Catch Curves (with selectivity) Installation
26 October, 2022
Catch curves (with selectivity)
Catch curve analysis is a method for estimating the total mortality of a stock (Z). The rate at which individuals die can be estimated from the slope ofa regression of the logarithms of relative numbers present in each age class on age. It can be used whenever there is one or more years of catch-at-age data (or at-length data, if they can be converted to age). The data can be fishery-dependent or -independent so long as they are representative of the population’s relative age/length structure. Fishing mortality (F) can be estimated as Z-M given an estimate of natural mortality (M) from another source (e.g., the literature, from a marine protected area, or from a tagging study).
The method is based on the equilibrium assumption (i.e., the total population receives constant recruitment and there is a constant mortality rate each year), which implies that all cohorts would be identical and the numbers in each age class would decline exponentially. Thus, if one assumes that fishing mortality and natural mortality have been constant, and one has the age-composition of the catch for a single year, if one plots the log-transformed numbers-at-age against age, then the slope of a linear regression through the data will provide an estimate of total mortality (Z).
Conventional catch-curve methods rely on the strong assumptions of constant fishing and natural mortality rates above some fully selected age that is usually estimated by visually inspecting a plot of log-catch at age vs age. Rather than just ignoring the earlier weakly selected ages below the (assumed) fully selected age, as in the classical catch-curve, it is possible to estimate selectivity parameters as well as the total mortality rate. However, this requires the use of an equilibrium based age-structured model rather than the log-transformed linear regression (see equations in Wayte and Klaer, 2010).
Catch curves with selectivity use a simple age-structured model to estimate selectivity from the age data. This provides an estimate of fully-selected fishing mortality rather than an average fishing mortality applied to all included age classes (as in the classic catch curve). This also circumvents the need to select a minimum age from which to fit the regression (to avoid the effects of selectivity), as is required for the classical catch curve.
As opposed to classic catch curves, catch curves with selectivity provide an estimate of fully-selected fishing mortality, rather than an average fishing mortality applied to all included age classes.
Although there are several packages that can be used to run catch curves with selectivity, the one by Dr Malcolm Haddon is supported here, because his version has been applied in several cases in Australia (refs needed).
See the Appendix for further description and equations, as supplied by Dr. Haddon and referencing Wayte and Klaer (2010) (Fisheries Research 106: 310-320).
Installation
This package is run from R as found on CRAN. R can be installed from https://cran.r-project.org/.
The recommended way to run R is using RStudio. After you have installed R, then install RStudio. It can be found https://www.rstudio.com/products/rstudio/download/.
The supported version of Catch Curve Analysis is part of a set of stock assessment tools within Dr Haddon’s datalowSA package. See the link https://github.com/haddonm/datalowSA.
Therefore one needs to first install datalowSA from within R.
Set up checking/installing packages functions
# Ordinary checking/installing function
Check_Install.packages <- function(pkg){
new.pkg <- pkg[!(pkg %in% installed.packages()[, "Package"])]
if (length(new.pkg))
install.packages(new.pkg,dependencies = TRUE) else print(paste0("'",pkg,"' has been installed already!"))
sapply(pkg, require, character.only = TRUE)
}
# github checking/installing function
Github_CheckInstall.packages <- function(github_pkg){
pkg <- tail(unlist(strsplit(github_pkg,"/")),1)
new.pkg <- pkg[!(pkg %in% installed.packages()[, "Package"])]
if (length(new.pkg))
devtools::install_github(github_pkg,build_vignettes=TRUE)
sapply(pkg, require, character.only = TRUE)
}Installing packages from GitHub directly.
Check and install “devtools”
Check_Install.packages("devtools")## [1] "'devtools' has been installed already!"## Loading required package: devtools## Loading required package: usethis## Error: package or namespace load failed for 'usethis' in loadNamespace(i, c(lib.loc, .libPaths()), versionCheck = vI[[i]]):
## namespace 'rlang' 1.0.5 is already loaded, but >= 1.0.6 is required## devtools
## FALSECheck and install datalowSA from github
# Check if existed and if not, install the required package TMBhelper:
github_pkg = "haddonm/datalowSA"
Github_CheckInstall.packages(github_pkg=github_pkg)## Loading required package: datalowSA## datalowSA
## TRUE#install_github("haddonm/datalowSA")
NB Once you have installed datalowSA, you don’t need to run the install.packages code chunk again. Luckily it will just give an error stating it won’t reinstall if your version is up to date.
Note that you may get an error, because the current version you have of Rcpp may preclude the installation of datalowSA. If this occurs, try reinstalling Rcpp from CRAN (Tools -> Install Packages).library(datalowSA)It should be noted that there are also good vignettes for this package by typing browseVignettes(package = “datalowSA”).
