# 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 logged catch-at-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.

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.

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.

The TropFishR package is a compilation of fish stock assessment methods for the analysis of length-frequency data in the context of data-poor fisheries. It includes methods and examples included in the FAO Manual by Sparre and Venema (1998), (Introduction to tropical fish stock assessment), as well as other more recent methods.

The function catchCurve in the TropFishR package applies the (length-converted) linearised catch curve to age-composition and length-frequency data, respectively, to estimate the instantaneous total mortality rate (Z). Optionally, gear selectivity can be estimated and a cumulative catch curve can be applied.

The main author of TropFishR’s catch curve package, Tobias Mildenberger, has an excellent online tutorial https://cran.r-project.org/web/packages/TropFishR/vignettes/tutorial.html.

This tutorial illustrates the application of the TropFishR package to perform a single-species fish stock assessment (incorporating a catch curve analysis) with length frequency data. This tutorial can be particularly useful if one lacks information on biological stock characteristics. It provides a detailed consideration of the following steps: (1) estimation of biological stock characteristics (growth and natural mortality), (2) exploration of fisheries aspects (exploitation rate and selectivity), and (3) assessment of stock size and status.

Otherwise, a good general description of the catch curve function can be found in the package vignette, at https://rdrr.io/cran/TropFishR/man/catchCurve.html.

This can also be found, together with descriptions of all the other functions in the package, in the “Package”TropFishR"" manual, at https://cran.r-project.org/web/packages/TropFishR/TropFishR.pdf

# 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 at https://www.rstudio.com/products/rstudio/download/.

The catch curve function is part of a set of stock assessment tools within the TropFishR package. The current version of TropFishR (v1.2) requires R >=3.0.0.

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)
}

Check_Install.packages("TropFishR")
## [1] "'TropFishR' has been installed already!"
## Loading required package: TropFishR
## Warning: package 'TropFishR' was built under R version 4.0.3
## TropFishR
##      TRUE
#install.packages("TropFishR", repos = "https://cran.rstudio.com/")

The package is loaded into the R environment with:

library(TropFishR)

## Obtaining life-history inputs

TropFishR uses von Bertalanffy growth parameters as inputs to its catch curve function.

The Mildenberger tutorial provides a detailed description and tutorial of a process for using TropFishR’s ELEFAN (ELectronic LEngth Frequency Analysis) methods to allow the user to estimate Linf and K from length-frequency data. This is undertaken by restructuring the data and fitting growth curves through the restructured length-frequency data (Pauly 1980). The tutorial then describes how to obtain an estimate of instantaneous natural mortality, M, using estimates of the VBGF growth parameters (Linf and K; Then et al. 2015). The estimation of M is challenging (Kenchington 2014; Powers 2014). When no controlled experiments or tagging data are available from which to estimate v, the main approach is to use empirical formulae. Overall, there are at least 30 empirical formulae for the estimation of this parameter (Kenchington 2014), relying on correlations with life history parameters and/or environmental information. TropFishR applies the most recent formula, which is based upon a meta-analysis of 201 fish species (Then et al. 2015), using its function M-empirical.

In our case, we will be using a simulated example dataset that has been created for the whole toolbox. Included in these test data are the following required life-history inputs:

• general (required) inputs:
• midLengths or age : midpoints of the length classes (length-frequency data) or ages (age composition data)
• Linf *: asymptotic length for the investigated species in cm [cm] (= 81.44505)
• K *: growth coefficent for the investigated species per year [1/year] (=0.15)
• t0 *: theoretical age at zero length, at which individuals of this species hatch (= -1.87856856036483 yr)

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.

For our worked example, we will use catch-at-age data and female growth rates.

The TropFishR catch curve function requires as its main input catch-at-length, or catch-at-age.

• catch: catches, vector or matrix with catches of subsequent years if the catch curve with constant time intervals should be applied;

A simulated example dataset has been created for the whole toolbox.

We will use the catch-at-age data, although we do also have available catch-at-length.

Let us now read in both the catch-at-age and catch-at-length 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, are for years (1960-2001) in rows (2-43), with 25 age bins (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.

The simulated catch-at-length data, LengthData.csv, are for years 1971-2001 in rows (2-32), with 17 length bins (10cm-90cm, by 5cm) in columns (E to U).

