1. A review of age and growth, reproductive biology, and demography of the white shark, Carcharodon carcharias Henry F. Mollet Moss Landing Marine Labs

2. 11/23/20152 http://homepage.mac.com/mollet/Cc/Cc_litters.html Taiwan Oct 1997 Japan Feb 1985 Med Sea Summer 1934

3. 11/23/20153 Update: White shark litter of 8; 50-60 cm TL Taiwan 13 October 1997 (Victor Lin p.c.)

4. 11/23/20154 5.55 m TL Kin(-Town) white shark caught on 16 Feb 1985 Uchida et al. (1987, 1996) assumed near-term embryos were aborted. 192 eggcases of 3 types (66 + 20 + 101; 104, 33, and 14 g) weighing 9 kg in left uterus. Early-term more likely (Mollet et al. 2000). No blastodisc eggcases? Overlooked? How old? 66 - 192 days?

5. 11/23/20155 Shortfin mako 3 cm TL embryos. How old? 4(L) +5(R) blastodisc eggcases (3x10 cm) with 1 embryo each > 40 (3x5 cm) eggcases in each uterus with 16-20 eggs per capsule; plus empty eggcases produced before blastodisc eggcases Shell gland produces one eggcase/day (Gilmore 1993 for sandtiger) Suggested age is 40-50 days (Mollet et al. 2000)

6. 11/23/20156 VBGF for 11 embryos and 20 free-swimmers L 0 = - 0.060 m ; L  = 1.837 m ; k = 0.977 yr -1 ; r 2 = 0.873

7. 11/23/20157 Gompertz for 11 embryos and 20 free-swimmers L 0 = 0.027 m; L  = 1.513 m; k = 2.649 yr -1 ; r 2 = 0.891

8. 11/23/20158 Comparison of VBGF and Gompertz for 11 embryos and 20 free-swimmers

9. 11/23/20159 von Bertalanffy (1938) dM/dt =  M 2/3 -  M, differential equation eta and kappa are anabolic catabolic parameters but no physiological interpretation needed Substitution method y = M 1/3 : dy/dt =  /3 -  /3y; (  = 3k will be used later) Separation of variable: dy/(  /3 -  /3y) = dt Integration: -3/  ln(  /3 -  /3y ) = t + constant *Constant to be determined by value of y at t = 0* Final result: y =  /  - (  /  - y o ) e -(x/3) t Back-substitution: M(t) = [  /  - (  /  - w 0 1/3 ) e -(x/3) t ] 3 [will become L(t) = L  - (L  - L 0 ) e -k t ]

10. 11/23/201510 L 0 versus t 0 VBGF using L 0 (von B. 1938) can be transformed mathematically to use x-axis intercept instead of y-axis intercept as 3rd parameter: L(t) = L  ( e -k (t - to) ); t 0 = 1/k ln[(L  - L 0 )/ L  ]. The two formulations are mathematically equivalent. However, L 0 has a biological meaning whereas t 0 has none. Therefore, unreasonable L 0 are easily detected but not unreasonable t 0.

11. 11/23/201511 1995 Edmonton in the Lion ’ s Den Slide von Bertalanffy was at U of Alberta in Edmonton. I said it in 1995 and now Tampa 2005: von Bertalanffy (1938) did not use t-zero (not in any of his publications). t-zero was introduced by Beverton (1954) to simplify yield calculations Came in widespread use after Beverton and Holt (1957) but it is not helpful for interpretation of VBGFs of elasmos with well defined size at birth (L 0 ).

12. 11/23/201512 k is a rate constant with units of reciprocal time Difficult to deal with/understand reciprocal time. Better to interpret k in terms of half-life (ln2/k) with units of time. The time it takes to reach fraction x of L  is given by t x = 1/k ln[(L  - L 0 )/L  (1- x)]. Can use x = 0.95 (Ricker 1979) and L 0 = 0.2L  : t 0.95 = longevity = 2.77/k = 4.0 ln2/k (4 half-life). Could use x = 0.9933 and L 0 = 0: t 0.9933% = longevity = 5.00/k = 7.21 ln2/k (~ 7.2 half-life, Fabens 1965) x = 0.95 vs. x = 0.9933 produces large range of longevities that differ by almost a factor of 2! (7.21/4 = 1.80)

13. 11/23/201513 L  is inversely proportional to k We started with dM/dt =  M 2/3 -  M Steady state means dM/dt =  M 2/3 -  M = 0 after a long time when t =  i.e. M (  ) = M  (L(  ) = L  ) Therefore M  1/3 =  /  (L  = q (  /  ) will become L  = 3q (  /k) L  is inversely proportional to k The steady state value (final mass or length) is determined by both  and k, the time it takes to get there is determined by k only.