Reading in the data
To run the Catch Curve Analysis with Selectivity requires the following inputs:
- Numbers-at-age * - for a single year, as a vector. This must be absolute numbers (as opposed to proportions-at-age) and a representative (across time) random sample.
- A single natural mortality estimate *
- The maximum age (within the available data) *
- Age-at-50%-selectivity, a50 *
- Delta (age-at-95% selectivity (a95) less
the age-at-50% selectivity (a50), ie
(a95-a50) *
- An estimate of current fishing mortality (used as an initial value) *
(Note that delta feeds into the equation for selectivity-at-age:
Selectivity at age is modelled using (1+exp(-ln(19)(a - a50)/delta))-1 instead of (1+exp(-ln(19)(a -a50)/(a95-a50)))-1 to reduce the correlation between two estimated parameters (a50 and a95), and to avoid a50 values larger than a95 values. It is relatively easy to get a a95 value less than the a50 value, and this leads to a selectivity pattern that declines instead of increasing with age.
In general, it goes without saying that one requires a minimum number of ages in the data set to ensure that there are more data points than parameter values. The selectivity-included method in Dr. Haddon’s package estimates three parameters and it is a reasonable rule of thumb to have at least three data points per parameter estimated - so, at least nine ages here.
A simulated example dataset has been created for the whole toolbox. Let us now read in these data. We assume you have saved the data to the same directory as your working directory. Otherwise you can change our code to point to the directory where the data file is and call it your working directory using the setwd command.
The simulated catch-at-age data, TotalCatchAge, has years (1960-2001) in rows (2-43), with 25 age bins (from ages 1 to 25) in columns (B to Z). The first 11 years of this file have no data, so we will focus on year 1971 onwards.
Let us read in the data:
TotCatAge <- read.csv("TotalCatchAge.csv")
print(TotCatAge[12:42,])## YearsOut A1 A2 A3 A4 A5
## 12 1971 1.910415 0.3925469 0.8258727 16.170486 1.877880
## 13 1972 5.840126 6.0808642 1.4250714 2.104864 33.731255
## 14 1973 62.394002 15.8097160 44.9978582 7.727642 9.657692
## 15 1974 24.236550 118.0074654 23.0015271 51.392508 8.639112
## 16 1975 55.103716 42.7568940 325.3436211 48.613787 102.768715
## 17 1976 12.188843 102.7895846 63.0607343 371.431144 55.189997
## 18 1977 55.643306 12.7038754 223.0947889 112.513335 527.059248
## 19 1978 76.643167 70.9197949 25.9093006 258.989382 109.240022
## 20 1979 81.047009 140.5462582 130.8785724 40.843807 371.659430
## 21 1980 26.807254 138.8902626 231.3314230 156.984856 37.997537
## 22 1981 113.402232 32.9076055 294.9740543 334.191571 210.562848
## 23 1982 183.714672 132.7821316 59.2440875 379.157799 387.005313
## 24 1983 148.349856 389.6158453 257.9485600 96.057167 504.854930
## 25 1984 85.834605 363.2626107 686.4943428 349.365257 95.529260
## 26 1985 133.084778 75.6188777 470.2082807 769.466339 298.351901
## 27 1986 101.715251 259.6285064 119.7158847 641.722041 785.347181
## 28 1987 286.446263 135.5081637 443.1192489 172.312856 668.293122
## 29 1988 308.013299 455.4089268 248.9493585 581.282370 167.764770
## 30 1989 596.622027 473.7542170 599.7932138 214.131427 384.245819
## 31 1990 1122.484109 373.2161377 623.8125995 511.509113 159.229529
## 32 1991 148.449770 1989.3673138 520.7658285 559.171268 354.863224
## 33 1992 201.162806 123.3779336 2675.6240388 462.689508 388.971831
## 34 1993 27.397110 329.6243051 144.2653213 2151.587635 235.998073
## 35 1994 443.581717 50.0309362 477.3122639 158.166694 1697.555787
## 36 1995 1265.225845 18.0096479 89.0893601 431.900777 100.770044
## 37 1996 44.238434 2373.0474340 30.0203166 78.552392 277.098045
## 38 1997 31.438088 8.8899254 845.7349405 