Elasticity Patterns for Elasmobranchs (in progress)

Abstract

Vital parameters (age-at-first-reproduction, maximum reproductive age, age-specific fertilities, and age-specific mortalities of lamnids were reviewed. Despite some progress in the last 20 years, the vital rates of lamnids and many other elasmobranch are not sufficiently well known to produce reasonably accurate population growth rates (λ). Luckily, elasticity patterns, which give the proportional changes of λ due to proportional changes of vital parameters, that are needed to evaluate management proposals (prospective analyses) are fairly robust and do not require precise λ. Furthermore, there is no need to actually calculate λ, because the elasticity pattern can be estimated from age-at-first-reproduction (α) and generation time (Ã) alone, with gestation period (GP) providing a refinement. If à is not known, it can be estimated from calculated Ã/αratios of other elasmobranch. The elasticities are given by the following formulas without the need of any life history tables or Leslie matrix calculations:

E(fertility) = E(m) = E1 = 1/Ã,
E(juvenile survival) = E(js) = E2 = (α - GP) E1,
E(adult survival) = E(as) = E3 = 1 - E2 = (Ã - α + GP) E1

The elasticity pattern as a function of age-at-first reproduction indicates that E2 is largest for all elasmobranchs if α > 1 yr (valid for most if not all elasmobranchs), Accordingly, it is the protection of juveniles, which will provide the most effective measure to reverse population declines if that has been observed for an elasmobranch species.

1. Introduction

1.1 Management guidelines for species with limited/minimal data were presented by Heppell et al. (2000) and Caswell (2001, p. 231/2) and were based on elasticity (E) patterns of E(fertility), E(juvenile survival), and E(adult survival). They showed that the elasticity pattern is determined by age-at-first-reproduction (α), adult survival (σ), and annual population growth (λ, largest eigenvalue). If λ is not known it can be assumed to be 1.0 or varied over some range. I am proposing a further simplification by showing that the elasticity pattern is determined by α and à alone (à is one of several measures of generation time). This facilitates the understanding and interpretation of elasticity patterns considerably. If à is not known it can be estimated. If adult survival is very low (high adult mortality) then à ~ α (i.e. one litter only); if adult survival is very high (low adult mortality) then à ~ (α + ω)/2 where ω = maximum age of the reproductive females.

1.2 Notes:
A) The elasticity pattern used by Heppell et al. (2000) was based on E(discounted fertility, F), E(juvenile survival excluding survival to age 1), and E(adult survival), whereas I am calculating the elasticity patterns for the lower level elastiscities E(fertility, m), E(juvenile survival including survival to age 1), and E(adult survival), i.e. the vital parameters as they appear in the life history table. In addition, Heppell et al. (2000) assumed that adult survival begins at age-at-first reproduction (α), whereas I include a gestation period, as is appropriate for elasmobranchs. Therefore the maturing "juveniles", which have their first litter at α, are adult females pregnant for the first time with associated adult survival.
B) The simplification is possible because E(fertility) = 1/<w,v> = 1/Ã. See eq. 9.96 in Caswell 2001 for the first part of this equation and Appendix A below for the second part. The scalar product <w,v> of the age-structure-vector (w) and reproductive-value-vector (v) is defined here as the scalar product when w₁ = 1.0 and v₁ = 1.0. This simplification is valid, not only for the stage-based model presented by Heppell et al. (2000), it applies to a Leslie matrix and any stage-based model as long as it begins with a stage of duration 1 yr (i.e. an age-class) (see Appendix A).
C) Yes, λ in the Heppell et al. (2000) model depends on juvenile survival and fertility , but it justified to assume that is around 1.0 (stationary population) as a first approximation. Yes, in my model, à depends on λ (thus juvenile survival and fertility) and adult survival , but the use of an approximation for à can be justified because adult survival is often not known and importantly using à (instead of λ and adult survival) provides a better understanding of the elasticity pattern.

2. Methods

2.1. I use an age-structured life history table (LHT) or Leslie matrix with age-at-first reproduction (α), maximum age of reproducing females (ω), fertility (m), juvenile survival (js), and adult survival (as). I assume that the maturing juveniles, which have their first litter after one projection interval are adults and use a gestation period (GP), i.e.α = age-at-maturity + GP. In the life history table it implies that the row preceeding the row with the first fertility (= rowα) uses adult survival rather than juvenile survival. As all calculations here are approximation using constant survival (mortality rates) this is not apparent when looking at the LHT. In the Leslie matrix, the first discounted fertility matrix element (F₁) is accordingly calculated as F₁ = m_α σ_α with σ_α = adult survival rather than juvenile survival. If one constructs a stage-based model with all the adult age-classes combined into one stage (Heppell et al. 2000), then a post-breeding census rather than a pre-breeding census is most appropriate. Again this is not apparent if one looks at the Leslie matrix or the stage-based matrix but it does affect the elasticity ratios because there are α - 1 rather than α juvenile age classes.

2.2. Finite geometrical series: a₁ + a₁ q + a₁ q² + ..... + a₁ q^n-1 = (a₁ - a₁ qⁿ)/(1 - q)

2.3. Net reproductive rate R₀ = l_α m_α + l_α+1 m_α+1 + .... + l_ω m_ω
If fertility m and adult survival rate are constant, then R₀ is a finite geometrical series and the sum can be calculated from 2. with a₁ = l_α m_α ; q = P = l_α+1 / l_α ; n = T = ω - α +1. Add defintion of T.