Let us read in and inspect 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  0.00000000
TotCatLen <- read.csv("LengthData.csv")
print(TotCatLen)
##    Year Fleet Sex Stage1_wght          L10         L15        L20       L25
## 1  1971     1   0         100 0.0108213780 0.056475611 0.52303724 1.0487370
## 2  1972     1   0         100 0.0064606981 0.340953751 2.80021169 2.3718087
## 3  1973     1   0         100 0.0057407915 0.111225471 0.59424943 2.5914423
## 4  1974     1   0         100 0.0064473956 0.146511374 0.93324804 1.5723084
## 5  1975     1   0         100 0.0296747223 0.008844181 0.23885398 1.0612991
## 6  1976     1   0         100 0.0006628005 0.101107619 0.39888185 0.4904583
## 7  1977     1   0         100 0.0128395440 0.137969740 0.75861347 0.8144316
## 8  1978     1   0         100 0.0085765087 0.113854048 0.78350802 1.3254273
## 9  1979     1   0         100 0.0023896198 0.015689117 0.19383656 0.8925509
## 10 1980     1   0         100 0.0067505990 0.046255415 0.57374226 0.8187733
## 11 1981     1   0         100 0.0007805095 0.140916856 1.13027842 1.3339034
## 12 1982     1   0         100 0.0087649180 0.134288853 1.05864903 1.9397389
## 13 1983     1   0         100 0.0303006063 0.020829199 0.24350730 1.2796627
## 14 1984     1   0         100 0.0144919112 0.073530596 0.76072038 0.7793930
## 15 1985     1   0         100 0.0207503456 0.058028038 0.32946319 0.9203682
## 16 1986     1   0         100 0.0027589273 0.132947714 0.87153618 1.0185650
## 17 1987     1   0         100 0.0064847347 0.126002489 1.53392943 2.5468354
## 18 1988     1   0         100 0.0032082537 0.129452506 1.45079128 2.7052436
## 19 1989     1   0         100 0.0183568058 1.159413335 7.76475864 7.3586842
## 20 1990     1   0         100 0.0119178694 0.122607575 0.80400678 7.0369343
## 21 1991     1   0         100 0.0012209778 0.189416400 1.24799759 1.4711444
## 22 1992     1   0         100 0.0089377102 0.045481461 0.23604117 1.2146968
## 23 1993     1   0         100 0.0038116158 0.003075382 0.06707352 0.3008998
## 24 1994     1   0         100 0.0213321210 1.057371193 8.10020596 5.7203531
## 25 1995     1   0         100 0.0200526711 0.029621320 0.58630474 8.5894651
## 26 1996     1   0         100 0.0011472076 0.073967120 0.58300176 0.8141074
## 27 1997     1   0         100 0.0002073036 0.071968796 0.45103523 0.9170445
## 28 1998     1   0         100 0.0052173898 0.031621196 0.18645062 0.4875539
## 29 1999     1   0         100 0.0079095602 0.091070745 0.77857343 0.6985086
## 30 2000     1   0         100 0.0015518027 0.067701012 0.26749233 0.7856025
## 31 2001     1   0         100 0.0012319084 0.195661977 1.78281979 1.6987257
##           L30       L35       L40       L45       L50       L55       L60
## 1   1.7660553  2.099976  6.094228 15.369595 18.167002 13.892845 11.946501
## 2   1.7565947  3.150852  4.903078  8.831542 16.997290 17.636103 13.590201
## 3   6.9579183  5.449736  5.306619  7.914746 12.266590 16.668806 14.671624
## 4   3.6953967  9.903884 12.768967  8.232849  9.933395 13.446945 13.676123
## 5   3.3818952  5.671647 13.223021 17.096120 13.226173 11.676958 11.120261
## 6   1.1521644  5.028154 10.619118 17.111300 19.371983 13.988815 10.904463
## 7   1.3901127  2.774636  7.494925 15.816378 20.253970 18.515553 12.540484
## 8   2.2628154  3.535581  4.876416 11.309055 19.610697 20.449917 14.890819
## 9   2.5611678  4.748444  6.719008  9.925508 15.714929 19.940495 16.900504
## 10  1.2333408  4.861463  8.677709 11.350535 13.917517 16.868445 16.465716
## 11  1.4202835  2.255932  7.016305 13.134027 16.147242 15.784386 16.445788
## 12  3.5791457  4.185768  5.689179 10.536051 16.448558 16.949436 14.843437
## 13  3.9060229  7.403979  9.104543  9.970630 13.752822 16.708818 14.488065
## 14  2.0229118  