CM_1969_H_08.pdf (438.0Kb)

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International Com"lcil for v. '._. ^t"I ⁰1

1969/'''.8'

J:_ 0

the 2xploration of' the Soa j?elagic Fish (H ortJ:lOrn) COIllllli

ttee

Ref.: Pelf Fish

(S)

Cttee.

A l~ethod of estiElating Lortality in Fish Stocks

by

Olav Aasen

Directorate of Fisheries

Institute of' Earine Research, Bergen, Norway

r. ^Theory

.i:'re::n.lli1ing a oonsta.'lt yearly recruitr:ent

(IT) a::..'ld

a constant yearly survival rate

(x)

¹ ti:le L1ngnitude

of'

the stock [:lay be I1ri tten:

" 'T "r -r 2

011. - 1 + kK +.Lx - 11 ,. • • +

n

._ i

T-L:--2f-

- 1 -

x

'\rJhere Ht Ex, and so forth is t::~e strengt:_ of tIlo 0 group ,-_. ,t:.:.e o:e.e year group,

and

so on up to tLe year gronp

(n - 1).

:tquation

(1)

may also be llri tteru 011.

:::: :n

1

(100

=

^a ^b +

r 11

-:- IT ^)C-:-1---^:;,

x

o + ^d

) 11(2)11

If' no", the stook in question is adequately sar:lpled, the per- oentage strength of the age groups

0

to

(p - 1), p

to

(q - 1),

q to

(r -

^{1 ),} ^and

r

^to

(1'1 -

¹⁾ ^{will be}YJlO'Vl1.. Let these be denotod by a, b, c, nnd d respectively

11(2)".

right hand side of' (2) nay be chosen arbitrarily. :11::"en four terms are prai'orred here, i t is only because this is a convenient form when comparing 1vi t11 other ll1ethoc1o.

(2)

-2-

Fror:l

(2)

may be coastructecl a:~1.ot:':~er equatio:1.:

H

1::>

:x ...

1

/ I

1.) n

=~''-J.: _ _ _ ",r

IT :::: ^~"- 1

Since 0

< ^x

^<,.1, ^Jell "dll normally be a smaller nt:u:lhor cOl.::pared

W1.d may be ignored.. 7tquatim.7.

(3)

takes th.oD. t1.:.e :forn:

c -+ d

.

b + c + d

to

(1.;.)

'l'his sir~lpli:fied expressio!.'l cannot be used 1,,1.:.e~:1

:.:3

[lElall, i::.1.

species t~.'le formula

(h)

should be applica:::Jle, insofar t:J.e basic assumption are approacl':~ed to SOLle ez:te:.1. t.

I t 'Hill also be realized t:::18.t if' ]( is large, say about O. [), this bias introduced by ol~litting xn w'ill net be negligiblE). I:~:l t ~ _'.i I']

case the equation:

" ox Il-p - (, 0 + c + a ,) x q-p -+ ( c + Q ') = 0

should be used. rl~his expres[lio~n is also a rearrengemant of

(3).

'1"11e difference 'tvill, however, be Gnall since a li.igh survival rate (x) also J.:leffilS a high age

(11.).

The same genernl idea can be appliecl to develop :forn:ulas expressing the relationship between t2.J.e life sp[m.(~l) and tiw mor- tality (x), i:'l which cnse also snaIl values of' ::'1. i'Till be included.

But this question ."ill not ~Je considered b.ere.

I t is a well know'l'1. :fact that tl:ce et,rength of the year classos I:lay vary considerabely in :fish stoeLs, a~1.d t?ie yearly recruitment is seldom, if' ever, a cons ta:;.·l t. Ii' several ·yea:<.~s of observation io at

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haD.d, tJ~i.e values may be averaged £lJ.1.d the disturbing ef'fects of' varying recruitment will be dal:1ped.

:~inar Lea >us calcula ted t:~,e ourvi vaI rate of' tl::.e H or1vof.gial1.

'I.finter Herring based on age :-ceaclingD :frolL 1907 to 1928. (Lea, 1⁰20 \ .7 ~.7 / •

Lea's algebruic method gives: IX

=

0.01 comprising t~e age clasneD

7 -

17 yearo. Lea does not prosent his basic material, but the age composition can be f'ound in a paper by Oscur 0w.J.d (19L~3). :!::~l. Table 1 is presented tl:e average age frequency distribution 1'01.' the period

1907 to 1928 based on the data give;~l by :::;l.u1.d.

Plotting the natural1.-olSnri--tm,w to tne values in Table

"l,-t'{--

17

years),

and

fitting a straight line, i t will be f'ou1.1.d that the 1 \,

slope is

-0.266

J. T~iis means that t}::.e yearly survival coef:ficient is ( Lo guri tl711i c method, Fig __

1) •

of a is

375

(tile DULl o:f the perco:tl tag-es up to

6

years). ;3imilarly, c

= ^{270 (10} ¹⁷

^{yearn ,}

⁾

^{lli1.d d}

= 10 (18 - 23

b

= 345 (7 - 9

years),

years). (Table

1).

The figures p, q and r may be chosen arbitrarily.

Since, hO'lilever, this new' met:~c.od is put to test on the same data Lea ilsed for his calculations, t:1e choice of' p =: 7 and r = 18 is obvious.

The choice of q ohould be made 00 that the root extraction will be siElple. ~ith (q - p) l~ke

3

or

4,

the operation uay be readily illude on a:a. ordinary Glide ruler. ~lit~:l the figures given, (L~) yield/]:

and

(5)

tak~3 the i'o:cm:

1)

Calculations by slido ruler, fitting by eye.

