A review of age and growth, reproductive biology, and

demography of the white shark, Carcharodon carcharias

Henry F. Mollet

Moss Landing Marine Labs

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http://homepage.mac.com/mollet/Cc/Cc_litters.html

Taiwan Oct 1997

Japan Feb 1985

Med Sea Summer 1934

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Update: White shark litter of 8; 50-60 cm TLTaiwan 13 October 1997 (Victor Lin p.c.)

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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?

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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)

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VBGF for 11 embryos and 20 free-swimmers L0 = - 0.060 m; L= 1.837 m; k = 0.977 yr-1; r2 = 0.873

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Gompertz for 11 embryos and 20 free-swimmers L0 = 0.027 m; L= 1.513 m; k = 2.649 yr-1; r2 = 0.891

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Comparison of VBGF and Gompertz for 11 embryos and 20 free-swimmers

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von Bertalanffy (1938)

• dM/dt = M2/3 - M, differential equationeta and kappa are anabolic catabolic parameters but no physiological interpretation needed

• Substitution method y = M1/3: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 = / - (/ - yo) e-(x/3) t

• Back-substitution: M(t) = [/ - (/ - w01/3) e-(x/3) t]3

[will become L(t) = L - (L - L0) e-k t ]

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L0 versus t0

• VBGF using L0 (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)); t0 = 1/k ln[(L - L0 )/ L].

• The two formulations are mathematically equivalent.

• However, L0 has a biological meaning whereas t0 has none. Therefore, unreasonable L0 are easily detected but not unreasonable t0.

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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 (L0).

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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

tx = 1/k ln[(L - L0)/L(1- x)]. • Can use x = 0.95 (Ricker 1979) and L0 = 0.2L:

t0.95 = longevity = 2.77/k = 4.0 ln2/k (4 half-life). • Could use x = 0.9933 and L0 = 0: t0.9933% = 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)

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L is inversely proportional to k

• We started with dM/dt = M2/3 - M

• Steady state means dM/dt = M2/3 - M = 0 after a long time when t = i.e. M () = M (L() = L )

• Therefore M1/3

= / (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.

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Various 2-parameter VBGF’s

• 2-parameter VBGF (using size at age data) by using observed size at birth for L0. 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.

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

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Maximum band count/age?Might be 30- 40 yearsWill 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).

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VBGF and Gompertz for combined data from around the world (mostly juvenile white sharks)

VBGF GompLo 1.5 1.6 mLoo 6.2 5.5 mk 0.079 0.144 y-1

t95% 35 22 y

n = 185

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Combining embryonic and post-natal data. Embryonic growth is expected to be larger than post-natal

growth; cannot use same growth function

VBGF GompLo 1.45 1.49 mLoo 5.92 5.2 mk 0.088 0.17 yr-1

t95% 31 19 yrt-zero -3.2 - yr

n = 216

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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!

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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 Fi = m .Si = 1.3737(same for pre- and post-breeding census because survival to age

1 (S1) was assumed to be same as juvenile and adult survival).

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Projection matrix (Leslie matrix) for white shark

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Elasticity matrix for white sharkE-pattern comprising E(m), E(js), and E(as) can be obtained by summing

over appropriate matrix elements but need to include discount

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Results: 1 = 1.082, (r1 = 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).

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, , and 3 Generation Times

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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.

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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 E3/E2 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!