Comparison of Elasticity Patterns of Elasmobranchs and Mammals

with Review of Vital Rates of Lamnids

Henry F. Mollet

Moss Landing Marine Labs

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1984 vs. 2001 Compagno FAO Catalogue, Lamnids Only

• Porbeagle: Much new info on reproduction (Francis and Stevens 2000; Natanson et al. 2002; Jenson et al. 2002; Campana et al. 2002)

• Salmon shark: Much new info in press (Ken Goldman pers. comm. in Compagno 2001)

• Longfin mako: Litter size 8 (Casey 1986 Abstract)

• Shortfin mako: Litter size as large as 25-30 (Mollet et al. 2002); Gestation period 18 months and reproductive cycle probably 3 years (Mollet et al 2000); Age-at-maturity ~14 yr rather than ~7 yr (1 growth band pair/year, Campana et al. 2002 based on bomb C14-analysis of 1 vertebrae from 1 specimen)

• White shark: Much new info on reproduction (Francis 1996; Uchida et al. 1996; both in Klimley and Ainley eds.). Gestation period 18? months with 3? yr repro cycle; length/age-at-maturity 5?m/ 15? yr (Mollet et al. 2000; Bruce et al. (in press). Crucial to get info on litters with early-term embryos.

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Sanzo (1912) and Uchida (1989) Shortfin Mako Embryos

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From Capture and ‘Recapture’ to Publications 90 years apart, 100 years to get correct ID

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?White Shark Litter of ~5, ~ 30 cm TLTaiwan ~ 10 years ago (Victor Lin p.c.)

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Summary

• Elasticities patterns for ALL Elasmos are nearly the same and can be done without a calculator, at the Rio Negro Beach.

• Can predict E-pattern for Elasmos with little or no data.

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Background I

• Elasticities give proportional changes of population growth () due to proportional change of vital rates (a): E(a) = dln()/dln(a) = (a/) ( d/da)

• Elasticities are robust, don’t need accurate

• Vital rates of elasmos, in particular lamnids, are poorly known. Therefore precise population growth rates () are difficult to obtain anyway.

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Background II

• Mollet and Cailliet (2003) reply to Miller et al. (2003): LHT or Leslie matrix are easier and safer than stage-based models.

• Here, I’ll cover elasticity patterns, not just for one species, but for all elasmos and all mammals using data for 60 elasmos and 50 mammals.

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LHT and corresponding Leslie Matrices for a Hypothetical Species

= 3 yr, m = 3, =5, P-juv = 0.631, P-adu = 1.0! (but m6 = 0)

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Elasticity Matrix and Elasticity Pattern by Summing Elasticities over Age-Classes

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, , and 3 Generation Times

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Elasticity pattern from and Abar (Gestation period GP provides refinement)

• E(fertility) = E(m)

• E(juvenile survival) = E(js)

• E(adult survival) = E(as)

• 1/E(m) = <w,v> = Abar ! Dynamite!(<w,v> with w1 = 1, v1 = 1)

• E(js) = ( - GP)/Abar• E(as) = (Abar - + GP)/Abar• Normalization is easy

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Why Abar? The Short Version!

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The Crux of the Matter

• E-pattern can be calculated from = age-at-first-reproduction,Abar = mean age of reproducing females at stable age distribution (= x -x lx mx = f (vital parameters and )), GP = gestation period provides refinement.

• Presents great simplification and allows better understanding of E-patterns even if we have to solve the characteristic equation to get Abar.

• Elasticity matrix no longer needed for age-structured species.

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A Potential Problem and Proposed Solution for Elasmos

• Catch 22 situation if we’d like to estimate the E-pattern of an Elasmo at the Rio Negro Beach?

• If Abar/alpha were roughly constant, we could estimate Abar from the mean ratio and the elasticity pattern could be easily estimated without the need to solve the characteristic equation.

• Example: = 7 yr, Abar/ ~ 1.3, thus Abar ~ 9 yr:E(m) normalized = 1/(Abar + 1) = 1/10 = 10%E(juvenile survival) = * E(m) = 7 * 10% = 70%E(adult survival) = 20% (from sum = 100%)

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(Abar/) Ratios for Elasmos and Mammals

• Mean (Abar/) of 60 Elasmos:1.31, CV = 9.3%, Range 1.1 (S. lewini, S. canicula) - 1.8 (C. taurus);

• Cortes (2002) Stochastic Calculation for n = 41 Elasmos, mean (Abar/) = 1.46, CV = 14.2%, range 1.1-2.0;

• Mean (Abar/ ) of the 50 Mammals in Heppell et al. (2000): 2.44, CV = 33.5%, Range 1.2 (Snowshoe Hare) - 5.0 (Thar, l. Brown Bat)

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Normalized Elasticity Pattern for Elasmos using mean (Abar/) Ratio of 60 Elasmos

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Area Plot of Elasticity Pattern for Elasmos using Mean (Abar/) Ratio of 60 Elasmos

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Theory and E-patterns of Elasmos from LHT are in good agreement

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Shortfin and Longfin Mako Elasticity Patterns

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Salmon shark, Porbeagle, and White Shark Elasticity Patterns

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Elasticity Patterns of 4 RaysDasyatis violacea, Narcine entemedor, Myliobatis californica, Dipturus laevis

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Elasmos with Extreme Elasticity Patterns?Scyliorhinus canicula (m =105/2) , Sphyrna lewini,(m = 26-35/2) Carcharias taurus (m = 2/2x2), Rhincodon typus (m = 300/2x2?)

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Potential Problems

• I used constant and fairly large mortalities for Elasmos.

• They could be more Mammal like, larger mortality for juveniles and smaller mortality for adults.

• Cortes (2002) used variable mortalities and Abar/ was still close to 1 and had low variability.

• Elasticities only applicable to stable age distribution. True but E-patterns are very robust. Would have to move far from stable age distribution for E-pattern to become unsuitable for making management proposals.

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