## 1. An Example using the Barndoor Skate

The vital rates of elasmobranchs and other long-lived marine species are not well known and population growth rates are mere guesstimates. However, elasticities are fairly robust even if the population growth rates are not very precise and do allow formulation of management proposal. My analysis indicates that the elasticity pattern can be predicted with a minimal amount of calculations and that it is determined by age-at-first reproduction (α, alpha) and the mean age of reproducing females (Ā, Abar) at the stable age-distribution. Knowledge of the gestation period (GP) may allow a refinement but here I assume GP = 0 in the formulas given in Mollet and Cailliet (2003, Appendix1b, normalized elasticities; i.e. sum is now 1.0 whereas before normalization the sum of the elasticities is 1+ E1):

E(fertility) = E1 = 1/(Ā + 1);
E(juvenile survival) = E2 = (α)/(Ā + 1);
E(adult survival) = E3 = (Ā - α)/(Ā + 1).

All we need to calculate the E-pattern of an elasmobranch is alpha and Abar. Easier said than done because the exact formula given below for Abar requires lambda. However, we can proceed with several approximations and compare them with the "correct" E-pattern. The barndoor skates shall serve as an example. The complete Leslie matrix with solution and the calculation of the E-pattern are included in the table below with a link already provided here. The "correct" normalized E-pattern is E1 = 0.08751, E2 = 0.7876, E3 = 0.1249. (E2/E1 = 9.0, E3/E1 = 1.43; E2/E1 + E3/E1 = 10.43 = Abar) . [Note that E2 >> E3 (E2/E3 = 6.3) as pointed out by Mollet and Cailliet (2003) when discussing Frick MG, Miller TJ, & Fogerty MJ (2002) who reported that E3 > E2. Accordingly, they were "off" by at least a factor of 6.3, almost an order of magnitude.

Abar can be calculated from the following 3-term formula which assumes constant fertility (m) and adult survival (Sa): while perhaps intimating at first, it is just a formula that is easily calculated.

1st Approximation. Abar (mean age of reproducing female) must be at least alpha (age at first reproduction), I will use Abar = 9.1 instead of 9.0 to avoid E3 = 0.0.
Using this value for the barndoor skate we obtain the following approximate E-pattern: E1 = 1/10.1 = 0.099; E2 = 9/10.1 = 0.891; E3 = 0.1/10.1 = 0.01. Note that this calculation predicts that E3 (elasticity of adult survival) is negligibly small. While certainly a rough approximation, it actually provides a much better approximation for the purpose of a management proposal than what was reported by Frick MG, Miller TJ, & Fogerty MJ (2002) who claimed that the elasticity of adult survival (E3) was larger than the elasticity of juvenile survival (E2).

2nd Approximation based on two terms in the above formula [Abar = alpha + Sa/(lambda-Sa)] with Sa =0.73564 and assuming lambda = 1.0 (stationary age distribution): Abar = 9.000 + 0.73564/(1.0 - 0.73564) = 11.783 which produces the following E-pattern: E1 = 1/12.783 = 0.078, E2 = 0.704, E3 = 0.218 which yields E2/E3 = 3.23.

3rd Approximation based on two terms in above formula but now using calculated lambda = 1.1624: Abar = 9.0000 + 0.7356/(1.1624 - 0.7356) = 10.7238 which produces the following E-pattern: E1 = 0.0883, E2 = 0.7677, E3 = 0.1470 which yields E2/E3 = 5.22.

4th "Approximation'" using all three terms in formula above gives Abar = 9.0000 + 1.7238 - 0.2967 = 10.4271 and the following E-pattern E1 = 0.08751, E2 = 0.7876, E3 = 0.1249 which yields E2/E3 = 6.31. "Approximation" in the sense that the parameters used are estimates. Using all 3 terms in the formula for Abar provides of course an exact result.

Note that Frick MG, Miller TJ, & Fogerty MJ (2002) used somewhat different vital rates (alpha = 15 yr, omega = 50 yr, S1 = 0.30000, S2-S50 = 0.91393 which produces a somewhat different E-pattern which would yield E2/E3 = 3.98 if the E-pattern is correctly calculated. They had used a stage-based model for the barndoor skate and incorrectly found that E3 > E2 which would lead to wrong management proposals.

The above formulas for the E-pattern can be used to calculate the requirement that E3 becomes at least as large as E2 which is Abar = 2 * alpha. If alpha used is 15 yr, then Abar would have to be at least 30 yr. This would require much larger survival rates, survival rates that would be more appropriate for a mammal than an elasmobranch.

## 2. A Table with Demographic Results for many Elasmobranchs

There is no need to fill in the gaps in this table because the E-pattern for all sharks has E3/E2 < 1 because Abar/alpha < 2 (see table below). In cases where the original calculations were carried out with a stage-based model, I have indicated that the calculated E-pattern is not correct unless it is calculated empirically.
Section 4. below refers to a pdf with calculated and graphed E-patterns for 66 shark populations which included the sharks listed in the table below. The calculation of Abar requires lambda (see above) and thus the Euler equation has to be solved. At a more basic level it can be shown that sharks with omega/alpha <= 3 must have Abar/alpha < 2 and for the sharks with omega/alpha > 3, x = Sa/lambda is sufficiently smaller than 1.0 such that E3/E2 is still <1.0 .