7.027043 11.366910
## 39 1998 16.823783 35.4404509 9.8173442 658.112337 3.784587
## 40 1999 10.469020 23.9587229 41.2237479 8.143751 520.974613
## 41 2000 26.843251 16.8107275 46.2265698 64.155627 8.694273
## 42 2001 39.994189 49.1654268 20.8476087 54.812648 67.803887
## A6 A7 A8 A9 A10 A11
## 12 0.4235131 1.0764674 5.620313 1.527880 3.0488564 2.074872e+00
## 13 3.6735822 0.6932627 1.827491 9.158322 2.2808116 4.245410e+00
## 14 156.4327838 14.2957686 2.746602 8.506518 39.8109582 1.177706e+01
## 15 8.5749914 133.8152545 11.737072 3.696920 8.0920327 2.796508e+01
## 16 16.0962080 11.2722323 215.224237 23.309309 3.0988263 9.996217e+00
## 17 86.1130907 13.8538426 13.588940 160.812182 15.5758392 4.213590e+00
## 18 67.2522226 103.2832831 20.280873 10.532464 188.4009945 1.636358e+01
## 19 475.3970046 51.7767864 73.055668 10.190807 14.6019250 1.449906e+02
## 20 131.0130449 540.0044436 58.489899 80.579764 14.4076450 1.141600e+01
## 21 324.3874383 121.9559262 455.457120 46.251419 81.7177133 1.382336e+01
## 22 39.6009276 378.1201915 130.873968 453.578522 49.0770492 6.207193e+01
## 23 207.4801095 39.2712029 300.558800 87.149002 357.1745328 3.692002e+01
## 24 435.9721487 206.2606835 40.758918 270.312910 81.1141696 3.124874e+02
## 25 454.1988904 346.5794458 164.564215 31.421909 190.2727395 5.145767e+01
## 26 70.7423726 289.9335413 214.307032 91.564524 15.5194472 1.275537e+02
## 27 239.0346777 52.9139342 217.612957 143.570593 46.8706752 1.121730e+01
## 28 657.9779811 174.7945636 39.602581 132.139531 86.9198092 3.954351e+01
## 29 500.6156585 405.2064002 117.069545 16.773959 76.8019004 3.785172e+01
## 30 89.7955500 201.4489260 170.623425 38.231450 8.0747136 2.196131e+01
## 31 192.4781307 34.0222976 71.736309 40.816773 9.8335650 1.415969e+00
## 32 81.9139204 83.1418089 11.063863 23.858871 14.1746997 9.953650e-01
## 33 156.6798065 23.9010080 30.579305 3.196548 4.0268565 2.220936e+00
## 34 198.3413233 60.9602678 13.961915 5.095099 0.5160038 1.962090e+00
## 35 167.8767647 81.8787208 32.601248 5.236099 2.4601533 7.193254e-02
## 36 888.0046035 57.3773141 29.131254 15.749575 0.3869628 3.428750e-01
## 37 46.2954214 342.7527245 21.530982 13.586527 3.1047126 3.315267e-02
## 38 39.1589728 5.9501188 32.787798 1.514699 0.9710801 2.169095e-03
## 39 7.2952123 21.4419043 3.213071 14.830530 1.5364099 5.703590e-01
## 40 2.8247573 5.2037954 13.468349 1.851832 8.3255664 6.372236e-01
## 41 630.0968344 1.9245927 4.117840 13.754784 1.8336675 1.018619e+01
## 42 11.0778164 500.8286221 2.348458 5.200383 9.1653060 1.822362e+00
## A12 A13 A14 A15 A16 A17
## 12 1.444049e+00 1.131696e+00 1.036129e+00 7.738659e-01 0.573972127 0.56727849
## 13 3.819904e+00 2.099353e+00 1.961948e+00 1.375408e+00 1.241416222 0.84977139
## 14 1.635968e+01 1.433114e+01 9.040685e+00 7.690224e+00 5.649773644 4.62796753
## 15 7.694873e+00 1.353481e+01 8.599677e+00 6.845094e+00 5.735573364 4.20218104
## 16 4.684429e+01 1.393139e+01 2.032851e+01 1.870078e+01 10.442120469 6.52826629
## 17 9.760003e+00 3.420136e+01 9.591182e+00 1.105961e+01 12.360166365 6.80116649
## 18 2.459466e+00 6.483654e+00 3.863917e+01 1.001368e+01 18.137117691 18.87582202
## 19 1.404280e+01 2.716079e+00 8.449055e+00 2.791701e+01 6.869512568 12.90826157
## 20 1.398043e+02 1.176627e+01 2.629969e+00 5.158894e+00 34.898452637 10.92050639
## 21 7.572340e+00 1.075001e+02 9.273752e+00 2.249562e+00 5.755758139 19.71812493
## 22 6.123684e+00 5.937859e+00 1.016002e+02 8.963431e+00 1.425079025 5.05897258
## 23 4.262454e+01 7.114140e+00 3.271413e+00 8.007298e+01 8.059648028 1.45956723
## 24 3.341098e+01 5.325094e+01 6.518383e+00 3.457130e+00 67.419743623 4.62477993
## 25 2.203049e+02 2.303407e+01 2.325284e+01 4.014672e+00 4.365335251 53.00089502
## 26 3.228934e+01 1.260496e+02 1.141339e+01 1.347403e+01 3.276615990 0.83420585
## 27 6.315319e+01 2.610839e+01 