6.389477 12.913859 15.488465 14.447025 14.536580 12.574110
## 15  2.7847841  4.054645 10.745356 19.373729 19.373770 15.262121 10.878707
## 16  2.0321802  4.659836  9.254394 15.680758 21.655199 19.146639 11.705528
## 17  3.7492156  4.721197  8.774132 14.982281 18.735942 19.085507 12.940282
## 18  5.4476534  9.305075 11.842704 13.698089 16.991765 15.696142 11.853683
## 19  5.8356650 10.444862 14.848370 15.491771 13.831440 10.819834  6.720715
## 20 19.5800092 15.462165 13.503343 16.024399 13.354907  7.593226  3.811495
## 21  6.3757988 23.586292 27.290475 16.746458 11.648937  6.958052  3.011219
## 22  3.5577272  8.360178 25.223312 31.988611 17.801857  7.426806  2.961044
## 23  1.3791468  6.157607 13.576645 28.145836 29.375833 15.196857  4.312335
## 24  0.7042392  2.634566  8.239599 18.121302 24.239228 19.876044  8.699990
## 25 22.2059555 11.217365  4.264894  9.639557 16.372865 14.730794  8.307461
## 26  7.6871774 28.441776 30.396696 10.365179  6.849961  7.023106  4.835885
## 27  1.8367047  8.465402 29.034346 35.233563 14.912990  4.652150  2.658960
## 28  1.3686358  3.412599 11.610144 29.925162 31.547730 15.627554  4.026052
## 29  0.8636623  2.617433  6.226606 17.341781 29.948790 26.165134 11.668761
## 30  1.8439275  2.022297  4.539734 12.037621 22.714896 27.874149 18.260569
## 31  1.8109995  3.696791  5.031228  8.462606 16.108794 23.217521 20.883664
##           L65       L70        L75         L80         L85          L90
## 1  10.6160860 8.1307503 5.42860983 3.031826287 1.145619499 6.718337e-01
## 2   9.9612501 7.7315914 5.20990554 2.911423708 1.280404851 5.203297e-01
## 3  10.9289855 7.0540302 4.99459558 2.598119475 1.309120437 5.764519e-01
## 4  10.6619651 7.0106354 4.09806064 2.404548314 1.036774059 4.719414e-01
## 5   9.3373387 6.6127377 3.87026447 2.118050500 0.903310653 4.235502e-01
## 6   8.1102375 5.6146556 3.92882347 2.001719819 0.828602312 3.488547e-01
## 7   7.9115894 5.4958835 3.36273317 1.635854914 0.782876266 3.011499e-01
## 8   9.6954151 5.2166799 3.01160684 1.832896763 0.763467804 3.132665e-01
## 9  10.5727960 6.0032687 3.28151147 1.606857768 0.592724191 3.283185e-01
## 10 12.0527908 6.6512667 3.61931638 1.534629780 1.002128560 3.196199e-01
## 11 11.7729835 7.0862326 3.71307381 1.712322532 0.587931658 3.176121e-01
## 12 10.9455666 7.4308588 3.63192617 1.734610385 0.663446532 2.205752e-01
## 13  9.9742020 6.4448189 3.92753575 1.862749448 0.650047533 2.314659e-01
## 14  9.2382967 5.5798516 3.15019058 1.260829530 0.642463922 1.278048e-01
## 15  7.3912687 4.5951091 2.31253193 1.369198686 0.332839736 1.973314e-01
## 16  6.7477290 3.8910842 1.94000088 0.807225476 0.339682240 1.139370e-01
## 17  6.9712109 3.3701522 1.37327570 0.768572814 0.168104788 1.468753e-01
## 18  6.2772412 2.9147568 0.99063984 0.500278247 0.151224078 4.205275e-02
## 19  3.5646885 1.4182600 0.45749751 0.191629562 0.060595441 1.345991e-02
## 20  1.7880517 0.5616333 0.16305025 0.095380503 0.076544021 1.032940e-02
## 21  1.0122216 0.3643558 0.06307546 0.015702093 0.013663043 3.969892e-03
## 22  0.8870635 0.1629651 0.08718290 0.015290448 0.012109277 1.069699e-02
## 23  1.0469921 0.3139969 0.05764882 0.017582663 0.012386327 3.227341e-02
## 24  2.1463427 0.3802908 0.03681364 0.015778062 0.006283302 2.614829e-04
## 25  3.2177883 0.6795673 0.09139539 0.039289001 0.004090429 3.534224e-03
## 26  1.9442661 0.8203258 0.11544283 0.006280026 0.027888791 1.379165e-02
## 27  1.1866159 0.3606157 0.09899023 0.090774928 0.022749392 5.882090e-03
## 28  1.1831513 0.3783228 0.17179407 0.028726202 0.005583188 3.701916e-03
## 29  2.8137292 0.5047504 0.18587561 0.049278078 0.038042767 9.391721e-05
## 30  7.1716229 1.8419431 0.47339454 0.079372394 0.006610140 1.151563e-02
## 31 11.8735441 4.1383061 0.84438656 0.167033730 0.047557225 3.913066e-02

