Estimation of Growth and Mortality Parameters for Use in ... - CiteSeerX

Estimation of Growth and Mortality Parameters for Use in LengthStructured Stock Production Models J .M.HOENIG * Minnesota D e p a r t m e n t of N a t u r a l R e s o u r c e s 500 C e d a r Avenue, St. P a u l Minnesota 55146, USA

Hoenig, J.M. 1987. Estimation of growth and mortality parameters for use in length-structured stock production models, p. 121-128.In D. Pauly and G.R. Morgan (eds.) Length-based methods in fisheries research. ICLARM Conference Proceedings 13,468p. International Center for Living Aquatic Resources Management, Manila, Philippines, and Kuwait Institute for Scientific Research, Safat, Kuwait.

Abstract The suggestion of some authors that linear methods of fitting von Bertalanffy growth curves generate biases may be premature, at least in one case discussed here, where the direction of the observed bias is explainable by the use of a predictive instead of a functional regression. When growth parameters are needed for length-converted catch curves or length-based cohort analysis, an inverse regression (age on length) may be appropriate. This can be computed through ordinary linear regression. An estimator for the total mortality, based on mean length, is derived for use when recruitment occurs periodically rather than continuously. Surplus-production models employing total mortality as the independent variable may become tilted t o the left if natural mortality is compensatory. When growth parameters are unknown, surplus production can be modelled as a function of Z/K. When available data are limited to a small portion of the surplus production curve, fitting curves by regression may give unreasonable results. In this case, one can constrain a parameter to the value of an independent estimate so that the other estimates become more reasonable.

Introduction Surplus production modelling as a function of total instantaneous mortality Z, or as a function of Z/K, was proposed by Csirke and Caddy (1983)as an alternative to collecting and calibrating data on fishing effort. Total mortality data can be obtained in a variety of ways, the simplest of which utilize data on mean length in a sample.

*Present address: RSMASICIMAS, University of Miami, 4600 Rickenbacker Causeway, Miami, Florida 33149, USA.

122 Linear Methods of Fitting Growth Curves

Vaughan and Kanciruk (1982) suggested that traditional linear methods of fitting the von Bertalanffy growth equation (e.g., the Ford-Walford plot) be abandoned in favor of nonlinear regression because of bias in the parameter estimates. If this is so, it places an added burden on fishery biologists ,without ready accessto appropriate computer facilities. However, Vaughan and Kanciruk (1982) used an ordinary predictive linear regression instead 9f a functional regression line , (D. Vaughan, pelS. comm.). This resulted in a)ower value ofL~ and a~~he~, value of K than if a functional line had been used. Thus, the ditection of the bias observed by Vaughan and Kanciruk (1982) is explainable by the choice of regression line. Monte Carlo simulation studies are needed to see if the magnitude of the bias is accounted for by the type of regression line. Until then, it seems premature to abandon the linear methods for "Iittmg the growth curve. ";';,,,\J

.,:, '..,'::'i.;':;'('

;

Predicting Age from Length

Estimates of the time ~t requited to grow through a given length interval are needed for applications such as length-converted catch curves and length-based cohort anal'jsis. Given unbiased estimates of the mean ages at the1imitsof the length interval, it followsthatah unbiased estimate of the average time to grow , through the interval ~tis givenQY th~dif(erenceo{th~ two -mean ages , " ' since ~t is a linear function of two random variables. The mean age at any length L ~an ~e determined by an inverse regression of the von Bertalanffy growth equation, i.e.,

-~

t -loge , ,

-c

(1

K "

'::" ::,,;,

..,;

..,

L/L~) + eL

,:c".:

...1) ': :.".'!',

,.

,:;, ':"

,

:

ic"

.'.

i'i.:ic

"",:.

.'c

:",

..,

, ' :

where eLc fsariU1;d9W,eqortermwhich can be a.function 9f L (ifc ~O).;This,canbefitted by ordinary least:squares lipearregression if L~ is fixed. One can then iteratively search for the value of L~ which gives the best fit. Since the linearization does not involve transformation ofthedependent variable, the assumed error structUre is unchanged. (The same is not true when linearizing the von Bertalanffy equation by loge (1- L/LOo) = -Kt+Kto'wheretistheindependentvarlable::ln this case, assuming the error stroctuieiridependent 91 t implies that thevariabilit'y'in leogthabout ,.. , age

decreases

with

age);

"

.,

"

','"

'

"c

To , examinethe1mportanceoftising inst~adoj the'u~t:i~r~gre~~Qn"Qf.1~n~60n age, growth parameterswefu estimated by equatiori'(i) both meth9ds ,C,", ~sin~n9nli~e~ re~essio~1!qr ,two"qat~ sets from the literature (Table 1). For the butterfly fish Chaetodon miliaris, K and L~ were both about 20% lower in the regression of age on length. For the shark Carcharhinus obscurus, the residuals increased with the dependent variable so that weights, proportional to the square of the independent variable, were used. In this case, K was about 60% lower, and L~ about 5% h~her in the regression of age on length than in the:regressiori"of length on age. To see the effect of the difference in growth parameter estimates on a length-converted catch curve, an example from Pauly (1984) was recomputed using new growth parameter estimates which differed from the original estimates by the percentages mentioned 'above. When K and L~ were , , " both decreased by about..20%, the values of~ t increased by 70% for the smallest length class to over 600% for the h~hest (Table 2),whilethe estimated value of'Z' changed 1roIrt1.8"to 1.0." '.

