + p2i + · · ·+ pdi−1

i

)pi

I

Gi =(1− pi)p

di

i

1− pdi

i

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

Calculating Pi

Table: Population and Survival

I

Pop. Years

M 0Mpi 1Mp2

i 2Mp3

i 3Mp4

i 4Mp5

i 5

Mp6i 6

I Mp5i are eligible to advance

I then M + Mpi + Mp2i + Mp3

i + Mp4i are ineligible.

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

Calculating Pi

Table: Population and Survival

I

Pop. Years

M 0Mpi 1Mp2

i 2Mp3

i 3Mp4

i 4Mp5

i 5

Mp6i 6

I Mp5i are eligible to advance

I then M + Mpi + Mp2i + Mp3

i + Mp4i are ineligible.

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

Calculating Pi

Table: Population and Survival

I

Pop. Years

M 0Mpi 1Mp2

i 2Mp3

i 3Mp4

i 4Mp5

i 5

Mp6i 6

I Mp5i are eligible to advance

I then M + Mpi + Mp2i + Mp3

i + Mp4i are ineligible.

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

Calculating Pi cont.

I M + Mpi + Mp2i + Mp3

i + Mp4i are ineligible to advance

I Only option for survival is remain in class.

I Proportion:

1 + pi + p2i + p3

i + p4i

1 + pi + p2i + p3

i + p4i + p5

i

I Hit this with piI Percentage that actually survive and remain.

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

Calculating Pi cont.

I M + Mpi + Mp2i + Mp3

i + Mp4i are ineligible to advance

I Only option for survival is remain in class.

I Proportion:

1 + pi + p2i + p3

i + p4i

1 + pi + p2i + p3

i + p4i + p5

i

I Hit this with piI Percentage that actually survive and remain.

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

Calculating Pi cont.

I M + Mpi + Mp2i + Mp3

i + Mp4i are ineligible to advance

I Only option for survival is remain in class.

I Proportion:

1 + pi + p2i + p3

i + p4i

1 + pi + p2i + p3

i + p4i + p5

i

I Hit this with piI Percentage that actually survive and remain.

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

Generalizing Pi

I

Pi =

(1 + pi + p2

i + p3i + p4

i

1 + pi + p2i + p3

i + p4i + p5

i

)pi

I Pi generalized: di represents stage duration.

I

Pi =

(1 + pi + p2

i + · · ·+ pdi−2i

1 + pi + p2i + · · ·+ pdi−1

i

)pi

I

Pi =

(1− pdi−1

i

1− pdi

i

)pi

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

Generalizing Pi

I

Pi =

(1 + pi + p2

i + p3i + p4

i

1 + pi + p2i + p3

i + p4i + p5

i

)pi

I Pi generalized: di represents stage duration.

I

Pi =

(1 + pi + p2

i + · · ·+ pdi−2i

1 + pi + p2i + · · ·+ pdi−1

i

)pi

I

Pi =

(1− pdi−1

i

1− pdi

i

)pi

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

Generalizing Pi

I

Pi =

(1 + pi + p2

i + p3i + p4

i

1 + pi + p2i + p3

i + p4i + p5

i

)pi

I Pi generalized: di represents stage duration.

I

Pi =

(1 + pi + p2

i + · · ·+ pdi−2i

1 + pi + p2i + · · ·+ pdi−1

i

)pi

I

Pi =

(1− pdi−1

i

1− pdi

i

)pi

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

Generalizing Pi

I

Pi =

(1 + pi + p2

i + p3i + p4

i

1 + pi + p2i + p3

i + p4i + p5

i

)pi

I Pi generalized: di represents stage duration.