8.053372e+01 7.914731e+00 10.810479916 0.88292178
## 28 2.966183e+00 4.114913e+01 1.193642e+01 4.375533e+01 2.168250729 4.60665875
## 29 1.225767e+01 2.186655e+00 1.274821e+01 5.100118e+00 20.649468952 1.64731808
## 30 1.353283e+01 3.374532e+00 4.749975e-01 3.941835e+00 1.428536084 3.24862709
## 31 6.161638e+00 2.600022e+00 2.052710e+00 6.968363e-02 0.647079702 0.03371932
## 32 1.603187e-01 9.486003e-01 5.317343e-02 3.653987e-01 0.233024542 0.85584157
## 33 1.163430e+00 0.000000e+00 2.860974e-01 2.753646e-02 0.000000000 0.00000000
## 34 5.221866e-02 3.854199e-02 0.000000e+00 0.000000e+00 0.000000000 0.00000000
## 35 9.900001e-03 7.005426e-01 2.928362e-04 8.694902e-02 0.000000000 0.00000000
## 36 6.112787e-01 3.601882e-03 1.368553e-03 5.492618e-03 0.000000000 0.00000000
## 37 5.974887e-02 0.000000e+00 0.000000e+00 1.087493e-04 0.000000000 0.00000000
## 38 2.553828e-04 0.000000e+00 0.000000e+00 0.000000e+00 0.000000000 0.00000000
## 39 4.262780e-01 0.000000e+00 3.206832e-05 3.062924e-02 0.000000000 0.00000000
## 40 2.337456e-01 2.467208e-02 1.322278e-04 7.243238e-05 0.000000000 0.00000000
## 41 1.295448e+00 9.046427e-02 9.394603e-03 0.000000e+00 0.004335718 0.00000000
## 42 5.584710e+00 6.179673e-01 2.976959e-02 1.099712e-05 0.005835920 0.00000000
## A18 A19 A20 A21 A22 A23
## 12 0.46707001 3.595825e-01 3.090172e-01 2.259975e-01 0.1676614955 1.446701e-01
## 13 0.81118788 7.796925e-01 5.400465e-01 3.191857e-01 0.3377839656 3.420404e-01
## 14 3.54970605 2.728570e+00 2.763659e+00 2.056554e+00 1.6854164047 1.310860e+00
## 15 3.98849818 3.057752e+00 2.824627e+00 2.501577e+00 1.1099254146 1.714971e+00
## 16 6.19360784 5.240554e+00 6.359342e+00 4.970663e+00 2.7921090204 2.941452e+00
## 17 5.56515872 4.332396e+00 4.527929e+00 3.956677e+00 2.5235344863 2.463087e+00
## 18 8.54163295 5.191591e+00 6.337373e+00 6.387457e+00 2.7464261142 2.100510e+00
## 19 8.67620315 6.959187e+00 5.840007e+00 4.688535e+00 3.6037951718 3.430368e+00
## 20 8.44497160 1.218222e+01 6.318749e+00 5.731237e+00 3.3787361919 3.267615e+00
## 21 7.17024165 6.457755e+00 7.691318e+00 7.184870e+00 3.1594607077 3.442483e+00
## 22 17.90600506 5.250411e+00 9.833281e+00 1.443184e+01 3.7422225763 2.533574e+00
## 23 1.50802738 1.253483e+01 4.699851e+00 6.034106e+00 5.4671850727 1.664063e+00
## 24 1.28162311 4.080661e+00 1.275567e+01 6.221721e+00 3.3483318441 4.909587e+00
## 25 3.30103442 2.984081e-01 3.401570e+00 8.858118e+00 3.1794387167 5.218498e+00
## 26 22.72123598 1.731936e+00 8.869921e-02 1.342427e+00 6.0933784180 1.239761e+00
## 27 0.70708219 1.851978e+01 8.016094e-01 4.756846e-02 0.6436520564 1.835686e+00
## 28 0.46554735 1.294484e-02 8.188379e+00 9.172146e-02 0.0004973671 6.203893e-01
## 29 1.17337409 3.994768e-02 1.588208e-01 1.670424e+00 0.1651683882 0.000000e+00
## 30 0.35092115 4.762157e-01 0.000000e+00 1.782347e-04 0.4652783693 1.126879e-04
## 31 0.39929430 4.009703e-03 2.264478e-01 0.000000e+00 0.0000000000 7.112896e-03
## 32 0.00000000 1.384126e-04 5.921586e-04 6.979977e-03 0.0000000000 0.000000e+00
## 33 0.05235793 0.000000e+00 0.000000e+00 0.000000e+00 0.0000000000 0.000000e+00
## 34 0.00000000 9.666143e-05 0.000000e+00 0.000000e+00 0.0000000000 0.000000e+00
## 35 0.00000000 0.000000e+00 0.000000e+00 0.000000e+00 0.0000000000 0.000000e+00
## 36 0.00000000 0.000000e+00 0.000000e+00 0.000000e+00 0.0000000000 0.000000e+00
## 37 0.00000000 0.000000e+00 0.000000e+00 0.000000e+00 0.0000000000 0.000000e+00
## 38 0.00000000 0.000000e+00 0.000000e+00 0.000000e+00 0.0000000000 0.000000e+00
## 39 0.00000000 0.000000e+00 0.000000e+00 0.000000e+00 0.0000000000 7.157566e-05
## 40 0.00000000 0.000000e+00 0.000000e+00 0.000000e+00 0.0000000000 0.000000e+00
## 41 0.00000000 0.000000e+00 0.000000e+00 0.000000e+00 0.0000000000 0.000000e+00
## 42 0.00000000 0.000000e+00 0.000000e+00 0.000000e+00 0.0000000000 