On 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.

In our case, we will focus on catch-at-age data from 1980-1995 (rows 21 to 36) inclusive. We must remove the first column containing the years.

For catch-at-length, we will remove the first 3 columns containing the year, fleet and sex.

We will also have to remove the last three length category columns, as they are greater than the specified Linf value and so result in the catch curve function taking the log of a negative number (log(1 - L/Linf)).

naa <- TotCatAge
yrs <- 21:36
naa <- naa[yrs,2:26] # We must remove the first column containing the years.
ages <- 1:25

nal <- TotCatLen
nal <- nal[,5:18]

Within the catch-at-age data, we can see there is 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 1988 in the first instance. These are years with high sample sizes (2840 and 2974, respectively) but without visual evidence of strong recruitment pulses.

To correctly format the data for input into the catchCurve function in TropFishR, catch-at-age or catch-at-length needs to be a matrix with each column being one year of data, and each row one age class. We will therefore have to transpose our data, and we will assign the column names to be the years:

catch_age_input <- t(naa)
colnames(catch_age_input)<-c(1980:1995)

catch_len_input <- t(nal)
colnames(catch_len_input)<-c(1971:2001)

For the catchCurve function in TropFishR, the input data are fed in as function argument param, which needs to be a list consisting of:

• midLengths or age : midpoints of the length classes (length-frequency data) or ages (age composition data),
• Linf : asymptotic length for the investigated species in cm [cm],
• K : growth coefficent for the investigated species per year [1/year],
• t0 : theoretical age at zero length, at which individuals of this species hatch,
• catch : catches, vector or matrix with catches of subsequent years if the catch curve with constant time intervals should be applied;
TestList_age <- list("age"=c(0.5:24.5),"Linf"=81.44505,"K"=0.15,"t0"=-1.87856856036483,catch=catch_age_input)

TestList_length <- list("midLengths"=seq(12.5,77.5,by=5),"Linf"=81.44505,"K"=0.15,"t0"=-1.87856856036483,catch=catch_len_input)

# Fitting the model to our data

We will now apply the catchCurve function to year 1988 (the 8th column) in the catch-at-age data.

The function catchCurve is

catchCurve( param, catch_columns = NA, cumulative = FALSE, calc_ogive = FALSE, reg_int = NULL, reg_num = 1, auto = FALSE, plot = TRUE )

with arguments

• param as described above
• catch_columns numerical; indicating the column of the catch matrix which should be used for the analysis.
• cumulative logical; if TRUE the cumulative catch curve is applied (Jones and van Zalinge method)
• calc_ogive logical; if TRUE the selection ogive is additionally calculated from the catch curve (only if cumulative = FALSE)
• reg_int instead of using the identity method a range can be determined, which is to be used for the regression analysis. If equal to NULL identity method is applied (default). For multiple regression lines provide list with the two points for the regression line in each element of the list.
• reg_num integer indicating how many separate regression lines should be applied to the data. Default 1.
• auto logical; no interactive functions used instead regression line is chosen automatically. Default = FALSE
• plot logical; should a plot be displayed? Default = TRUE

This function includes the identify function, which asks you to manually choose two points from a graph. The two points, which you choose by clicking on the plot in the graphical device, represent the start and end of the data points, which bracket the negative-linear section of the scatterplot and hence which data should be used for the analysis. Based on these points the regression line is calculated.

Here we tell the model (using the reg_int argument) to select points 5 and 24, as bracketing the most (negative) linear section of the scatterplot. For this demonstration, we will use the reg_int argument of the function to provide the interval used for the regression analysis.

catchCurve(TestList_age,reg_int=c(5,24),catch_columns=8)

