Several methods can be used to estimated natural mortality (M) from other types of data, including parameters from the von Bertalanffy growth equation, maximum age, and temperature. These relationships have been developed from meta-analyses of a large number of populations. Several of these methods are implemented in this function.
Usage
Mmethods(method = c("all", "tmax", "K", "Hoenig", "Pauly", "FAMS"))
metaM(
method = Mmethods(),
tmax = NULL,
K = NULL,
Linf = NULL,
t0 = NULL,
b = NULL,
L = NULL,
Temp = NULL,
t50 = NULL,
Winf = NULL,
PS = NULL,
verbose = TRUE
)Arguments
- method
A string that indicates what grouping of methods to return (defaults to all methods) in
Mmethods()or which methods or equations to use inmetaM(). See details.- tmax
The maximum age for the population of fish.
- K
The Brody growth coefficient from the fit of the von Bertalanffy growth function.
- Linf
The asymptotic mean length (cm) from the fit of the von Bertalanffy growth function.
- t0
The x-intercept from the fit of the von Bertalanffy growth function.
- b
The exponent from the weight-length relationship (slope from the logW-logL relationship).
- L
The body length of the fish (cm).
- Temp
The temperature experienced by the fish (C).
- t50
The age (time) when half the fish in the population are mature.
- Winf
The asymptotic mean weight (g) from the fit of the von Bertalanffy growth function.
- PS
The proportion of the population that survive to
tmax. Should usually be around 0.01 or 0.05.- verbose
Logical for whether to include method name and given inputs in resultant data.frame. Defaults to
TRUE.
Value
Mmethods returns a character vector with a list of methods.
metaM returns a data.frame with the following items:
M: The estimated natural mortality rate.cm: The estimated conditional natural mortality rate (computed directly fromM).method: The name for the method within the function (as given inmethod).name: A more descriptive name for the method.givens: A string that contains the input values required by the method to estimate M.
Details
One of several methods is chosen with method. The available methods can be seen with Mmethods() and are listed below with a brief description of where the equation came from. The sources (listed below) should be consulted for more specific information.
method="HoenigNLS": The “modified Hoenig equation derived with a non-linear model” as described in Then et al. (2015) on the third line of Table 3. This method was the preferred method suggested by Then et al. (2015). Requires onlytmax.method="PaulyLNoT": The “modified Pauly length equation” as described on the sixth line of Table 3 in Then et al. (2015). Then et al. (2015) suggested that this is the preferred method if maximum age (tmax) information was not available. RequiresKandLinf.method="PaulyL": The “Pauly (1980) equation using fish lengths” from his equation 11. This is the most commonly used method in the literature. Note that Pauly used common logarithms as used here but the model is often presented in other sources with natural logarithms. RequiresK,Linf, andT.method="PaulyW": The “Pauly (1980) equation for weights” from his equation 10. RequiresK,Winf, andT.method="HoeingO",method="HoeingOF",method="HoeingOM",method="HoeingOC": The original “Hoenig (1983) composite”, “fish”, “mollusc”, and “cetacean” (fit with OLS) equations from the second column on page 899 of Hoenig (1983). Requires onlytmax.method="HoeingO2",method="HoeingO2F",method="HoeingO2M",method="HoeingO2C": The original “Hoenig (1983) composite”, “fish”, “mollusc”, and “cetacean” (fit with Geometric Mean Regression) equations from the second column on page 537 of Kenchington (2014). Requires onlytmax.method="HoenigLM": The “modified Hoenig equation derived with a linear model” as described in Then et al. (2015) on the second line of Table 3. Requires onlytmax.method="HewittHoenig": The “Hewitt and Hoenig (2005) equation” from their equation 8. Requires onlytmax.method="tmax1": The “one-parameter tmax equation” from the first line of Table 3 in Then et al. (2015). Requires onlytmax.method="K1": The “one-parameter K equation” from the fourth line of Table 3 in Then et al. (2015). Requires onlyK.method="K2": The “two-parameter K equation” from the fifth line of Table 3 in Then et al. (2015). Requires onlyK.method="JensenK1": The “Jensen (1996) one-parameter K equation”. Requires onlyK.method="JensenK2": The “Jensen (2001) two-parameter K equation” from their equation 8. Requires onlyK.method="Gislason": The “Gislason et al. (2010) equation” from their equation 2. RequiresK,Linf, andL.method="AlversonCarney": The “Alverson and Carney (1975) equation” as given in equation 10 of Zhang and Megrey (2006). RequirestmaxandK.method="Charnov": The “Charnov et al. (2013) equation” as given in the second column of page 545 of Kenchington (2014). RequiresK,Linf, andL.method="ZhangMegreyD",method="ZhangMegreyP": The “Zhang and Megrey (2006) equation” as given in their equation 8 but modified for demersal or pelagic fish. Thus, the user must choose the fish type withgroup. Requirestmax,K,t0,t50, andb.method="RikhterEfanov1": The “Rikhter and Efanov (1976) equation (#2)” as given in the second column of page 541 of Kenchington (2014) and in Table 6.4 of Miranda and Bettoli (2007). Requires onlyt50.method="RikhterEfanov2": The “Rikhter and Efanov (1976) equation (#1)” as given in the first column of page 541 of Kenchington (2014). Requirest50,K,t0, andb.method="QuinnDeriso": The “Quinn & Derison (1999)” equation as given in the FAMS manual as equation 4:18. RequiresPSandtmax. Included only for use withrFAMSpackage.method="ChenWatanabe": The “Chen & Watanabe (1989)” equation as given in the FAMS manual as equation 4:24. As suggested in FAMS manual usedtmaxfor final time and 1 as initial time. Requirestmax,K, andt0. Included only for use withrFAMSpackage.method="PetersonWroblewski": The “Peterson & Wroblewski (1984)” equation as given in the FAMS manual as equation 4:22. As suggested in FAMS manual usedWinffor weight. RequiresWinf. Included only for use withrFAMSpackage.
Conditional mortality (cm) is estimated from instantaneous natural mortality (M) with 1-exp(-M). It is returned with M here simply as a courtesy for those using the rFAMS package.
Testing
Kenchington (2014) provided life history parameters for several stocks and used many models to estimate M. I checked the calculations for the PaulyL, PaulyW, HoenigO, HoenigOF, HoenigO2, HoenigO2F, "JensenK1", "Gislason", "AlversonCarney", "Charnov", "ZhangMegrey", "RikhterEfanov1", and "RikhterEfanov2" methods for three stocks. All results perfectly matched Kenchington's results for Chesapeake Bay Anchovy and Rio Formosa Seahorse. For the Norwegian Fjord Lanternfish, all results perfectly matched Kenchington's results except for HoenigOF and HoenigO2F.
Results for the Rio Formosa Seahorse data were also tested against results from M.empirical from fishmethods for the PaulyL, PaulyW, HoenigO, HoenigOF, "Gislason", and "AlversonCarney" methods (the only methods in common between the two packages). All results matched perfectly.
References
Ogle, D.H. 2016. Introductory Fisheries Analyses with R. Chapman & Hall/CRC, Boca Raton, FL.
Alverson, D.L. and M.J. Carney. 1975. A graphic review of the growth and decay of population cohorts. Journal du Conseil International pour l'Exploration de la Mer. 36:133-143.
Chen, S. and S. Watanabe. 1989. Age dependence of natural mortality coefficient in fish population dynamics. Nippon Suisan Gakkaishi 55:205-208.
Charnov, E.L., H. Gislason, and J.G. Pope. 2013. Evolutionary assembly rules for fish life histories. Fish and Fisheries. 14:213-224.
Gislason, H., N. Daan, J.C. Rice, and J.G. Pope. 2010. Size, growth, temperature and the natural mortality of marine fish. Fish and Fisheries 11:149-158.
Hewitt, D.A. and J.M. Hoenig. 2005. Comparison of two approaches for estimating natural mortality based on longevity. Fishery Bulletin. 103:433-437. [Was (is?) from http://fishbull.noaa.gov/1032/hewitt.pdf.]
Hoenig, J.M. 1983. Empirical use of longevity data to estimate mortality rates. Fishery Bulletin. 82:898-903. [Was (is?) from http://www.afsc.noaa.gov/REFM/age/Docs/Hoenig_EmpiricalUseOfLongevityData.pdf.]
Jensen, A.L. 1996. Beverton and Holt life history invariants result from optimal trade-off of reproduction and survival. Canadian Journal of Fisheries and Aquatic Sciences. 53:820-822. [Was (is?) from .]