I

Pi =

(1 + pi + p2

i + · · ·+ pdi−2i

1 + pi + p2i + · · ·+ pdi−1

i

)pi

I

Pi =

(1− pdi−1

i

1− pdi

i

)pi

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

Final Matrix

0 0 0 0 127 4 80.6747 .7371 0 0 0 0 0

0 .0486 .6611 0 0 0 00 0 .0147 .6907 0 0 00 0 0 .0518 0 0 00 0 0 0 .8091 0 00 0 0 0 0 .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

Population After 50 Years

I

0 10 20 30 40 5010

−2

100

102

104

106

Time (years)

Pop

ulat

ion

1st2nd3rd4th5th6th7th

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

Diagonalizing

I Call matrix model L

I

Lk = V ΛkV−1

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

Diagonalizing

I Call matrix model L

I

Lk = V ΛkV−1

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

Diagonalizing

I ~xk = Lk ~x0

I ~xk =(V ΛkV−1

)~x0

I V−1~x0 = ~c

I~xk = c1λ

k1 ~v1 + · · · + cnλ

kn ~vn

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

Diagonalizing

I ~xk = Lk ~x0

I ~xk =(V ΛkV−1

)~x0

I V−1~x0 = ~c

I~xk = c1λ

k1 ~v1 + · · · + cnλ

kn ~vn

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

Diagonalizing

I ~xk = Lk ~x0

I ~xk =(V ΛkV−1

)~x0

I V−1~x0 = ~c

I~xk = c1λ

k1 ~v1 + · · · + cnλ

kn ~vn

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

Diagonalizing

I ~xk = Lk ~x0

I ~xk =(V ΛkV−1

)~x0

I V−1~x0 = ~c

I~xk = c1λ

k1 ~v1 + · · · + cnλ

kn ~vn

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

Eigenvalues

Table: Eigenvalues

Gi = 1 λ1 λ2 λ3 λ4 λ5 λ6 λ7

Stage 1 −.100 −.100 .317 .317 .750 .750 .963Stage 2 −.230 −.230 .320 .320 1.16 .778 .778Stage 3 −.320 −.320 1.30 .791 .791 .336 .336Stage 4 −.225 −.225 .320 .320 1.15 .775 .775Stage 5 −.090 −.090 .314 .314 .753 .753 .953Stage 6 −.095 −.095 .314 .314 .753 .753 .953

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

Stage 3 Eigenvalues

I Using λi for Stage 3 from eigenvalue table

I k = 50

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

I When λi < 1 then λki → 0

I ~x (50) ≈ c3λ503 ~v3

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

Stage 3 Eigenvalues

I Using λi for Stage 3 from eigenvalue table

I k = 50

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

I When λi < 1 then λki → 0

I ~x (50) ≈ c3λ503 ~v3

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

Stage 3 Eigenvalues

I Using λi for Stage 3 from eigenvalue table

I k = 50

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

I When λi < 1 then λki → 0

I ~x (50) ≈ c3λ503 ~v3

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

Stage 3 Eigenvalues

I Using λi for Stage 3 from eigenvalue table

I k = 50

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

I When λi < 1 then λki → 0

I ~x (50) ≈ c3λ503 ~v3

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

Stage 3 Eigenvalues

I Using λi for Stage 3 from eigenvalue table

I k = 50

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

I When λi < 1 then λki → 0

I ~x (50) ≈ c3λ503 ~v3

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

G3 Set to 100%

I

0 10 20 30 40 5010

0

102

104

106

108

1010

Time(years)

Pop

ulat

ion

Rat

io

1st2nd3rd4th5th6th7th

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

G3 Set to 4%

I

0 10 20 30 40 5010

0

101

102

103

104

105

Time(years)

Pop

ulat

ion

Rat

io

1st2nd3rd4th5th6th7th

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

G3 Set to 5%

I

0 10 20 30 40 5010

0

101

102

103

104

105

Time(years)

Pop

ulat

ion

Rat

io

1st2nd3rd4th5th6th7th

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

Population After 50 Years

I

0 10 20 30 40 5010

−2

100

102

104

106

Time (years)

Pop

ulat

ion

1st2nd3rd4th5th6th7th

Ben Marshall and Alex Varner Loggerhead Sea Turtle Conservation

The Bibliography

Deborah Crouse, Larry Crowder, Hal Caswell, 1987 A StageBased Population Model For Loggerhead Sea Turtles AndImplications 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