## $age ## [1] 0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.5 11.5 12.5 13.5 14.5 ## [16] 15.5 16.5 17.5 18.5 19.5 20.5 21.5 22.5 23.5 24.5 ## ##$Linf
## [1] 81.44505
##
## $K ## [1] 0.15 ## ##$t0
## [1] -1.878569
##
## $catch ## 1980 1981 1982 1983 1984 1985 ## A1 26.807254 113.402232 183.714672 148.349856 85.8346051 133.08477784 ## A2 138.890263 32.907605 132.782132 389.615845 363.2626107 75.61887769 ## A3 231.331423 294.974054 59.244087 257.948560 686.4943428 470.20828072 ## A4 156.984856 334.191571 379.157799 96.057167 349.3652568 769.46633876 ## A5 37.997537 210.562848 387.005313 504.854930 95.5292599 298.35190052 ## A6 324.387438 39.600928 207.480110 435.972149 454.1988904 70.74237257 ## A7 121.955926 378.120191 39.271203 206.260684 346.5794458 289.93354127 ## A8 455.457120 130.873968 300.558800 40.758918 164.5642152 214.30703200 ## A9 46.251419 453.578522 87.149002 270.312910 31.4219089 91.56452353 ## A10 81.717713 49.077049 357.174533 81.114170 190.2727395 15.51944722 ## A11 13.823355 62.071930 36.920017 312.487417 51.4576662 127.55372029 ## A12 7.572340 6.123684 42.624543 33.410977 220.3048954 32.28934345 ## A13 107.500149 5.937859 7.114140 53.250937 23.0340709 126.04958235 ## A14 9.273752 101.600200 3.271413 6.518383 23.2528446 11.41338915 ## A15 2.249562 8.963431 80.072985 3.457130 4.0146717 13.47403288 ## A16 5.755758 1.425079 8.059648 67.419744 4.3653353 3.27661599 ## A17 19.718125 5.058973 1.459567 4.624780 53.0008950 0.83420585 ## A18 7.170242 17.906005 1.508027 1.281623 3.3010344 22.72123598 ## A19 6.457755 5.250411 12.534829 4.080661 0.2984081 1.73193645 ## A20 7.691318 9.833281 4.699851 12.755674 3.4015702 0.08869921 ## A21 7.184870 14.431843 6.034106 6.221721 8.8581183 1.34242688 ## A22 3.159461 3.742223 5.467185 3.348332 3.1794387 6.09337842 ## A23 3.442483 2.533574 1.664063 4.909587 5.2184976 1.23976099 ## A24 2.454345 2.324539 2.041543 3.042586 3.5105308 3.92435221 ## A25 10.495536 13.207998 10.500432 17.195259 9.2887474 6.35022780 ## 1986 1987 1988 1989 1990 ## A1 101.71525140 2.864463e+02 308.01329936 5.966220e+02 1.122484e+03 ## A2 259.62850639 1.355082e+02 455.40892679 4.737542e+02 3.732161e+02 ## A3 119.71588468 4.431192e+02 248.94935853 5.997932e+02 6.238126e+02 ## A4 641.72204084 1.723129e+02 581.28236991 2.141314e+02 5.115091e+02 ## A5 785.34718070 6.682931e+02 167.76477039 3.842458e+02 1.592295e+02 ## A6 239.03467769 6.579780e+02 500.61565845 8.979555e+01 1.924781e+02 ## A7 52.91393420 1.747946e+02 405.20640019 2.014489e+02 3.402230e+01 ## A8 217.61295704 3.960258e+01 117.06954536 1.706234e+02 7.173631e+01 ## A9 143.57059311 1.321395e+02 16.77395913 3.823145e+01 4.081677e+01 ## A10 46.87067516 8.691981e+01 76.80190036 8.074714e+00 9.833565e+00 ## A11 11.21729739 3.954351e+01 37.85171579 2.196131e+01 1.415969e+00 ## A12 63.15319038 2.966183e+00 12.25767033 1.353283e+01 6.161638e+00 ## A13 26.10838594 4.114913e+01 2.18665527 3.374532e+00 2.600022e+00 ## A14 80.53372129 1.193642e+01 12.74820878 4.749975e-01 2.052710e+00 ## A15 7.91473118 4.375533e+01 5.10011753 3.941835e+00 6.968363e-02 ## A16 10.81047992 2.168251e+00 20.64946895 1.428536e+00 6.470797e-01 ## A17 0.88292178 4.606659e+00 1.64731808 3.248627e+00 3.371932e-02 ## A18 0.70708219 4.655473e-01 1.17337409 3.509212e-01 3.992943e-01 ## A19 18.51978373 1.294484e-02 0.03994768 4.762157e-01 4.009703e-03 ## A20 0.80160937 8.188379e+00 0.15882083 0.000000e+00 2.264478e-01 ## A21 0.04756846 9.172146e-02 1.67042422 1.782347e-04 0.000000e+00 ## A22 0.64365206 4.973671e-04 0.16516839 4.652784e-01 0.000000e+00 ## A23 1.83568558 6.203893e-01 0.00000000 1.126879e-04 7.112896e-03 ## A24 0.60345904 1.259690e+00 0.16992485 0.000000e+00 0.000000e+00 ## A25 9.05873048 3.921233e+00 0.92499674 8.638601e-01 4.374913e-02 ## 1991 1992 1993 1994 1995 ## A1 1.484498e+02 2.011628e+02 2.739711e+01 4.435817e+02 1.265226e+03 ## A2 1.989367e+03 1.233779e+02 3.296243e+02 5.003094e+01 1.800965e+01 ## A3 5.207658e+02 2.675624e+03 1.442653e+02 4.773123e+02 8.908936e+01 ## A4 5.591713e+02 4.626895e+02 2.151588e+03 1.581667e+02 4.319008e+02 ## A5 3.548632e+02 3.889718e+02 2.359981e+02 1.697556e+03 1.007700e+02 ## A6 8.191392e+01 1.566798e+02 1.983413e+02 1.678768e+02 8.880046e+02 ## A7 8.314181e+01 2.390101e+01 6.096027e+01 8.187872e+01 5.737731e+01 ## A8 1.106386e+01 3.057931e+01 1.396191e+01 3.260125e+01 2.913125e+01 ## A9 2.385887e+01 3.196548e+00 5.095099e+00 5.236099e+00 1.574957e+01 ## A10 1.417470e+01 4.026857e+00 5.160038e-01 2.460153e+00 3.869628e-01 ## A11 9.953650e-01 2.220936e+00 1.962090e+00 7.193254e-02 3.428750e-01 ## A12 1.603187e-01 1.163430e+00 5.221866e-02 9.900001e-03 6.112787e-01 ## A13 9.486003e-01 0.000000e+00 3.854199e-02 7.005426e-01 3.601882e-03 ## A14 5.317343e-02 2.860974e-01 0.000000e+00 2.928362e-04 1.368553e-03 ## A15 3.653987e-01 2.753646e-02 0.000000e+00 8.694902e-02 5.492618e-03 ## A16 2.330245e-01 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 ## A17 8.558416e-01 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 ## A18 0.000000e+00 5.235793e-02 0.000000e+00 0.000000e+00 0.000000e+00 ## A19 1.384126e-04 0.000000e+00 9.666143e-05 0.000000e+00 0.000000e+00 ## A20 5.921586e-04 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 ## A21 6.979977e-03 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 ## A22 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 ## A23 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 ## A24 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 ## A25 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 ## ##$tplusdt_2
##  [1]  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 NA
##
## $lnC_dt ## A1 A2 A3 A4 A5 A6 A7 ## 5.6575510 4.9090319 6.0938389 5.1493118 6.5047269 6.4891715 5.1636114 ## A8 A9 A10 A11 A12 A13 A14 ## 3.6788943 4.8838584 4.4649860 3.6774015 1.0872761 3.7172028 2.4795939 ## A15 A16 A17 A18 A19 A20 A21 ## 3.7786134 0.7739207 1.5275028 -0.7645415 -4.3470581 2.1027159 -2.3889989 ## A22 A23 A24 A25 ## -7.6061821 -0.4774082 0.2308658 NA ## ##$reg_int
## [1]  5 24
##
## $linear_mod ## ## Call: ## lm(formula = yvar ~ xvar, data = df.CC.cut) ## ## Coefficients: ## (Intercept) xvar ## 8.883 -0.492 ## ## ##$Z
## [1] 0.4920218
##
## $se ## [1] 0.08470289 ## ##$confidenceInt
## [1] 0.3140676 0.6699760
##
## attr(,"class")
## [1] "catchCurve"