Jensen, A.L. 2001. Comparison of theoretical derivations, simple linear regressions, multiple linear regression and principal components for analysis of fish mortality, growth and environmental temperature data. Environometrics. 12:591-598. [Was (is?) from http://deepblue.lib.umich.edu/bitstream/handle/2027.42/35236/487_ftp.pdf.]
Kenchington, T.J. 2014. Natural mortality estimators for information-limited fisheries. Fish and Fisheries. 14:533-562.
Pauly, D. 1980. On the interrelationships between natural mortality, growth parameters, and mean environmental temperature in 175 fish stocks. Journal du Conseil International pour l'Exploration de la Mer. 39:175-192. [Was (is?) from http://innri.unuftp.is/pauly/On%20the%20interrelationships%20betwe.pdf.]
Peterson, I. and J.S. Wroblewski. 1984. Mortality rate of fishes in the pelagic ecosystem. Canadian Journal of Fisheries and Aquatic Sciences. 41:1117-1120.
Quinn III, T.J. and R.B. Deriso. 1999. Quantitative Fish Dynamics. Oxford University Press, New York.
Rikhter, V.A., and V.N. Efanov. 1976. On one of the approaches for estimating natural mortality in fish populations (in Russian). ICNAF Research Document 76/IV/8, 12pp.
Slipke, J.W. and M.J. Maceina. 2013. Fisheries Analysis and Modeling Simulator (FAMS 1.64). American Fisheries Society.
Then, A.Y., J.M. Hoenig, N.G. Hall, and D.A. Hewitt. 2015. Evaluating the predictive performance of empirical estimators of natural mortality rate using information on over 200 fish species. ICES Journal of Marine Science. 72:82-92.
Zhang, C-I and B.A. Megrey. 2006. A revised Alverson and Carney model for estimating the instantaneous rate of natural mortality. Transactions of the American Fisheries Society. 135-620-633. [Was (is?) from http://www.pmel.noaa.gov/foci/publications/2006/zhan0531.pdf.]
See also
See M.empirical in fishmethods for similar functionality.
Author
Derek H. Ogle, DerekOgle51@gmail.com
Examples
## List names for available methods
Mmethods()
#> [1] "HoenigNLS" "HoenigO" "HoenigOF"
#> [4] "HoenigOM" "HoenigOC" "HoenigO2"
#> [7] "HoenigO2F" "HoenigO2M" "HoenigO2C"
#> [10] "HoenigLM" "HewittHoenig" "tmax1"
#> [13] "PaulyLNoT" "PaulyL" "PaulyW"
#> [16] "K1" "K2" "JensenK1"
#> [19] "JensenK2" "Gislason" "AlversonCarney"
#> [22] "Charnov" "ZhangMegreyD" "ZhangMegreyP"
#> [25] "RikhterEfanov1" "RikhterEfanov2" "QuinnDeriso"
#> [28] "ChenWatanabe" "PetersonWroblewski"
Mmethods("tmax")
#> [1] "tmax1" "HoenigNLS" "HoenigO" "HoenigOF" "HoenigOM"
#> [6] "HoenigOC" "HoenigO2" "HoenigO2F" "HoenigO2M" "HoenigO2C"
#> [11] "HoenigLM" "HewittHoenig"
## Simple Examples
metaM("tmax",tmax=20)
#> M cm method name givens
#> 1 0.25545 0.2254321 tmax1 Then et al. (2015) tmax equation tmax=20
metaM("HoenigNLS",tmax=20)
#> M cm method name
#> 1 0.3150387 0.2702394 HoenigNLS Then et al. (2015) Hoenig (NLS) equation
#> givens
#> 1 tmax=20
metaM("HoenigNLS",tmax=20,verbose=FALSE)
#> M cm method
#> 1 0.3150387 0.2702394 HoenigNLS