0.000000e+00
## A24 A25
## 12 0.08024316 0.45846617
## 13 0.35627317 1.11742859
## 14 0.80927973 4.71588408
## 15 0.72236562 3.50056611
## 16 2.71981575 9.06333014
## 17 2.04740905 5.55253487
## 18 2.81083035 10.01730059
## 19 1.98934474 10.12539818
## 20 1.10508659 9.83712957
## 21 2.45434518 10.49553551
## 22 2.32453925 13.20799849
## 23 2.04154336 10.50043192
## 24 3.04258550 17.19525948
## 25 3.51053076 9.28874743
## 26 3.92435221 6.35022780
## 27 0.60345904 9.05873048
## 28 1.25969016 3.92123258
## 29 0.16992485 0.92499674
## 30 0.00000000 0.86386005
## 31 0.00000000 0.04374913
## 32 0.00000000 0.00000000
## 33 0.00000000 0.00000000
## 34 0.00000000 0.00000000
## 35 0.00000000 0.00000000
## 36 0.00000000 0.00000000
## 37 0.00000000 0.00000000
## 38 0.00000000 0.00000000
## 39 0.00000000 0.00000000
## 40 0.00000000 0.00000000
## 41 0.00000000 0.00000000
## 42 0.00000000 0.00000000On inspection, there is no catch within many older ages from 1988. On the other hand, the earlier years also appear to have very few counts relative to those in later years.
We can visualise the age composition for specified years using the function compyear. In our case, we will focus on 1980-1995 (rows 21 to 36) inclusive. (Note that, when plotting, we will have to remove the year column from the data).
naa <- TotCatAge
yrs <- 21:36
naa <- naa[yrs,2:26] # We must remove the first column containing the years
ages <- 1:25
#plotprep()
compyear(naa,1980:1995,ages,plots=c(4,4),freq=TRUE,border=3) 
As per the advice in Dr. Haddon’s vignettes: when our simulated example age data are considered using these frequency plots, it is clear that the notion of a exponential decline is only approximated by these data. Such deviations from the theoretical expectation derive from the fact that real populations are not in equilibrium and recruitment is not constant. So the dynamics appear to be realistic although the assumptions about equilibrium do not appear to be valid. Despite this we can use these data to illustrate the use of catch curves.
The sample sizes from each year are printed on the above plot, but can also be found by using the following.
sample_sizes <- apply(TotCatAge[,2:26],1,sum)
sample_sizes## [1] 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
## [8] 0.0000 0.0000 0.0000 0.0000 42.6888 87.0125 451.4760
## [15] 485.1910 1010.6400 1013.5600 1475.8700 1429.9300 1856.3300 1835.7300
## [22] 2297.7000 2357.5100 2965.2500 3184.0100 2787.1800 2840.9700 2957.8000
## [29] 2974.6300 2826.8400 3152.8000 3790.3900 4073.9600 3169.8000 3117.5700
## [36] 2896.6100 3230.3200 984.8420 773.3230 637.3400 826.0440 769.3050We can see a strong year cohort progressing through from 1990 onwards, which violates the assumption of constant recruitment. Despite the large sample sizes in the 1990s, let us instead focus our analysis on years 1986 and 1988 in the first instance. These are years with high sample sizes (2840 and 2974, respectively), but without visual evidence of strong recruitment pulses.
Our dataset was created using Stock Synthesis based on an age-at-maturity of roughly 3-4 years and a rate of natural mortality of 0.2 yr -1 for females, and 0.25 yr -1 for males.
Regarding the input parameters:
- Natural mortality estimate = 0.225 yr -1 (average of males and females)
- The maximum age in the sample data = 25.
- Age-at-50%-selectivity = 4.45 years (average of males and females: 3.97 for females, 4.93 for males)
- Age-at-95%-selectivity = 19.07 years (average of males and females: 13.14 for females, 19.07 for males)
- Delta = 19.07-4.45 = 14.62 years
- An estimate of current fishing mortality (used as an initial value)
=
???STILL NEEDED
Fitting the model to our data
selectCC is the function that sets up and runs the catch curve (with selectivity) model and simulations.