14. 11/23/201514 Various 2-parameter VBGF ’ s 2-parameter VBGF (using size at age data) by using observed size at birth for L 0. Justification: If one had age versus length data for neonates that ’ s what the data would have to be. Fabens for tag-recapture data, also a 2-parameter VBGF (k and L  ). Length at tagging and recapture and time-at-large data are used. Gulland-Holt, yet another 2-parameter VBGF (k and L  ) which is usually used in graphical form by plotting annualized growth versus average size.

15. 11/23/201515 Observed # of band-pairs in 3 populations. White sharks in SA appear to be younger for given TL?

16. 11/23/201516 Maximum band count/age?Might be 30- 40 years Will show later (2nd talk) that longevity (  ) is important Gans Baai 23 bands, CD not available? TL ~ 5.6-6 m (Dave Ebert, Mollet et al. 1996). North Cape, 22+-1 bands, CD 63 mm only but not max size. TL ~ 5.36 m; (bands were stained by mud, disappeared in air) (Francis 1996). Cojimar Cuba 1945 6.4 m TL? specimen had 80x37 mm vertebrae, no band counts given. Taiwan ~5.6 m TL specimen had 83x36 mm vertebrae, 16-22 bands (Sabine Wintner).

17. 11/23/201517 VBGF and Gompertz for combined data from around the world (mostly juvenile white sharks) VBGF Gomp Lo 1.5 1.6 m Loo 6.2 5.5 m k 0.079 0.144 y -1 t 95% 35 22 y n = 185

18. 11/23/201518 Combining embryonic and post-natal data. Embryonic growth is expected to be larger than post-natal growth; cannot use same growth function VBGF Gomp Lo 1.45 1.49 m Loo 5.92 5.2 m k 0.088 0.17 yr -1 t 95% 31 19 yr t-zero -3.2 -  yr n = 216

19. 11/23/201519 Embryonic and wild vs. captive 1st year growth Assume gestation period of 1.5 yr, growth will be 1.3-1.5/1.5 = 0.81 - 1.0 m/yr. Calculated 1st year growth from existing VBGF ’ s much smaller at 0.25-0.35 m/yr However, captive growth at MBA with feeding to satiation was 0.8 m/yr. Female would mature in 5 instead of 15 yr? Captive neonate pelagic stingrays (male and female) may have reached maturity within 1 year!

20. 11/23/201520 Preparation for Elasticity talk: Vital rates for white shark Age at first and last reproduction:  = 15 yr,  = 60 yr;  /  = 4.0; Mollet and Cailliet (2002) but   45 yr;  /   3.0. Annualized female fertility m = 8.9/2*3 = 1.4833; Mortality = - ln(0.01)/60 = 0.07675 S = exp(-M) = 0.9261; Discounted fertilities F i = m. S i = 1.3737 (same for pre- and post-breeding census because survival to age 1 (S 1 ) was assumed to be same as juvenile and adult survival).

21. 11/23/201521 Projection matrix (Leslie matrix) for white shark

22. 11/23/201522 Elasticity matrix for white shark E-pattern comprising E(m), E(js), and E(as) can be obtained by summing over appropriate matrix elements but need to include discount

23. 11/23/201523 Results: 1 = 1.082, (r 1 = ln( 1 ) = 0.07869); Abar = 20.91 yr (Abar/  = 1.394), T = 23.11 yr,  1 = 26.15 yr. E-pattern: E(m) = 0.04564 E(Sj) = 0.6846, E(Sa) = 0.2697 (E(Sa)/ E(Sj) = 0.394).

24. 11/23/201524 , , and 3 Generation Times

25. 11/23/201525 Summary Slide We know very little about details of reproductive biology of white shark. The good news is that we know enough. We can formulate management proposals using prospective elasticity analysis with available inadequate data. Thanks to Sabine Wintner and Barry Bruce for sharing raw vertebrae count data.

26. 11/23/201526 Recipe for management proposals based on E-pattern from prospective elasticity analysis (To be proved in afternoon presentation) Use E-triangle to graph E-pattern comprising elasticities of fertility, juvenile survival, and adult survival. E1 = E(m), E2 = E(Sj), E3 = E(Sa). E-pattern is determined by  and Abar alone (don ’ t even need E-matrix). Most important is ratio E3/E2 = Abar/  - 1. If  /   3.0 then E 3 /E 2  3.0 without further considerations (most elasmos). E2/E1 =  (obviously large  means that E2 >> E1) One complication, if repro cycle is not 1 yr. White shark as example: E2/E1 =  = 15 yr say. If repro cycle is 3 yr then E2/E1 =  = 15/3 = 5 (3-yr units). E3/E2 stays about the same!