## Example Patagonian Sprat ... from Table 2 in Cerna et al. (2014)
## http://www.scielo.cl/pdf/lajar/v42n3/art15.pdf
Temp <- 11
Linf <- 17.71
K <- 0.78
t0 <- -0.46
tmax <- t0+3/K
t50 <- t0-(1/K)*log(1-13.5/Linf)
metaM("RikhterEfanov1",t50=t50)
#> M cm method name
#> 1 1.050009 0.6500656 RikhterEfanov1 Richter & Evanov (1976) equation #1
#> givens
#> 1 t50=1.3818805
metaM("PaulyL",K=K,Linf=Linf,Temp=Temp)
#> M cm method name
#> 1 1.14058 0.6803665 PaulyL Pauly (1980) length equation
#> givens
#> 1 K=0.78, Linf=17.71, Temp=11
metaM("HoenigNLS",tmax=tmax)
#> M cm method name
#> 1 1.602862 0.7986805 HoenigNLS Then et al. (2015) Hoenig (NLS) equation
#> givens
#> 1 tmax=3.3861538
metaM("HoenigO",tmax=tmax)
#> M cm method name
#> 1 1.274125 0.7203245 HoenigO Hoenig (1983) combined equation (OLS)
#> givens
#> 1 tmax=3.3861538
metaM("HewittHoenig",tmax=tmax)
#> M cm method name
#> 1 1.246252 0.7124193 HewittHoenig Hewitt & Hoenig (2005) tmax equation
#> givens
#> 1 tmax=3.3861538
metaM("AlversonCarney",K=K,tmax=tmax)
#> M cm method name
#> 1 1.35398 0.7417895 AlversonCarney Alverson & Carney (1975) equation
#> givens
#> 1 tmax=3.3861538, K=0.78
## Example of multiple calculations
metaM(c("RikhterEfanov1","PaulyL","HoenigO","HewittHoenig","AlversonCarney"),
K=K,Linf=Linf,Temp=Temp,tmax=tmax,t50=t50)
#> M cm method name
#> 1 1.050009 0.6500656 RikhterEfanov1 Richter & Evanov (1976) equation #1
#> 2 1.140580 0.6803665 PaulyL Pauly (1980) length equation
#> 3 1.274125 0.7203245 HoenigO Hoenig (1983) combined equation (OLS)
#> 4 1.246252 0.7124193 HewittHoenig Hewitt & Hoenig (2005) tmax equation
#> 5 1.353980 0.7417895 AlversonCarney Alverson & Carney (1975) equation
#> givens
#> 1 t50=1.3818805
#> 2 K=0.78, Linf=17.71, Temp=11
#> 3 tmax=3.3861538
#> 4 tmax=3.3861538
#> 5 tmax=3.3861538, K=0.78
## Example of multiple methods using Mmethods
# select some methods
metaM(Mmethods()[-c(15,20,22:24,26:29)],K=K,Linf=Linf,Temp=Temp,tmax=tmax,t50=t50)
#> M cm method
#> 1 1.6028619 0.7986805 HoenigNLS
#> 2 1.2741252 0.7203245 HoenigO
#> 3 1.2562214 0.7152721 HoenigOF
#> 4 1.2401272 0.7106526 HoenigOM
#> 5 0.8835623 0.5866920 HoenigOC
#> 6 1.4786176 0.7720474 HoenigO2
#> 7 1.5784652 0.7937085 HoenigO2F
#> 8 1.4266652 0.7598917 HoenigO2M
#> 9 1.4625421 0.7683534 HoenigO2C
#> 10 1.6243510 0.8029605 HoenigLM
#> 11 1.2462517 0.7124193 HewittHoenig
#> 12 1.5087915 0.7788229 tmax1
#> 13 1.3304645 0.7356455 PaulyLNoT
#> 14 1.1405804 0.6803665 PaulyL
#> 15 1.3197600 0.7328006 K1
#> 16 1.3070000 0.7293693 K2
#> 17 1.1700000 0.6896331 JensenK1
#> 18 1.3566000 0.7424651 JensenK2
#> 19 1.3539801 0.7417895 AlversonCarney
#> 20 1.0500095 0.6500656 RikhterEfanov1
#> name givens
#> 1 Then et al. (2015) Hoenig (NLS) equation tmax=3.3861538
#> 2 Hoenig (1983) combined equation (OLS) tmax=3.3861538
#> 3 Hoenig (1983) fish equation (OLS) tmax=3.3861538
#> 4 Hoenig (1983) mollusk equation (OLS) tmax=3.3861538
