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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;

• Durban 2001 (MFR .53)

• La Paz 2000

• Penn State 1999 (Jim Bourdon)

• Guelph 1998

• Seattle 1997

• New Orleans 1996

• Edmonton 1995

• Pelagic StingrayDemography

• Kansas City 2002• Durban 2001 (MFR 53)

• Shortfin MakoDemography

• Durban 2001 (Manuscript withdrawn)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/poptools/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 Matrixto 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

• 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

• 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 y8.2%/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 SharkRelative change of due to relative changes of fertility or survival

ei,,j = (dln / dlnai,,j) = (ai,,j /) (d/dai,,j) = (ai,,j /) si,,j

• 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 = 6.9Interpretation 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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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)

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Triangle Graph after Heppell et al. (2000) of Elasticities of 4 Elasmobranchs (normalized to 1)