Estimating trophic levels using length and Ecopath

9.3.3 Estimating trophic levels using length and Ecopath

Pauly et al. (2000) have developed a simple an apparent k of 0.96 year -1 . Mean lengths at age method of calculating change in trophic level calculated from this apparent growth curve will be (DTL) where mean body sizes have been reduced by biased if fish migrate out of the sampling area. The fishing. The estimation depends on an empirical actual mean lengths at these ages is given by the relationship between body size and TL in 180 true growth curve. The next assumption is that species of fishes: the true cohorts have the same COV as the youngest cohort in the otolith samples (in this ex-

DTL b = log 10 { [ L • + { [ Lc M k ( ) ( 1 - E ) ] ample, 0.23). If this is true, then ‘ideal’ cohorts

[ ( Mk ) ( 1 - E )+ 1 ]}]

with normally distributed lengths, including both [ L • + { ( LcM k ) ( Mk + 1 ) } ] , (9.15) unmigrated and migrated fish, can be set up using the mean lengths from the true growth curve. A where Lc = length at first capture; E = exploitation

distribution of fish remaining is then set up by rate, F/Z; M = natural mortality rate; b = slope pf multiplying the ‘ideal’ cohort vector by the proba- relationship between TL and body length in cm bility of migrating for each length class. If the cor- which is 0.63 for large carnivores like cod, and 0.24 rect migration line is chosen, the mean length of for small pelagic and demersal fishes; and other pa- this distribution should be equal to the observed rameters are as above. mean length for the apparent growth curve.

This method has been used in analysis of de- Goodness-of-fit is judged by least squares of the cline of TL in Canadian fisheries; the length-based two pairs of mean lengths and standard deviations. method provided very similar answers to an age- The analysis is repeated for all combinations of based method.

Size-based Methods in Fisheries Assessment

207 Growth and age structure of the catch are es-

9.4 CONCLUSIONS

timated simultaneously with population pa- rameters such as recruitment, selectivity,

After over 120 years of length–frequency analysis catchability and natural mortality . . . we are justified in asking ‘Are we there yet?’, and

Spatial structure can be included in the ‘Where do we go from here?’

model. . . . Missing data and data of different At one time, length-based methods were regard-

temporal resolutions are allowable . . .

ed as being suitable only for the tropics. This has Auxiliary data (such as tagging data) can be been exposed for the myth that it is (e.g. Pauly

incorporated into the model as appropriate. 1994). Since that time, the discovery of daily

Various structural hypotheses, such as otolith rings has meant that, at least for the young,

density-dependent growth, trends . . . and tropical fish can be aged. And length–frequency

seasonal catchability, can be incorporated analysis has become a cost-effective way of mini-

into the model and tested. mizing expensive ageing for many temperate fishes. An unexpected practical use for length– An example of Monte Carlo simulations exploring frequency analysis has arisen from Froese and the sensitivities of such a complex model is given Binholan’s (2000) technique, which has led to the by Fu and Quinn (2000). printing of posters allowing buyers at markets to

In a different direction, length-based short reject fishes that are too small.

cuts that can be programmed on a spreadsheet can

For conventional length–frequency analysis to tackle a range of interesting problems – doubtless determine growth and mortality rates, it seems at many of these still remain to be invented and used. first sight that there is nowhere much left to go.

But there may be completely new techniques on Parametric methods are now fully matured, and the horizon. One recent development may point to classic analyses such as mixture analysis may an exciting way to deal with our opening example

be carried out, complete with uncertainty, on a of estimating the growth and mortality of humans spreadsheet. Among the non-parametric methods, massed in St Peter’s Square. Smith and Botsford most of which may also be performed on spread- (1998), who bravely say that it is ‘analytically sheets, the projection matrix stands out as being more challenging to analyze size frequency distri- deserving of further development and rigour. For butions that lack multiple age pulses’, show how most normal fisheries work in both the tropics and von Bertalanffy parameters estimated from tagging higher latitudes, in the past decade these analyses of individuals can be informative about the shapes have become robust and reliable standard tools. A of length–frequency distributions. Through over- word of caution is always to map the goodness-of- layed Monte Carlo simulations, which can very fit surface rather than rely on optimization rou- quickly be graphed on modern computers; they tines built into spreadsheets. A further warning is demonstrate how characteristic length–frequency always to perform several different analyses and shapes arise even in continuously recruiting compare the results.

animals. The technique has been applied to fish

In one direction, the research frontier of con- such as lingcod (Ophiodon elongatus), and inver- ventional length–frequency analysis is now mov- tebrates such as whelks (Buccinium undatum), ing into highly complex integrated catch-at-length surf clams (Macromeris polynyma), and urchins models that are far moved from analyses that most (Strongylocentrus franciscanus). fisheries scientists can perform on a spreadsheet.

Long-lived, slow-growing fish that are difficult For example, MULTIFAN-CL (Fournier et al. 1998) to age and assess using conventional means is described as a length-based, age-structured, like- currently remain intractable even with complex lihood model that circumvents many of the diffi- length-based methods. This problem has not culties associated with sequential analyses such as changed. But it is a gap in the fisheries assessment VPA (Shepherd and Pope, Chapter 7, this volume):

toolbox now that serial depletion by area, depth,

Chapter 9

species and taxon proceeds apace in the world’s Froese, R. and Binholan, C. (2000) Empirical relation- overexploited oceans. It seems as though the criti-

ships to estimate asymptotic length, length at first cal problems for the fisheries of the early 21st cen-

maturity and length at maximum yield per recruit in tury have bypassed single-species analyses such as fishes, with a simple method to evaluate length fre- quency data. Journal of Fish Biology 56, 758–73.

those presented here. Are length-based ecosystem Fu, C. and Quinn, T.J. (2000) Estimability of natural mor- analyses the hope of the future?

tality and other population parameters in a length- based model: Pandalus borealis in Kachemak Bay, Alaska. Canadian Journal of Fisheries and Aquatic

ACKNOWLEDGEMENTS

Sciences

Gayanilo, F.C., Sparre, P. and Pauly, D. (1996) FAO-

I am grateful to Peter Macdonald who, a long time ICLARM Stock Assessment Tools (FiSAT): User Man-

ual . Rome: FAO.

ago, initiated me into formal length–frequency Gulland, J.A. (1983) Fish Stock Assessment: A Manual of analysis using what was then a revolutionary new

Methods . London: Wiley.

4 K computer, the Elliot 803. I also thank Daniel Gulland, J.A. and Rosenberg, A.A. (1990) A Review of Pauly, who, not so long ago, provided critical

Length-based Approaches to Assessing Fish Stocks . comments on a draft of this manuscript.

FAO Fisheries Technical Paper, No. 323. Rome: FAO. Hald, A. (1952) Statistical Theory with Engineering Applications. New York: Wiley.

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