#> 5 Hoenig (1983) cetacean equation (OLS) tmax=3.3861538
#> 6 Hoenig (1983) combined equation (GM) tmax=3.3861538
#> 7 Hoenig (1983) fish equation (GM) tmax=3.3861538
#> 8 Hoenig (1983) mollusk equation (GM) tmax=3.3861538
#> 9 Hoenig (1983) cetacean equation (GM) tmax=3.3861538
#> 10 Then et al. (2015) Hoenig (LM) equation tmax=3.3861538
#> 11 Hewitt & Hoenig (2005) tmax equation tmax=3.3861538
#> 12 Then et al. (2015) tmax equation tmax=3.3861538
#> 13 Then et al. (2015) Pauly_NLS-T equation K=0.78, Linf=17.71
#> 14 Pauly (1980) length equation K=0.78, Linf=17.71, Temp=11
#> 15 Then et al. (2015) one-parameter K equation K=0.78
#> 16 Then et al. (2015) two-parameter K equation K=0.78
#> 17 Jensen (1996) one parameter K equation K=0.78
#> 18 Jensen (2001) two parameter K equation K=0.78
#> 19 Alverson & Carney (1975) equation tmax=3.3861538, K=0.78
#> 20 Richter & Evanov (1976) equation #1 t50=1.3818805
# select just the Hoenig methods
metaM(Mmethods("Hoenig"),K=K,Linf=Linf,Temp=Temp,tmax=tmax,t50=t50)
#> M cm method name
#> 1 1.6028619 0.7986805 HoenigNLS Then et al. (2015) Hoenig (NLS) equation
#> 2 1.2741252 0.7203245 HoenigO Hoenig (1983) combined equation (OLS)
#> 3 1.2562214 0.7152721 HoenigOF Hoenig (1983) fish equation (OLS)
#> 4 1.2401272 0.7106526 HoenigOM Hoenig (1983) mollusk equation (OLS)
#> 5 0.8835623 0.5866920 HoenigOC Hoenig (1983) cetacean equation (OLS)
#> 6 1.4786176 0.7720474 HoenigO2 Hoenig (1983) combined equation (GM)
#> 7 1.5784652 0.7937085 HoenigO2F Hoenig (1983) fish equation (GM)
#> 8 1.4266652 0.7598917 HoenigO2M Hoenig (1983) mollusk equation (GM)
#> 9 1.4625421 0.7683534 HoenigO2C Hoenig (1983) cetacean equation (GM)
#> 10 1.6243510 0.8029605 HoenigLM Then et al. (2015) Hoenig (LM) equation
#> 11 1.2462517 0.7124193 HewittHoenig Hewitt & Hoenig (2005) tmax equation
#> givens
#> 1 tmax=3.3861538
#> 2 tmax=3.3861538
#> 3 tmax=3.3861538
#> 4 tmax=3.3861538
#> 5 tmax=3.3861538
#> 6 tmax=3.3861538
#> 7 tmax=3.3861538
#> 8 tmax=3.3861538
#> 9 tmax=3.3861538
#> 10 tmax=3.3861538
#> 11 tmax=3.3861538
## Example of computing an average M
# select multiple models used in FAMS (example only, these are not best models)
( res <- metaM(Mmethods("FAMS"),tmax=tmax,K=K,Linf=Linf,t0=t0,
Temp=Temp,PS=0.01,Winf=30) )
#> M cm method name
#> 1 1.3600003 0.7433393 QuinnDeriso Quinn & Deriso (1999) from FAMS
#> 2 1.2562214 0.7152721 HoenigOF Hoenig (1983) fish equation (OLS)
#> 3 1.1700000 0.6896331 JensenK1 Jensen (1996) one parameter K equation
#> 4 0.8203911 0.5597405 PetersonWroblewski Peterson & Watanabe (1984) from FAMS
#> 5 1.1405804 0.6803665 PaulyL Pauly (1980) length equation
#> 6 0.1403480 0.1309443 ChenWatanabe Chen & Watanabe (1989) from FAMS
#> givens
#> 1 PS=0.01, tmax=3.3861538
#> 2 tmax=3.3861538
#> 3 K=0.78
#> 4 Winf=30
#> 5 K=0.78, Linf=17.71, Temp=11
#> 6 tmax=3.3861538, K=0.78, t0=-0.46
( meanM <- mean(res$M) )
#> [1] 0.9812569
( meancm <- mean(res$cm) )
#> [1] 0.5865493