Comparison of Elasticity Patterns of Elasmobranchs and Mammals with Review of Vital Rates of Lamnids Henry F. Mollet Moss Landing Marine Labs

10/16/ vs 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 (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 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.

10/16/20153 Sanzo (1912) and Uchida (1989) Shortfin Mako Embryos

10/16/20154 From Capture and ‘ Recapture ’ to Publications 90 years apart, 100 years to get correct ID

10/16/20155 ?White Shark Litter of ~5, ~ 30 cm TL Taiwan ~ 10 years ago (Victor Lin p.c.)

10/16/20156 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.

10/16/20157 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.

10/16/20158 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.

10/16/20159 LHT and corresponding Leslie Matrices for a Hypothetical Species  = 3 yr, m = 3,  =5, P-juv = 0.631, P-adu = 1.0! (but m 6 = 0)

10/16/ Elasticity Matrix and Elasticity Pattern by Summing Elasticities over Age-Classes

10/16/ , , and 3 Generation Times

10/16/ 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) = = Abar ! Dynamite! ( with w 1 = 1, v 1 = 1) E(js) = (  - GP)/Abar E(as) = (Abar -  + GP)/Abar Normalization is easy

10/16/ Why Abar? The Short Version!

10/16/ 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 l x m x = 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.

10/16/ 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%)

10/16/ (Abar/  ) Ratios for Elasmos and Mammals Mean (Abar/  ) of 60 Elasmos: 1.31, CV = 9.3%, Range 1.1 (S. lewini, S. canicula) (C. taurus); Cortes (2002) Stochastic Calculation for n = 41 Elasmos, mean (Abar/  ) = 1.46, CV = 14.2%, range ; Mean (Abar/  ) of the 50 Mammals in Heppell et al. (2000): 2.44, CV = 33.5%, Range 1.2 (Snowshoe Hare) (Thar, l. Brown Bat)

10/16/ Normalized Elasticity Pattern for Elasmos using mean (Abar/  ) Ratio of 60 Elasmos

10/16/ Area Plot of Elasticity Pattern for Elasmos using Mean (Abar/  ) Ratio of 60 Elasmos

10/16/ Theory and E-patterns of Elasmos from LHT are in good agreement

10/16/ Shortfin and Longfin Mako Elasticity Patterns

10/16/ Salmon shark, Porbeagle, and White Shark Elasticity Patterns

10/16/ Elasticity Patterns of 4 Rays Dasyatis violacea, Narcine entemedor, Myliobatis californica, Dipturus laevis

10/16/ 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?)

10/16/ 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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