2.4. Euler Sum = λ^-α l_α m_α + λ^-(α+1) l_α+1 m_α+1 + .... + λ^-ω l_ω m_ω
Again, if fertility m and adult survival rate are constant, then the Euler sum is a finite geometrical series and can be calculated. Lambda (λ) is used as an adjustable parameter until the Euler Sum is 1.0 (e.g. by using Solver in Excel).
The advantage of this procedure is that there is no need to set up a life history table with a large number of rows (= ω). Lambda (λ) of a large number of species or one species with a large number of different vital rates can be calculated in a single spreadsheet with each species/vital rates on one row.

Note that juvenile survival determines l_α (the fraction of individuals left at the age-of-first-reproduction). Therefore juvenile survival is not required to be constant and can be different from adult survival. In addition, λ (and R₀₎ do not change if fertility is increased (e.g. millions of eggs in the case of long-lived rockfish) with a corresponding decrease of first-year survival to keep l_α m_α constant.

2.5. The calculation of elasticities requires à (Abar, the mean age of the reproducing females at the stable age distribution).
à = α λ^-α l_α m_α + (α+1) λ^{- (α +1)} l_α+1 m_α+1 + .... + ω λ^-ω l_ω m_ω
This is not a geometric series even if fertility m and adult survival rate are assumed to be constant because the quotient includes the term (α + 1)/α, which is not constant.
I need a closed formula for à if I do not want to calculate it with a LHT or Leslie matrix in order to be able to it include it with a calculation of R₀ and λ on one row for each species in a spreadsheet and then use it to calculate the elasticity pattern. Therefore, I used the following approximation for à based on the model for calculation of elasticities from minimal data (Heppell et al. 2000; Caswell 2001, p. 231/2).
à ~ α- 1 + λ/(λ - P_α) ~ α + P_α/(λ - P_α) with P_α = σ (1- γ) where σ = adult survival and γ = fraction of adults graduating to next stage which is given by eq. 6.103 in Caswell 2002 on p. 161 with the parameters σ/γ and T = ω -α +1.

Note: The à approximation using Heppell et al. (2000) stage-based model can be shown to be always be larger than à using LHT or Leslie matrix (see Appendix B). Therefore have to check how large the difference is or produce a better approximation for à if I do not want to do the LHT or Leslie matrix calculation (i.e. calculation of R₀ and λ for each species on one line). Preliminary calculations have indicated that for elasmobranch, the à approximation is reasonably close to the à calculated from a LHT or Leslie matrix.

2.6. Elasticities of the "lower level" parameters fertility, juvenile survival, and adult survival (note that the Leslie matrix parameters are discounted fertility. juvenile survival, and adults survival). The formulas assume a gestation period of duration GP and that the maturing "juveniles" are pregnant adults with adult survival

E(fertility) = E(m) = E1 = 1/Ã;
E(juvenile survival) = E(js) = E2 = (α - GP) E1, Ratio E2/E1 = ER2 = α - GP;
E(adult survival) = E(as) = E3 = 1 - E2 = ( Ã - α + GP) E1, Ratio E3/E1 = ER3 = Ã - α + GP;
E1 + E2 + E3 = 1 + E1 (or E2 + E3 = 1), ER2 + ER3 = Ã;

(E1, E2, E3 are lower level elasticities and don't sum to 1, instead the survival elasticities E2 and E3 sum to 1 and the sum of the two elasticity ratios is Ã).

ER2 interpretation: ER2 is a measure of how much more effective a management proposal is that increases juvenile survival (e.g. turtle excluder devices) compared to increasing fertility (head start programs). It is not really surprising that ER2 is large (=α - GP) if one considers a cohort. Changing fertility by a proportional amount will have the same effect as changing juvenile survival of a ANY juvenile age class by the same amount. If there are many juvenile age classes, and if one considers a proposal which increases survivorship of ALL juvenile age classes, then of course this is far more effective than increasing fertility by the same proportional amount (e.g. 10%).

ER3 interpretation: ER3 gives the number of juvenile age classes that can be fished (Z = M + F(ish)), which will have the same effect on population growth (λ) as fishing all the adult age classes. This follows because E1 is the same as the elasticity of each juvenile age class. Please note that one has to use absolute changes of fishing corresponding to the proportional changes of the corresponding survival probabilities in order to produce the relative changes of λ (see Mollet and Cailliet 2002).

2.7. In the following row, I am calculating the elasticity patterns as a function of age-first-reproduction α and a = Ã/α. This will help to better understand the graphs of the elasticity patterns for different a = Ã/α ratios in the results section.

A. Gestation period = 1 yr

E1 = 1/(a α)
E2 = (α - 1)/(/a α)
E3 = 1 - (α - 1)/(/a α)

B. Gestation period = 0 yr

E1 = 1/(a α)
E2 = 1/a = constant
(i.e. independent of α)
E3 = 1 - 1/a = constant

C. Excluding survival to age 1 (Heppell et al. 2000)

E1 = 1/(a α)
E2 = (α - 1)/(/a α)
E3 = a/(1 - a) = constant
(i.e. independent of α)

3. Results (cannot get it published because reviewers insist that my results are wrong wheras without a shadow of a doubt, I have proved the the current theory is wrong).