.';

:

"...,

'.:.1

'

"C

...23 parison of two procedures for fitting growth

" c',:}

erfly fish data from Pauly (1984), shark data

ciC ~.":rJ:c;,:'

c,';;,

-.c"

;',;;,.;';'"ii!:;~.

(1979).

utterfly fish (Chaetodon miliaris) ge -1 ay

Age on length

% changea

-1 K .0024 day L~ 101 mm

s

-22.6 -20.5

to 6.0 days

-

Lower class limit (cm)

(Carcharhinus obscurus) eb K ~ to

Age on lengthb

-1 .029 year 481 cm -6.9 year

01 -1 .2 year Loa 502 cm to -6.9 year

-58.6 +4.4 c,"-,.: -10

A

07 a /0 change

I hc ge on engt

N

lita

% increase in litC

litb

% changea

K

h C Lengt on age

Table 2. Effects of different procedures for obtaining growth parameter estimates on length-converted catch curve computations. Data are for banded grouper (Epinephelus sexfasciatus) from Pauly (1984).

-1 -1 K .035 year K .013 year ~ 439 cm L~ 482 cm to -6.3 year to -6.4 year "a 100 (2nd col.-1st col)/lst col. squared. bRegression weighted by length

-62.9 +9.8 -22

4 6 8

5 29" 114" 161

.15 .16 .18 .20

.26 .29 .33 .37

71.5 76.2 82.1 89.8

12 14

143 118

.22 .25

.44 .53

100.5 115.4

16 18 20

61 50 32 17 4 4

.28 .33 .40 .50 .67 1 . 03

.67 .92 1.46 3.83 ---

138.2 178.8 267.8 667.7

24 ,.' 26

CUnweighted regression.." r I' .,. ..".C ti ' " ", ",;~, ,:t,jl.~ '1;.,!,i,:t,I.".'--! ""-..,! .,;";-:; ,.,..", ;"-i:'""i;\_'(c~;,,i.,,, " , c-.' ',j...!.", .lIt.. ,;LI.o..,,' :/.";,,(i,,.I...«.1,.. !J.:fj-;~~;:;:~:; (" ';;'"c

\

aBased on L~ = 30.9 cm; K = .51 year-1. d . K by. 22 '6 0770and L~ 'by. 20 507/0', bRa d se on39474L ecreasmg= 245655 ' K = I.e., ., ~ ... C% .increase in, lit = 100, (2nd estimat~lst)/lst. ;1'.:,[J

c't':.f,j~~,.

;. (; j\!;;;"",::,

Estimation

~(~::;.::'lji

of Total Mortality

When Reproduction

from Mean Length

Occurs Periodically

,

Methods for estimating the total instantaneQus mortality rate, Z, from length-frequency . have been available since Beverton and Holt (1956) derived the formula

..Z

=.'

K (L~ -L) ~,=~;,:",..,;

data

" ,c,'.iC,.

, ;

tt ~ f

where K and L~ are parameters of the von Bertalanffy growth equation, L is the mean length of fish above L " while L I is the lower limit of length class in which the animals are fully .yulnerable to

~ f ,

the sampling gear. This. and other approaches, which ass~me contmuous recruitmen~(i..e.., out the year), were reviewed by Pauly (1983) and Hoemg et al.. (1983) (see alsoWetherall Part I, this vol.).

throughet al.,

When recruitment occurs once a year as a discrete event, an analogous model can be derived by equating the mean length with a function of the growth parameters and mortality rate. The

,,-

~-

~

~c

124

~':"..

.-method

can also be used when recruitment occurs with other periodicity if the quantities are expressedin appropriate time units (seealso Damm, Part I, this vol.). Thus,

t max L = L::: NjLj i = t'

L =~

/

e-Zj Lj

t ffidX L Nj i = t'

/

.1= t'

=

~

e-Zi

.1 = t'

(~e-Zj

Lj) (1 -e-Z)

,

e-Zt -e-Z(tmax+1)

~

...3)

where Lj is the mean length at agei, N j is the number of animals at ~e i, and t' and tmax are the youngestand oldest agesfully representedin the sample.Note that L, the samplemean, is the mean of those fish whose agesare fully representedin the sample.This implies that one should choose a left truncation point (L') that lies in a "trough" betweentwo peaksin a length-frequencydistribution. Different growth models may be substituted for Lj in equation (3), notably the seasonally oscillating growth model of Pauly and Gaschiitz(1979). If the simple, nonoscillating von Bertalanffy growth equation is selected,one obtains

Loo(l-e-Z)

L=L

00

(e-t' (Z + K)-e-(tmax + 1) (Z + K» eKto

-. (l-e-(Z

+ K» (e-Zt ,-e-Z(tmax + 1»

Finally, if there is no reasonto believethe older agegroupsare underrepresented,then tmax can be taken to be infinite and

,

-Loo

L=L -.

(1- e-Z) e-k (t -to)

00

1-e-