The main input data expected is a vector of numbers-at-age for a single year. As stated above, we will try years 1986 and 1988 (rows 27 and 29 of TotCatAge), separately.
We will assign the same names to the numbers-at-age vector and the other inputs as per datalowSA.
In creating our catch-at-age vector, there are a few potential trip-points. The first is that the data (numbers-at-age, numaa) MUST be a vector, not a list. The second is that the rownames (or names) of the vector should equate to a character list of the actual ages.
numaa <- TotCatAge[27,2:26]
names(numaa) <- as.character(1:25)
numaa <- unlist(numaa)
print(numaa)## 1 2 3 4 5 6
## 101.71525140 259.62850639 119.71588468 641.72204084 785.34718070 239.03467769
## 7 8 9 10 11 12
## 52.91393420 217.61295704 143.57059311 46.87067516 11.21729739 63.15319038
## 13 14 15 16 17 18
## 26.10838594 80.53372129 7.91473118 10.81047992 0.88292178 0.70708219
## 19 20 21 22 23 24
## 18.51978373 0.80160937 0.04756846 0.64365206 1.83568558 0.60345904
## 25
## 9.05873048The model needs the natural mortality and maximum sampled age as a globals list which we can call glb, copying the example in datalowSA. Let’s create the list for glb (which we could of course have instead read in).
NatM <- 0.225
maxage <- 25
glb <- list(M = NatM,maxage = maxage)
print(glb)## $M
## [1] 0.225
##
## $maxage
## [1] 25Finally, the parameters relating to selectivity:
pars <- c(A50=4.45,delta=14.62,fcur=0.6)
print(pars)## A50 delta fcur
## 4.45 14.62 0.60Running the model
Now that we have the data and inputs set up, the next is to run the model itself, using selectCC.
library(datalowSA)
out <- selectCC(glb$M,glb$maxage,numaa,pars,plot=TRUE)## Warning in cbind(counts, pcount, logcount, predcount, pCAA, pp, sel, FaA):
## number of rows of result is not a multiple of vector length (arg 6)
This shows the fitted catch curve.
We can examine the model output to check the fitted selectivity parameters (age-at-50%-maturity, A50, delta, and current fishing mortality, fcur) and predicted catch-at-age:
print(out)We can see that, for 1986, this results in a fully selected fishing mortality estimate of 0.1342896 yr-1 (and a logistically-increasing selectivity-at-age, as indicated in the resultant plot).
Using instead the catch-at-age data from 1988:
numaa <- TotCatAge[29,2:26]
names(numaa) <- as.character(1:25)
numaa <- unlist(numaa)
print(numaa)## 1 2 3 4 5 6
## 308.01329936 455.40892679 248.94935853 581.28236991 167.76477039 500.61565845
## 7 8 9 10 11 12
## 405.20640019 117.06954536 16.77395913 76.80190036 37.85171579 12.25767033
## 13 14 15 16 17 18
## 2.18665527 12.74820878 5.10011753 20.64946895 1.64731808 1.17337409
## 19 20 21 22 23 24
## 0.03994768 0.15882083 1.67042422 0.16516839 0.00000000 0.16992485
## 25
## 0.92499674out <- selectCC(glb$M,glb$maxage,numaa,pars,plot=TRUE)## Warning in cbind(counts, pcount, logcount, predcount, pCAA, pp, sel, FaA):
## number of rows of result is not a multiple of vector length (arg 6)
print(out)## $best
## $best$par
## A50 delta fcur
## 4.0913906 3.6463649 0.2494271
##
## $best$value
## [1] 7061.287
##
## $best$counts
## function gradient
## 244 NA
##
## $best$convergence
## [1] 0
##
## $best$message
## NULL
##
##
## $result
## counts predcounts logcount logpredcount pCAA predpCAA
## 1 308.01329936 196.54044710 5.73014296 5.2808682 1.035468e-01 3.945625e-02
## 2 455.40892679 306.30674909 6.12119576 5.7245870 1.530977e-01 6.607223e-02
## 3 248.94935853 420.17552380 5.51724950 6.0406725 8.369087e-02 1.029731e-01
## 4 581.28236991 478.40472749 6.36523665 6.1704571 1.954133e-01 1.412530e-01
## 5 167.76477039 442.66904428 5.12256282 6.0928224 5.639853e-02 1.608283e-01
## 6 500.61565845 344.91370445 6.21583866 5.8432943 1.682951e-01 1.488148e-01
## 7 405.20640019 240.59619239 6.00439657 5.4831200 1.362208e-01 1.159518e-01
## 8 117.06954536 158.07088457 4.76276816 5.0630436 3.935600e-02 8.088273e-02
## 9 16.77395913 100.84666966 2.81982763 4.6136012 5.639007e-03 5.313968e-02
## 10 76.80190036 63.46394799 4.34122938 4.1504720 2.581898e-02 3.390226e-02
## 11 37.85171579 39.69062322 