Transcript Comparative population demography of elasmobranchs using

Comparative Population Demography
of Elasmobranchs using Life History Tables,
Leslie Matrices, and Stage Based Matrix Models
• Henry F. Mollet and
Gregor C. Cailliet
• Moss Landing
Marine Laboratories
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Pelagic Stingray 1995-2002
Dasyatis (Pteroplatytrygon)violacea
• Pelagic Stingray
Distribution ;
Captive Biology;
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Durban 2001 (MFR .53)
La Paz 2000
Penn State 1999 (Jim Bourdon)
Guelph 1998
Seattle 1997
New Orleans 1996
Edmonton 1995
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• Pelagic Stingray
Demography
• Kansas City 2002
• Durban 2001 (MFR 53)
• Shortfin Mako
Demography
• Durban 2001
(Manuscript withdrawn)
• Noumea 1997
• Seattle 1997
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Shortfin Mako Demography ?
• Withdrew Durban 2001 manuscript
• Based on new vertebrae analysis by Lisa Natanson and
• Radiocarbon (atomic bomb) dating by Steve Campana et
al. (in press)
• 1 band-pair/year (Cailliet et al. 1983) rather than
2 (Pratt & Casey 1983).
Age-at-maturity ~ 14 y rather than 7 y
• Review with 3 of Greg’s 1997 Seattle slides
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Demography of the Pelagic Stingray
• Good example for demonstration
because short-lived, thus small Leslie
matrix (-Lewis 1942)
• Won’t discuss Life history table and
Euler-Lotka equation
• Stage-based matrix models
• Difficulties are concepts of
discounted fertility in
pre-breeding or. post-breeding census,
which won’t be discussed in detail.
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Pelagic Stingray Vital Rates
• Mollet et al. (2002)
• Age-at-first-reproduction 3 y
• Longevity ~ 10 y
• Mortality -ln(0.01)/10 =
0.460 y-1 (S = 63.1%)
• Fertility 6/2 = 3 female
pups/year
• Seasonal parturition i.e.
birth pulse approximation
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• Good Tools were already
available in the Middle
Ages
• Today’s Outlaw
Demographers use
Greg Hood’s PopTools to
Shoot for Solution of
matrix population models.
• Free DownShoots at
http://www.dwe.csiro.au/vbc/po
ptools/index.htm
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Life Cycle Graph and 10 x10
Leslie Matrix for Pelagic Stingray
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Matrix Multiplication of
State vector (n) with Transition Matrix (A)
• For pelagic stingray age-at-first-reproduction = 3,
thus discounted Fertilities F1 = F2 = 0
• P1, P2, .... P9 = survival probabilities, we use G1, G2, ....G9 to get
agreement with terminology for stage-based models where Pi’s are
used for in-stage survival
• Once/if age-distribution is stable, then n(t + 1) =  n(t)
• (A is assumed to be constant, no environmental nor density effects)
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PopTools Solution (i.e.long term stable behavior)
of 10 x10 Leslie Matrix for Pelagic Stingray
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Stable Age Distribution and Reproductive Values
for Pelagic Stingray 10 x10 Leslie Matrix
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Converting Age-based 10x10 Leslie Matrix
to 3x3 Stage-based Matrix
• Adult age-classes (8) are put into 1 stage (stage duration T3 = 8 y)
(Heppell et al. 2000)
• Assume that age-structure is maintained within stage
• Can calculate fraction in stage 3 that graduate to next stage (=death)
= G3 = 0.0038 ( not needed); P3 = (3 -G3) = 0.6271
(3 = 0.6309 is survival probability in stage 3)
(P3 is in-stage survival probability)
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Heppell et al. (2000) Model for Pelagic Stingray
(3x3 matrix because only 2 juvenile age-classes)
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PopTools Solution of 3x3 Age/Stage
Based Matrix for Pelagic Stingray
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Sandtiger Shark
(Carcharias taurus) Vital Rates
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Branstetter and Musick (1994)
Age-at-first-reproduction 6 y
Longevity ~ 25 y
Mortality -ln(0.01)/25
= 0.1842 y-1 (S = 83.2%)
• Effective Fertility of 0.5 female
pups every year vs. actual
fertility of 1 female pup every