The function value is a list with the input parameters and following list objects:

• classes.num, tplusdt_2, t_midL, or ln_Linf_L : age, relative age or subsitute depending on input and method,
• lnC or lnC_dt : logarithm of (rearranged) catches,
• reg_int : the interval used for the regression analysis,
• linear_mod : linear model used for the regression analysis,
• Z : instantaneous total mortality rate, confidenceInt
• se : standard error of the total mortality;
• confidenceInt : confidence interval of the total mortality;

in case calc_ogive == TRUE, additionally:

• intercept : intercept of regression analysis,
• linear_mod_sel : linear model used for the selectivity analysis,
• Sobs : observed selection ogive,
• ln_1_S_1 : dependent variable of the regression analysis for selectivity parameters,
• Sest : estimated selection ogive,
• t50 : age at first capture (age at which fish have a 50% probability to be caught),
• t75 : age at which fish have a 75% probability to be caught,
• L50 : length at first capture (length at which fish have a 50% probability to be caught),
• L75 : length at which fish have a 75% probability to be caught

This function applies the (length-converted) linearised catch curve to age composition or length-frequency data, respectively. It allows estimation off the instantaneous total mortality rate (Z). Optionally, the gear selectivity can be estimated and the cumulative catch curve can be applied.

The assumption is made that Z is constant for all year classes or length groups, respectively, when the selection ogive is calculated by means of the catch curve. According to Sparre and Venema (1998) this assumption might be true, because F is smaller for young fish (selectivity) while M is higher for young fish (high natural mortality). The selectivity for old fish that are not fully exploited (e.g. due to being caught in a gillnet fishery) can not be calculated using the catch curve method. Based on the format of the list argument catch and whether the argument catch_columns is defined, the function automatically distinguishes between the catch curve with variable parameter system (if catch is a vector) and the one with constant parameter system (if catch is a matrix or a data.frame and catch_columns = NA). In the case of the variable parameter system the catches of one year are assumed to represent the catches during the entire life span of a so called pseudo-cohort.

We reiterate that the cumulative catch curve does not allow for the estimation of the selectivity ogive.

Let’s now run the model to also estimate the selectivity ogive, again in the year 1988.

(In this case we will use the length data, because we suspect a bug with estimating selectivity using the catch-at-age data).

#catchCurve(TestList,reg_int=c(1,26),calc_ogive=TRUE,catch_columns=8)

catchCurve(TestList_length,reg_int=c(9,13),calc_ogive=TRUE,catch_columns=18)