3.63367631 3.6811150 1.272485e-02 2.133507e-02
## 12 12.25767033 24.75321317 2.50615189 3.2089553 4.120738e-03 1.334305e-02
## 13 2.18665527 15.41807736 0.78237310 2.7355407 7.351016e-04 8.321443e-03
## 14 12.74820878 9.59810183 2.54539077 2.2615654 4.285645e-03 5.183192e-03
## 15 5.10011753 5.97354035 1.62926358 1.7873398 1.714538e-03 3.226654e-03
## 16 20.64946895 3.71731841 3.02768960 1.3130026 6.941861e-03 2.008162e-03
## 17 1.64731808 2.31316225 0.49914856 0.8386155 5.537892e-04 1.249674e-03
## 18 1.17337409 1.43937111 0.15988344 0.3642063 3.944605e-04 7.776302e-04
## 19 0.03994768 0.89564348 -3.22018458 -0.1102128 1.342946e-05 4.838824e-04
## 20 0.15882083 0.55730846 -1.83997855 -0.5846364 5.339179e-05 3.010941e-04
## 21 1.67042422 0.34678096 0.51307762 -1.0590619 5.615570e-04 1.873539e-04
## 22 0.16516839 0.21578163 -1.80078979 -1.5334883 5.552569e-05 1.165795e-04
## 23 0.00000000 0.13426831 NA NA 0.000000e+00 7.254066e-05
## 24 0.16992485 0.08354732 -1.77239897 -2.4823421 5.712470e-05 4.513782e-05
## 25 0.92499674 0.13761861 -0.07796507 -1.9832691 3.109619e-04 2.808663e-05
## Selectivity FaA
## 1 0.03544116 0.01898593
## 2 0.07611816 0.03889418
## 3 0.15593405 0.07305943
## 4 0.29290894 0.12011385
## 5 0.48155891 0.16851723
## 6 0.67561712 0.20543799
## 7 0.82363936 0.22768479
## 8 0.91283097 0.23923861
## 9 0.95915238 0.24477812
## 10 0.98136131 0.24733216
## 11 0.99160094 0.24848846
## 12 0.99623675 0.24900763
## 13 0.99831819 0.24923986
## 14 0.99924926 0.24934357
## 15 0.99966505 0.24938985
## 16 0.99985059 0.24941049
## 17 0.99993336 0.24941970
## 18 0.99997028 0.24942381
## 19 0.99998675 0.24942564
## 20 0.99999409 0.24942646
## 21 0.99999736 0.24942682
## 22 0.99999882 0.24942698
## 23 0.99999948 0.24942706
## 24 0.99999977 0.24942709
## 25 0.99999990 0.24942710For 1988, fully selected fishing mortality is estimated to be 0.2494271 yr-1 (again with a logistically-increasing selectivity-at-age, indicated in the resultant plot).
(Note: using 1990’s data (row 31) results in a fishing mortality estimate of 0.1949766 yr-1 for all ages. The fitted proportion-at-age plot shows a negative exponential decline, which equates to all age classes being fully selected. This is the result of the strong cohort in the earliest age class. Using 1992’s data (row 33), we see an increase, a peak and then a decline in proportion-at-age (reflecting the strong cohort having since aged by 2 years), which yielded a more logistic increase in selectivity with age. The fully-selected fishing mortality was estimated to be very high, at 0.952 yr-1, again as a result of this strong year cohort).
Other years can be tested, as desired. It is important to reiterate that the data do not result in a constant fishing mortality over time.
Interpreting results in a management context
Catch curve analysis results in an estimate of total mortality, and, assuming a fixed natural mortality, fishing mortality. It does not provide an estimate of stock status per se, except where consistently low fishing mortality is used as a proxy for sustainability (although this would require many years of age-composition data). The estimates of fishing mortality could be compared to a reference point such as the fishing mortality at maximum sustainable yield, if this were available.
As only a single year of numbers-at-age is required, catch curve analysis suits fisheries which have occasional age-structured data, and whose catch-at-age distribution is not expected to vary strongly through time. As such, catch-curve analysis is frequently applied to data-poor fisheries, where a variety of disparate information sources (each of low quality for stock assessment) may be available. Future work could formalize a process by which catch curves are combined with expert opinion (i.e., interviews with fishers) and other data-poor analyses (e.g., changes in mean length) into a quantitative estimate of stock status (Thorson and Prager 2011).
Key references
Catch Curve Methodology
Chapman, D.G. and D.S. Robson. 1960. The analysis of a catch curve. Biometrics 16:354-368.