other year
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Brewster-Geisz & Miller (2000) Model for
Sandtiger Shark (resting stage for mature females)
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Sandtiger Shark
Demography Results
• Population is decreasing by -0.40%/year
(using effective annual fertility with 0.5
female pups every year)
• Population is increasing by 0.69%/year
(using actual reproductive cycle with 1
female pup every other year)
• Due to compounding. Better to put $100 in the bank now
compared to $50 now and $50 one year later
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Pelagic Thresher Shark
(Alopias pelagicus) Vital Rates
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Liu et al. (1999); Age-at-first-reproduction 8 y
Longevity ~ 30 y
Mortality -ln(0.01)/30 = 0.1535 y-1(S = 85.8%)
Fertility 1 female pup/year
We consider Seasonal vs. Year-round Parturition
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Pelagic Thresher Demography Results
(Birth-pulse vs. Birth-flow)
• Birth-pulse (distinct seasonal parturition)
Population is increasing at 5.5%/year
• Birth-flow (= year-round parturition)
Population is increasing at 6.4%/year
• Intermediate results (5.9%, 6.1%, 6.3%)
can be calculated by using shorter
projection intervals of 1/2, 1/4, 1/12 years
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White Shark Vital Rates
(Carcharodon carcharias)
• Cailliet et al. (1985); Francis (1996);
Wintner and Cliff (1999);
Mollet et al. (2000)
• Age-at-first reproduction
15 y (~ 5 m TL)
• Longevity ~ 60 y
(36 y in some calculations)
• Mortality -ln(0.01)/60
= 0.077 y-1 (S = 92.6%)
• Fertility 8.9/2 fem. pups every 3 y
(annual effective fertility 1.483)
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White Shark Results (Comparison of Step-Like
(aka knife-edge) vs. Logistic Fertility Function)
• LHT to age 60 y
8.2%/y (step-like)
8.0%/y (logistic)
• 3x3 (1-13-46)
8.2%/y (fixed stage
distribution)
8.7%/y (variable stage
distribution)
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Elasticities for White Shark
Relative change of  due to relative changes of fertility or survival
ei,,j = (dln / dlnai,,j) = (ai,,j /) (d/dai,,j) = (ai,,j /) si,,j
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E1 = E (fertility) =  e1, j = 0.048
E2 = E (juvenile survival) =  ej+1, j = 0.670
E3 = E (adult survival) =  ej+1, j + E1 = 0.331 (with E1 = 0.048 added)
Ratios: ER2 = E2/E1 = 14 ( -1) and ER3 = E3/E1 = 6.9
Interpretation of ER3: Fishing of ~ 7 juvenile age classes has same
effect as fishing all 48 adult age classes (because E1 = ej+1, j ,j < 15)
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Recovery Time Estimates ln(10)/ln()
where  = damping ratio = 1 / |2|
(have to be cautious when using stage-based models with few stages)
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Recovery time (y)
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L-matrix
Heppell
3x3
2x2
30
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10
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25
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"Generation time" (x)
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Future Outlook?
• Need better vital rates for
elasmobranchs
• Stage based models have
great potential (e.g. 20 x20
matrix could deal with 5
populations and both sexes)
• Elasticities are best tool
for management of
elasmobranchs (prospective
analysis as per Caswell,2001)
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Exponential, Logistic, and Modified Logistic Population
Growth for White Shark (r = 0.08 y-1, K = 1000, No = 3.6842)
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Sustainable Yield (first derivative) for White Shark
(r = 0.08 y-1, K = 1000, No = 3.6842)
(can “fish” with F = r ( Z = M + r) to N ~ K)
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Area plots showing stage-specific elasticities after
Heppell et al. (2000) and Cortes (in press)
1.00
0.80
0.60
E (Fer)
1-E(AS)
0.40
0.20
0.00
0
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Triangle Graph after Heppell et al. (2000)
of Elasticities of 4 Elasmobranchs (normalized to 1)
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