## $midLengths ## [1] 12.5 17.5 22.5 27.5 32.5 37.5 42.5 47.5 52.5 57.5 62.5 67.5 72.5 77.5 ## ##$Linf
## [1] 81.44505
##
## $K ## [1] 0.15 ## ##$t0
## [1] -1.878569
##
## $catch ## 1971 1972 1973 1974 1975 ## L10 0.01082138 0.006460698 0.005740792 0.006447396 0.029674722 ## L15 0.05647561 0.340953751 0.111225471 0.146511374 0.008844181 ## L20 0.52303724 2.800211687 0.594249426 0.933248039 0.238853983 ## L25 1.04873705 2.371808684 2.591442305 1.572308413 1.061299110 ## L30 1.76605527 1.756594705 6.957918261 3.695396670 3.381895167 ## L35 2.09997632 3.150852145 5.449736214 9.903884228 5.671646821 ## L40 6.09422836 4.903077955 5.306619287 12.768967386 13.223021244 ## L45 15.36959470 8.831541673 7.914745646 8.232849114 17.096119959 ## L50 18.16700222 16.997289547 12.266589748 9.933394508 13.226173090 ## L55 13.89284542 17.636103379 16.668805906 13.446945061 11.676958221 ## L60 11.94650082 13.590200537 14.671623874 13.676122857 11.120261214 ## L65 10.61608598 9.961250058 10.928985463 10.661965100 9.337338736 ## L70 8.13075033 7.731591404 7.054030178 7.010635418 6.612737742 ## L75 5.42860983 5.209905540 4.994595580 4.098060642 3.870264466 ## 1976 1977 1978 1979 1980 1981 ## L10 6.628005e-04 0.01283954 0.008576509 0.00238962 0.006750599 7.805095e-04 ## L15 1.011076e-01 0.13796974 0.113854048 0.01568912 0.046255415 1.409169e-01 ## L20 3.988818e-01 0.75861347 0.783508017 0.19383656 0.573742265 1.130278e+00 ## L25 4.904583e-01 0.81443156 1.325427312 0.89255088 0.818773330 1.333903e+00 ## L30 1.152164e+00 1.39011269 2.262815393 2.56116781 1.233340831 1.420283e+00 ## L35 5.028154e+00 2.77463644 3.535581397 4.74844428 4.861463027 2.255932e+00 ## L40 1.061912e+01 7.49492488 4.876416041 6.71900829 8.677708928 7.016305e+00 ## L45 1.711130e+01 15.81637761 11.309055165 9.92550828 11.350534714 1.313403e+01 ## L50 1.937198e+01 20.25396982 19.610696908 15.71492948 13.917517290 1.614724e+01 ## L55 1.398881e+01 18.51555312 20.449917246 19.94049529 16.868445470 1.578439e+01 ## L60 1.090446e+01 12.54048399 14.890819093 16.90050382 16.465716012 1.644579e+01 ## L65 8.110238e+00 7.91158940 9.695415141 10.57279597 12.052790808 1.177298e+01 ## L70 5.614656e+00 5.49588354 5.216679866 6.00326870 6.651266655 7.086233e+00 ## L75 3.928823e+00 3.36273317 3.011606837 3.28151147 3.619316381 3.713074e+00 ## 1982 1983 1984 1985 1986 1987 ## L10 0.008764918 0.03030061 0.01449191 0.02075035 0.002758927 0.006484735 ## L15 0.134288853 0.02082920 0.07353060 0.05802804 0.132947714 0.126002489 ## L20 1.058649031 0.24350730 0.76072038 0.32946319 0.871536176 1.533929434 ## L25 1.939738858 1.27966275 0.77939297 0.92036817 1.018565023 2.546835426 ## L30 3.579145661 3.90602288 2.02291181 2.78478413 2.032180185 3.749215577 ## L35 4.185767796 7.40397941 6.38947721 4.05464470 4.659835933 4.721196858 ## L40 5.689178523 9.10454328 12.91385884 10.74535553 9.254393797 8.774131549 ## L45 10.536050800 9.97063010 15.48846481 19.37372904 15.680758498 14.982281486 ## L50 16.448558308 13.75282160 14.44702504 19.37377003 21.655198583 18.735941777 ## L55 16.949436431 16.70881792 14.53657982 15.26212066 19.146638582 19.085506944 ## L60 14.843437144 14.48806544 12.57410952 10.87870657 11.705527819 12.940282006 ## L65 10.945566646 9.97420196 9.23829671 7.39126874 6.747729017 6.971210889 ## L70 7.430858778 6.44481889 5.57985159 4.59510907 3.891084153 3.370152225 ## L75 3.631926174 3.92753575 3.15019058 2.31253193 1.940000882 1.373275699 ## 1988 1989 1990 1991 1992 1993 ## L10 0.003208254 0.01835681 0.01191787 0.001220978 0.00893771 0.003811616 ## L15 0.129452506 1.15941333 0.12260758 0.189416400 0.04548146 0.003075382 ## L20 1.450791284 7.76475864 0.80400678 1.247997586 0.23604117 0.067073519 ## L25 2.705243589 7.35868421 7.03693435 1.471144369 1.21469682 0.300899769 ## L30 5.447653439 5.83566497 19.58000924 6.375798770 3.55772720 1.379146798 ## L35 9.305075120 10.44486158 15.46216525 23.586292025 8.36017800 6.157607377 ## L40 11.842703787 14.84836952 13.50334337 27.290475240 25.22331220 13.576644521 ## L45 13.698088769 15.49177093 16.02439875 16.746458250 31.98861063 28.145836449 ## L50 16.991764690 13.83143980 13.35490735 11.648937469 17.80185711 29.375832954 ## L55 15.696142476 10.81983406 7.59322572 6.958051652 7.42680552 15.196856638 ## L60 11.853683256 6.72071518 3.81149459 3.011219450 2.96104406 4.312334734 ## L65 6.277241165 3.56468854 1.78805171 1.012221563 0.88706345 1.046992119 ## L70 2.914756751 1.41826000 0.56163326 0.364355762 0.16296505 0.313996898 ## L75 0.990639839 0.45749751 0.16305025 0.063075459 0.08718290 0.057648822 ## 1994 1995 1996 1997 1998 1999 ## L10 0.02133212 0.02005267 0.001147208 2.073036e-04 0.00521739 0.00790956 ## L15 1.05737119 0.02962132 0.073967120 7.196880e-02 0.03162120 0.09107075 ## L20 8.10020596 0.58630474 0.583001761 4.510352e-01 0.18645062 0.77857343 ## L25 5.72035309 8.58946513 0.814107447 9.170445e-01 0.48755393 0.69850864 ## L30 0.70423921 22.20595548 7.687177362 1.836705e+00 1.36863579 0.86366230 ## L35 2.63456559 11.21736539 28.441776214 8.465402e+00 3.41259947 2.61743324 ## L40 8.23959878 4.26489364 30.396695803 2.903435e+01 11.61014361 6.22660636 ## L45 18.12130174 9.63955722 10.365179499 3.523356e+01 29.92516183 17.34178084 ## L50 24.23922759 16.37286498 6.849961453 1.491299e+01 31.54773039 29.94878954 ## L55 19.87604422 14.73079378 7.023105706 4.652150e+00 15.62755410 26.16513421 ## L60 8.69999049 8.30746101 4.835885262 2.658960e+00 4.02605217 11.66876120 ## L65 2.14634273 3.21778829 1.944266093 1.186616e+00 1.18315127 2.81372917 ## L70 0.38029079 0.67956730 0.820325768 3.606157e-01 0.37832285 0.50475038 ## L75 0.03681364 0.09139539 0.115442835 9.899023e-02 0.17179407 0.18587561 ## 2000 2001 ## L10 0.001551803 0.001231908 ## L15 0.067701012 0.195661977 ## L20 0.267492331 1.782819794 ## L25 0.785602536 1.698725748 ## L30 1.843927479 1.810999535 ## L35 2.022296915 3.696790885 ## L40 4.539733592 5.031227555 ## L45 12.037621277 8.462605525 ## L50 22.714896258 16.108793810 ## L55 27.874149294 23.217521321 ## L60 18.260568808 20.883663593 ## L65 7.171622877 11.873544050 ## L70 1.841943108 4.138306126 ## L75 0.473394541 0.844386558 ## ##$t_midL
##  [1] -0.7677769 -0.2658723  0.2769175  0.8678489  1.5163002  2.2346886
##  [7]  3.0399440  3.9560016  5.0182938  6.2825444  7.8440039  9.8867911
## [13] 12.8469531 18.3045443
##
## $lnC_dt ## L10 L15 L20 L25 L30 L35 L40 ## -5.0157151 -1.3934848 0.9415568 1.4758834 2.0784547 2.5058617 2.6259297 ## L45 L50 L55 L60 L65 L70 L75 ## 2.6336292 2.6890754 2.4189679 1.9017746 0.9546319 -0.2726343 NA ## ##$reg_int
## [1]  9 13
##
## $linear_mod ## ## Call: ## lm(formula = yvar ~ xvar, data = df.CC.cut) ## ## Coefficients: ## (Intercept) xvar ## 4.7998 -0.3894 ## ## ##$Z
## [1] 0.3893934
##
## $se ## [1] 0.02253122 ## ##$confidenceInt
## [1] 0.3176890 0.4610978
##
## $intercept ## [1] 4.799812 ## ##$linear_mod_sel
##
## Call:
## lm(formula = ln_1_S_1 ~ t_ogive, na.action = na.omit)
##
## Coefficients:
## (Intercept)      t_ogive
##       5.858       -1.865
##
##
## $Sobs ## L10 L15 L20 L25 L30 L35 ## 4.048843e-05 1.842143e-03 2.350774e-02 5.048929e-02 1.187280e-01 2.408027e-01 ## L40 L45 ## 3.715216e-01 5.348644e-01 ## ##$ln_1_S_1
##        L10        L15        L20        L25        L30        L35        L40
## 10.1144539  6.2949819  3.7266369  2.9341855  2.0045311  1.1482837  0.5256946
##        L45
## -0.1396843
##
## $Sest ## [1] 0.0006817419 0.0017368853 0.0047663242 0.0142158860 0.0461128027 ## [6] 0.1558585003 0.4533247083 0.8207694838 0.9707778829 0.9971611338 ## [11] 0.9998453603 0.9999965766 0.9999999863 1.0000000000 ## ##$t50
## [1] 3.140323
##
## $t75 ## [1] 3.729264 ## ##$t95
## [1] 4.71877
##
## $L50 ## [1] 43.08199 ## ##$L75
## [1] 46.32565
##
## \$L95
## [1] 51.16988
##
## attr(,"class")
## [1] "catchCurve"

# 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 is appropriate for fisheries that 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.

## Other

Kenchington, T.J. 2014. Natural mortality estimators for information-limited fisheries. Fish and Fisheries 15(4):533-62. https://doi.org/10.1111/faf.12027.

Pauly, D. 1980. On the interrelationships between natural mortality, growth parameters, and mean environmental temperature in 175 fish stocks. ICES Journal of Marine Science 39(2):175-192 https://doi.org/10.1093/icesjms/39.2.175.

Powers, J.E. 2014. Age-specific natural mortality rates in stock assessments: size-based vs. density-dependent. ICES Journal of Marine Science 71(7):1629-37. https://doi.org/10.1093/icesjms/fst226

Sparre, P. and S.C. Venema. 1998. Introduction to tropical fish stock assessment. Part 1. Manual. FAO Fisheries Technical Paper (306.1, Rev. 2). 407 p.

Then, A.Y., Hoenig, J.M., Hall, N.G. and D.A. Hewitt. 2015. Evaluating the prodicitve performance of emiprical estimators of natural mortality rate using information on over 200 fish species. ICES Journal of Marine Science 72(1):82-92. https://doi.org/10.1093/icesjms/fsu136.