Dunn, A., Francis, R.I.C.C. and I.J. Doonan. 2002. Comparison of the Chapman-Robson and regression estimators of Z from catch-curve data when non-sampling stochastic error is present. Fisheries Research 59:149-159. https://doi.org/10.1016/S0165-7836(01)00407-6
Gulland, J.A. 1971. The fish resources of the ocean. West Byfleet, UK: Fishing News Books.
Smith, M.W., Then, A.Y., Wor, C., Ralph, G., Pollock, K.H. and J.M. Hoenig. 2012. Recommendations for catch-curve analysis. North American Journal of Fisheries Management 32:956-967. http://dx.doi.org/10.1080/02755947.2012.711270
Thorson, J.T. and M.H. Prager. 2011. Better catch curves: Incorporating age-specific natural mortality and logistic selectivity. Transactions of the American Fisheries Society 140:356-366. http://dx.doi.org/10.1080/00028487.2011.557016
Wayte, S.E. and N.L. Klaer. 2010. An effective harvest strategy using improved catch-curves. Fisheries Research 106:310-320. https://doi.org/10.1016/j.fishres.2010.08.012
Catch Curve Evaluation
Allen, M.S. 1997. Effects of variable recruitment on catch-curve analysis for crappie populations. North American Journal of Fisheries Management 17(1):202-205. https://doi.org/10.1577/1548-8675(1997)017<0202:EOVROC>2.3.CO;2
Griffiths, S.P. 2010. Stock assessment and efficacy of size limits on longtail tuna (Thunnus tonggol) caught in Australian waters. Fisheries Research 102(3):248-257. https://doi.org/10.1016/j.fishres.2009.12.004
Oyarzún, C., Cortés, N. and E. Leal. 2013. Age, growth and mortality of southern rays bream Brama australis (Bramidae) off the southeastern Pacific coast. Revista de biología marina y oceanografía 48(3). http://dx.doi.org/10.4067/S0718-19572013000300014
See references in http://derekogle.com/fishR/examples/oldFishRVignettes/CatchCurve.pdf.
Appendix: Catch Curve Equations
Catch-Curves with Selectivity
(reproduced with permission from Dr. Haddon’s vignettes)
A simple model of the expected explonential decline of numbers-at-age is needed:
\[\begin{matrix} {{N}_{0}}= & 1 & a=0 \\ {{N}_{a}}= & {{N}_{a-1}}\exp \left( -{{s}_{a-1}}F-M \right) & 0<a<{{a}_{\max }} \\ {{N}_{a}}= & \frac{{{N}_{a-1}}\exp \left( -{{s}_{a-1}}F-M \right)}{\left( 1-\exp \left( -{{s}_{a}}F-M \right) \right)} & a={{a}_{\max }} \\ \end{matrix} \]
where \(N_a\) is the numbers at age \(a\), \(s_a\) is the selectivity at age \(a\), \(F\) is the fully- selected average fishing mortality over the years for which agedtaa are available, \(M\) is the assumed rate of natural mortality, and \(a_{max}\) is the maximum age. Selectivity is assumed to be asymptotic (so this method may not be suitable for data collected using gillnets or hook gear):
\[s_a = \frac{1}{1+exp(-\ln(19)(a-a_{50})/\delta)}\] where \(a\) is age, \(a_{50}\) is the age-at-50%-selectivity and \(\delta\) is equal to \(a_{95} - a_{50}\).
The use of 19 is something of an enigma: if you do not use 19, the model does not predict the 95% of the selectivity, but it is not a simple relationship.
It should also be noted that the selectivity curve used does not need to be the particular version of the logistic that is used herein. Instead one can use other logistic equations, and as they do not use the rather arcane -Ln(19) they are sometimes easier to understand
Three parameters are estimated, both the selectivity paramters, \(a_{50}\) and \(\delta\), and the average fully-selected \(F\) value. \(\delta\) is used instead of \(a_{95}\) as this reduces the correlation between the two selectivity paarameters and helps avoid \(a_{50}\) values larger than the \(a_{95}\) values. These parameters can be estimated through minimizing the multinomial negative log-likelihood:
\[-veLL=-\sum\limits_{y}\sum\limits_{a}{{O}_{y,a}}log\left(\frac{{\hat{N}_{a}}}{\sum{{\hat{N}_{a}}}} \right)\]
where \(O_{y,a}\) is the observed proportion of numbers-at-age \(a\) in the catch during year \(y\), and \({\hat{N}_{a}}\) is the expected or predicted proportion of numbers-at-age \(a\) in the catch during year \(y\). The predicted numbers-at-age are calculated using the Baranov catch equation:
\[{\hat{N}_a}=\frac{{N_a}{s_a}F}{(s_aF+M)}\left(1-e^{-(s_aF+M)} \right )\]