gnomonicM-package {gnomonicM}R Documentation

Estimate Natural Mortality for Different Life Stages.

Description

Estimate natural mortality (M) throughout the life history for organisms, mainly fish and invertebrates, based on gnomonic interval approach. It includes estimation of duration of each gnomonic interval (life stage) and the constant probability of death (G).

Details

Package: gnomonicM

Type: Package

The natural mortality (M) estimation throughout different life stages is based on the gnomonic approach (Caddy, 1991, 1996), including new features in this package-version.

In the gnomonic model, the estimation of MiM_{i} for each gnomonic interval Δi\Delta_{i} requires -at least- information about: (i) the number of development stages throughout the life cycle ii inin 1,2,3,n1,2,3, …n. (ii) the duration of the first life stage corresponding to first gnomonic interval (Δ1\Delta_{1}, egg stage), (iii) the mean lifetime fecundity MLFMLF, and (iv) the longevity of the species. As additional information, the duration of the other developments stages or gnomonic intervals (larval, juvenile, adults) could be provided.

According to Caddy (1996) and Martinez-Aguilar (2005), the gnomonic method is supported by a negative exponential function, where the independent variable is Δi\Delta_{i} representing the number of gnomonic intervals from ii inin 1,2,3,n1,2,3, …n, the equation is expressed as follows:

Ni=MLFe(MiΔi);fori=1N_{i} = MLF \cdot e^{-(M_{i} \cdot \Delta_{i})}; for i = 1

Ni=Ni1e(MiΔi);fori>1N_{i} = N_{i-1} \cdot e^{-(M_{i} \cdot \Delta_{i})}; for i > 1

where:

MiM_{i} is the average value for natural mortality rate, that integrates the declining death rate through the short time interval duration Δi\Delta_{i}. The NiN_{i} is the survivors from previous interval, only for the first interval (Δ1\Delta_{1}) is assumed that the numbers of hatching eggs (initial population) is equivalent to the mean lifetime fecundity (MLFMLF).

The duration of first gnomonic interval Δ1\Delta_{1} is equal to the time elapsed after the moment of hatching t1t_{1}. The duration of the subsequent gnomonic intervals (i>1i > 1) are estimated following:

Δi=Δ1α(α+1)i2\Delta_{i} = \Delta_{1} \cdot \alpha (\alpha + 1)^{i-2}

where,

Δi\Delta_{i}: Duration of the gnomonic interval when i>1i > 1.

Δ1\Delta_{1}: Duration of the first gnomonic interval t1t_{1}.

α\alpha: Proportionality constant.

ii: ithi^{th} gnomonic interval.

The MiM_{i} is estimated as follows:

Mi=GΔi,i1M_{i} = \frac{G}{\Delta_{i, i-1}}

where GG is the constant proportion of the overall natural death rate. The GG value is calculated so that the number of individuals surviving to the last gnomonic time-interval is Nn=2N_{n} = 2 following the assumption of stable population replacement (Caddy, 1996; Martinez-Aguilar, 2005). The new equation for GG is expressed:

G=ln((2MLF)1n)G = -ln((\frac{2}{MLF})^{\frac{1}{n}})

The final solution is to estimate the proportionality constant (α\alpha) parameter by iterative solution via univariate (1-dim.) minimization.

Author(s)

Josymar Torrejon-Magallanes <ejosymart@gmail.com>

References

Caddy JF (1991). Death rates and time intervals: is there an alternative to the constant natural mortality axiom? Rev Fish Biol Fish 1:109–138. doi:10.1007/BF00157581.

Caddy JF (1996). Modelling natural mortality with age in short-lived invertebrate populations: definition of a strategy of gnomonic time division. Aquat Living Resour 9:197–207. doi:10.1051/alr:1996023.

Martínez-Aguilar S, Arreguín-Sánchez F, Morales-Bojórquez E (2005). Natural mortality and life history stage duration of Pacific sardine (Sardinops caeruleus) based on gnomonic time divisions. Fish Res 71:103–114. doi:10.1016/j.fishres.2004.04.008.

Examples

#See examples for functions gnomonic() and gnomonicStochastic().

[Package gnomonicM version 1.0.1 Index]