11/25/20151 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

11/25/20152 Pelagic Stingray 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1995 Pelagic Stingray Demography Kansas City 2002 Durban 2001 (MFR 53) Shortfin Mako Demography Durban 2001 (Manuscript withdrawn) Noumea 1997 Seattle 1997

11/25/20153 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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7 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.

11/25/20158 Pelagic Stingray Vital Rates Mollet et al. (2002) Age-at-first-reproduction 3 y Longevity ~ 10 y Mortality -ln(0.01)/10 = y-1 (S = 63.1%) Fertility 6/2 = 3 female pups/year Seasonal parturition i.e. birth pulse approximation

11/25/20159 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 ptools/index.htm

11/25/ Life Cycle Graph and 10 x10 Leslie Matrix for Pelagic Stingray

11/25/ Matrix Multiplication of State vector (n) with Transition Matrix (A) For pelagic stingray age-at-first-reproduction = 3, thus discounted Fertilities F 1 = F 2 = 0 P 1, P 2,.... P 9 = survival probabilities, we use G 1, G 2,....G 9 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)

11/25/ PopTools Solution (i.e.long term stable behavior) of 10 x10 Leslie Matrix for Pelagic Stingray

11/25/ Stable Age Distribution and Reproductive Values for Pelagic Stingray 10 x10 Leslie Matrix

11/25/ Converting Age-based 10x10 Leslie Matrix to 3x3 Stage-based Matrix Adult age-classes (8) are put into 1 stage (stage duration T 3 = 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) = G 3 = ( not needed); P 3 = (  3 -G 3 ) = (  3 = is survival probability in stage 3) (P 3 is in-stage survival probability)

11/25/ Heppell et al. (2000) Model for Pelagic Stingray (3x3 matrix because only 2 juvenile age-classes)

11/25/ PopTools Solution of 3x3 Age/Stage Based Matrix for Pelagic Stingray

11/25/ Sandtiger Shark (Carcharias taurus) Vital Rates Branstetter and Musick (1994) Age-at-first-reproduction 6 y Longevity ~ 25 y Mortality -ln(0.01)/25 = y-1 (S = 83.2%) Effective Fertility of 0.5 female pups every year vs. actual fertility of 1 female pup every other year

11/25/ Brewster-Geisz & Miller (2000) Model for Sandtiger Shark (resting stage for mature females)

11/25/ 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

11/25/ 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 = y-1(S = 85.8%) Fertility 1 female pup/year We consider Seasonal vs. Year-round Parturition

11/25/ 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

11/25/ 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 = y-1 (S = 92.6%) Fertility 8.9/2 fem. pups every 3 y (annual effective fertility 1.483)

11/25/ 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 ( ) 8.2%/y (fixed stage distribution) 8.7%/y (variable stage distribution)

11/25/ Elasticities for White Shark Relative change of due to relative changes of fertility or survival e i,,j = (dln / dlna i,,j ) = (a i,,j / ) (d /da i,,j ) = (a i,,j / ) s i,,j E 1 = E (fertility) =  e 1, j = E 2 = E (juvenile survival) =  e j+1, j = E 3 = E (adult survival) =  e j+1, j + E 1 = (with E 1 = added) Ratios: ER 2 = E 2 /E 1 = 14 (  -1) and ER 3 = E 3 /E 1 = 6.9 Interpretation of ER 3 : Fishing of ~ 7 juvenile age classes has same effect as fishing all 48 adult age classes (because E 1 = e j+1, j,j < 15)

11/25/ Recovery Time Estimates ln(10)/ln(  ) where  = damping ratio = 1 / | 2 | (have to be cautious when using stage-based models with few stages)

11/25/ 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)

11/25/ Exponential, Logistic, and Modified Logistic Population Growth for White Shark (r = 0.08 y -1, K = 1000, N o = )

11/25/ Sustainable Yield (first derivative) for White Shark (r = 0.08 y -1, K = 1000, N o = ) (can “ fish ” with F = r ( Z = M + r) to N ~ K)

11/25/ Area plots showing stage-specific elasticities after Heppell et al. (2000) and Cortes (in press)

11/25/ Triangle Graph after Heppell et al. (2000) of Elasticities of 4 Elasmobranchs (normalized to 1)