[Alpha = age-at-first-reproduction, Abar = mean age of reproducing females. at the stable age distribution, omega = maximum age (longevity estimate)]

Elasmobranch
Vital Rates or Preliminary Demography
Elasticity Pattern & Determining Parameters
Demography Publications
E1 (%)

E2 (%)

E3 (%)

Abar/alpha Omega/alpha
Angel shark, Squatina californica
Ayres, 1859
~E-pattern based on Mu1 =14.5 yr given in Cailliet et al. 1992
~6.4
~64.5
~29.0
~1.5
3.5
Barndoor skate,
Dipturus laevis Mitchill, 1817
8.75
78.8
12.5
1.159
1.667
Frisk et al (2002) but see Mollet and Cailliet (2003) for errors made because stage-based model was used.
Cetorhinus maximus (Gunnerus, 1765)
3.94
71.0
25.1
1.353
2.222

Bat ray, Myliobatis californica (Gil~l 1865)
~10.0
~60.0
~30.0
assumed 1.50

Blue shark, Prionace glauca
(Linnaeus, 1758)

Bull shark

Catshark, brown, Apristurus brunneus (Gilbert, 1892)

Catshark, filetail, Parmaturus xaniuru (Gilbert, 1892)

Dogfish, spiny, Squalus acanthia
(Linnaeus, 1758)

Leopard shark, Triakis semifasciata  Girard, 1854
5.74
68.9
25.3
1.368
check
Horn shark, Heterodontus francisci (Girard, 1854)

Mako, shorfin , Isurus oxyrinchus
Rafinesque, 1809
9.34
65.4
25.3
1.387
3.0
Mako, longfin , Isurus paucus
Guitard, 1966

Oceanic whitetip shark, Carcharhinus longimanus
(Poey, 1861)

Pacific sleeper shark, Somniosus pacificus  Bigelow and Schroeder, 1944

Pelagic stingray, Dasyatis (Pteroplatytrygon) violacea
(Bonaparte, 1832)
19.6
58.8
21.7
1.368
3.333
(Porbeagle, NWA, Lamna nasus
(Bonnaterre, 1788))
5.58
75.1
19.1
1.255
2.308
Cortes 2002; Campana et al. (in press)
Prickly shark, Echinorhinus cookei  Pietschmann, 1928

(Sandbar shark, NWA)

Salmon shark, Lamna ditropis
Hubbs & Follett, 1947
7.50
67.5
25.0
1.371
check

Sandtiger Shark, NWA,
Carcharias taurus
(Rafinesque, 1810)
9.17
55.0
35.8
1.650

4.167

School shark, Galeorhinus galeus

Sevengill shark, Notorynchus cepedianus  (Peron, 1807)
4.97
79.4
15.6
1.196
2.813
Smith et al. 1999;
Sixgill shark, Hexanchus griseus (Bonnaterre, 1788)

Smoothhound, brown
Mustelus henlei (Gill, 1863)
17.6
52.8
29.6
1.560
3.000 check
Smoothhound, gray Mustelus californicus Gill, 1864
17.2
51.7
31.0
1.600
4.333 check
Soupfin shark Mustelus californicus
Gill, 1864

Swell shark
Cephaloscyllium ventriosum (Garman, 1880)

Thresher shark, bigeye Alopius supersiliosus  (Lowe, 1839)

Thresher shark, common Alopius vulpinus (Bonnaterre, 1788)

Thresher shark, pelagic Alopius pelagicus Nakamura, 1935

Tiger shark Galeocerdo cuvier
(Peron & LeSueur, 1822)

Whale Shark
Rhincodon typusSmith, 1828
4.29
77.3
18.5
1.239
3.055
Bradshaw, Mollet, Meekan (2007) Tables 3 &4 have E-patterns for 4 scenarios using 1-yr & 2-yr projection interval
White Shark Carcharodon carchariasÊ (Linnaeus, 1758)

Elasmobranch
Vital Rates or Preliminary Demography
Elasticity Pattern & Determining Parameters
Publications
E1 (%)

E2 (%)

E3 (%)

Abar/alpha Omega/alpha

## 4. The Complete Works on Elasticities for the 21st Century The Whole Enchilada comprising 47 double-spaced 'layers' plus two Appendices. Caution: Paper includes so much data, graphs, and complete discussion for 240 populations of elasmobranchs, mammals, turtles, and birds that it might cause appendicitis.

Abstract. The elasticity pattern (E-pattern) of an age-structured animal comprising the elasticities E(fertility, m), E(juvenile survival, Sj), and E(adult survival, Sa) is determined by age at first reproduction (alpha) and mean age of the reproducing females at the stable age distribution (Abar). The E-patterns of sharks and marine turtles are characterized by Abar/alpha ratios < 2.0 [E(Sa)/E(Sj) < 1] and a proportional change in juvenile survival has the largest effect on population growth (lambda1). Marine mammals and birds generally have Abar/alpha ratios > 2.0 and adult survival has the largest effect on population growth. Terrestrial turtles, mammals, and birds show a large range of Abar/alpha ratios. The fast-slow continuum concept is not useful to understand E-patterns of these animals in the context of a prospective elasticity analysis with the goal of providing management proposals. It is important to include the survival part in the discounted fertilities of the Leslie matrix when calculating the E-pattern, otherwise post- and pre-breeding censuses will yield different and biased E-patterns. This bias is largest for animals with alpha = 1 yr. The sum of the E- pattern is 1 + E(m) and has to be normalized when graphed in an elasticity triangle for easy interpretation. Assuming age-independent m and Sa, a new 3-term algebraic equation for Abar facilitates the understanding and interpretation of E-patterns.

Created February 1997; updated April 2014. Back to previous page