1. Henry F. Mollet Moss Landing Marine Labs
Elasticity patterns for sharks, turtles, mammals, and birds: the importance of age at first reproduction, mean age of reproducing females, and survival in the discounted fertilities of the Leslie matrix
Henry F. Mollet
Moss Landing Marine Labs

2. LHT and corresponding Leslie Matrices for a Hypothetical Species  = 3 yr, m = 3, =5, S-juv = 0.631, S-adu = 1.0! (but m6 = 0)
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3. Elasticity pattern by summing elasticities in elasticity matrix over age-classes Need to fish for survival term in the discounted fertilities!
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4. E(Sa)>E(Sj)
E(Sa<E(Sj)
Abar/ = 2.0
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5. Why are Elasmos so”Easy”
Main purpose of this talk is to provide a mathematical foundation why the E-patterns of Elasmos are all similar. Elasticity of juvenile survival E(Sj)  E2 has to be largest without further considerations for most elasmos.
Will also cover use of Abar in E-patterns and that sum of elasticities in E-pattern is not 1.0 and requires normalization for graph using E-triangle.
Mollet and Cailliet (2003, Reply; in Appendix) more in Mollet (MFR, in review)
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6. Clarification I: Stable Age Distribution
Elasticity patterns apply to the stable age distribution (as does population growth ).
Why should this be useful if we may never be at the stable age distribution?
Solubility is useful to the chemist because it reveals something about the arrangement of electron in the substance and the solvent, even if the substance never encounters the solvent (Caswell 2001, p. 109).
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7. Clarification II: Prospective vs. Retrospective
E-pattern gives summary of the E-matrix and like E-matrix applies to the stable age distribution.
Management proposals based on E-pattern are prospective.
Management proposals based on retrospective analysis are actually studying life history variation in an evolutionary perspective (Caswell 2001).
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8. Clarification III: Productivity vs. Elasticities
 can be used as measure of productivity. Age at maturity () has large effect on productivity.
Elasticities are robust with respect to changes in  Population growth () has little effect on elasticities.
Will show that most important ratio in the E-pattern is the E3/E2 ratio (= Abar/ - 1). Abar/ is robust to changes in . Therefore not surprising that  has little effect on the E3/E2 ratio.
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10. Clarification IIII: Deterministic versus Stochastic
All my calculation are deterministic assuming constant vital rates and no density dependence.
E-patterns are also robust with respect to variations of vital rates (Nakaoka 1996; Dixon et al 1997).
However, I can calculate pseudo-stochastic E-patterns by using 1 instead of Abar. Use  = 1.0 approximation (Abar = 1 = T = ln /lnR0) arrived at by changing juvenile survival and/or fertility.
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12. Elasticities of vital rates (x) from the Characteristic Function
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13. E-pattern is determined by  and Abar alone (Mollet and Cailliet 2003 Appendix)
E(m) = E1 = 1/Abar E(Sj) = E2 = /Abar [E2/E1 = ] E(Sa) = E3 = (Abar - )/Abar [E3/E2 = (Abar/) -1]
Normalization needed for graph using E-triangle: En(m) = En,1 = 1/(Abar + 1) En(Sj) = En,2 = /(Abar + 1) En(Sa) = En,3 = (Abar - )/(Abar + 1)
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14. Abar, why not used in Caswell (2001)
in the formulation of E-patterns of age-structured animals?
It is simple: Abar and 1 are not well-defined for plants with complex reproductive cycles that require stage-based matrix population models.
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15. Sum of elasticities in the E-pattern, 1 or 1 + E(m)? .
Heppell et al. (2000); Caswell (2001) technically correct but….
1. E-pattern calculated empirically from LHT is only the same as that from the corresponding Leslie matrix if we include contribution of survival in the discounted fertility matrix elements.
2. E-pattern from post- and pre-breeding census Leslie matrix should give same E-pattern.
3. E-triangle needs to provide space for hypothetical  < 1 animals.
Calculations tedious, best illustrated with a graph:
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17. Formula for Abar with fewer terms
Formula for Abar with fewer terms ? Would help understanding and interpretation
Definition of Abar =  x -x lx mx, has ( -  +1) terms.
Abar = <w,v> if we choose w1 = 1 and v1 = 1. Easy to calculate but has many terms () (Mollet and Cailliet 2003).
3-term formula would be helpful even if only exactly correct species with age-independent Sa and m.
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18. Abar from the characteristic function (assuming age-independent Sa and m)
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19. New Abar Formula (Mollet 2005) “Bock Brew”: Everything was checked 2x
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20. E3/E2 from the characteristic function (assuming age-independent Sa and m for Abar)
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22. * * * * * * * *
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23. / 3 *
E3/E2  1.0 *
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24.  / ratio for elasmos
mean / = 2.68 (CV = 45%), n = 66, range ~ 5.0.
70% have /  3.0, thus E3/E2  1.0 without further considerations;
30% have / > 3.0 but Sa/1 sufficiently < 1.0 so that E3/E2  1.0.
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25. Recipe for management proposals based on E-pattern from prospective elasticity analysis
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!
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27. (Abar/) Ratios for Elasmos
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
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28. Elasticities of  and 
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