(4)

-4-

This equatio:l:1. :t.ao a oolution for

0<x.<:1

o:f' x :.:

0.77 1) •

• ^{But Lea}

has excluded a l l fish over

17

years in. ::.!.in calcul.atioas. It:. the SCLllle is done J:;,ore, the equation 1·1ill be:

( r'

\0)

at""1.d in this case:

x ::; 0.78

( l"iP'. <;>

2) •

I f t:Lle nili.1pli:fied forraula

(

~,

)

is used, the result Hill ^1,)0

x ::::

^JL;.5:z:¹¹

is not negligible for s·u.c::.'. :::::.ig:L::. values of

x.

when the oldest age groups are excluded.

for a series of yoars t is t:Lle a1:'ctic cod ~lIdc:·.'. spavr.:."1o i:'"1 t::.e Lo:::ota:"'l area. For thin :fish Rollofsen (1933) has introduced read~2gs of spm·ming zones in the otolitiw D..:.."ld instead of age clistriln:tio:l, ono gets in this case a distribution of spal.]:i1.iag groups. Tl-::.o so data r;-laY, of' course, be 'broated i:::::. 'cho Oo.co w:::I.y aD ^'/:;:.10 :::I.ge dict:.~i~::·~:·tio:.::: as far as

calculations of r;:;ortali t-:;.r rates are concer:'lod. Here is used the dato.

In Ta0lo 2 is prosen ted t:le average dis tri'bution in per mille of the spm·m.i:i.1.g groups of :feDale cod in Lofoten for the yearn

1932 - 52.

The Lea nethod, applied on the spavmi:c.g groups I -

VIr:,

givelJ

x

==

0.506.

is in ^..1-.'td .. 1.l..D ^• caGe very simple • Th.a three f i r s t apmming groups constitute

082

per millet Going bacl;: to the original i'orll11 .. 11a t~:1.is uay be Vlri ttmu

1 - x J / 1 _ x

13 _

882/1000

in whic:L:, case

x

¹J ^q lllo.y be ignored for value of :.K.

q

1 - 1(-" = 0.882 or

3 r -

x

:.~

YO.11G :::

O.1.~91

1) Solved grnphico.1.ly by 0. Llothod illustrated in. Fig. 2.

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Using the

"logarit~'ll~ic

l:letl:.od" tL.e result will be 1) ];: - O,L:·)5.

(Fig.

3).

Both scales and otolit~s cay b~ ~ifficult to read for older ages, ':lit:::1. the ¹¹pOller series Le'i:;1::od", ~101-.'ey.3r, where certai:<l i:~1. ter- vale of age groups are pooled, i t will be suX~iciont i f the individual readineS ca:;:1 be carried up to a certain 2.ge

(q - 1)

with reasonable exactitude. ~n~_at is above that age goes into tIle last group. For the f i r s t group is o1-:.osen the age (p -

1)

.-ft,ere full recruitl':lent is attained. I t will be realised that also in this case equation

(4)

is valid, and this is perhaps th.e gr0atest r:.:erit of' the ::'1.e11 J:1ethod.

The Atlantic Eacl;:erel frm.l t1-::.e NO]Jt~l 0ea area presento a case ^C"

as described above :--:.) The otoli ths are generally hard to read clearly above eight years (q

::-:9)

\

and p) full recruitmei'lt is attai:2ed at the age of five years

(p =

Reliable lTortvegian data for the ege distribution are available only f'or the last t3.1.ree years,

the otoli t:--w years of aGo.

O^/',~./

.

^I^\.i..^r,..,^aD_e^" ¹

^3).

iLlpossible to

."4',

In cases

read even to

8 .'

according to t:210 rule that :for

L>"38

^cr;~ tho age is Over

C

years, a:i.'lcl for 1,:;38 CEl t:'::'.e age is beb'Jeen 5 llil.d 8 years.

For

1967 t:i.e

condensed data i'rOl;I Table

3

sho'l,n

9'----__

L~ __

.5"---___

G__._-·;, (3

157 512 331

similarly 1~68:

o - h ,5 -

⁽³ ^,^{. ( )}⁽⁾

533

I

312 155

x I;.

= .l1:!. =

8¹;.3

0.393 0.332 •

1) Calculations by slide ruler, fitting by eye.

x = 0.791

x = 0.761

(6)

o - h JJ4

.5 ... 8

L~ 10

)8

Or::

- -

• •

f

x .- 0.789

For the pGrioc1

1967 - 69,

tILe average 81..:rvival rate is:

0

-

^L!. ^J^~.

^-

^{( )}⁰ ^I ^:.;~.^uⁿ ^J~^y ^L~

=

^{2 L}

_6!59 :·7 = _0.3'/5

^"'-^~r ^:::;; O.70 L!.

JL~ 1 L!·12 2L~7

III. 002."lclusio:ns

'fho three methods for calculation ol' t3.,-e survival rate shoH reasonable correspondenco -vfheIl applied to the age data of tLe

lJ or1'ITegian -;'Jinter ~~erring in tiLe period

1907

I t will be seen that Lea's moti:cod gives the

~::.~igher

value of' x.

iy.81J.

"lhich may be termed 11 power series r;~etl"i.Od" 1 gives practical13^T the same result as tIle "logari t};.r:lic E:.ot:::iod" both for the abbreviated

and for equation (-,)

r o. 77j

For equation

(6)

the value of x is a l i t t l e hitjl!.er

1''0.70-;

\...~ ....

For the case of the Lofotel1. cod, i t 'dill be seen t;".:.at t1:::.e re-

Here

too the Loa method gives t~e highest value of x

0.506 i.

^The

11 power series n,et~lOd"

give practicall:?" the same reoul to, as \"Tas aLso the case :for the l:.erring l11aterio.l.

The hIO examples for the herring fu"J.d cod data shou t~-:at tl-:e

"poller series E1ethocl^ll is applicable )':"'or calculating survival rates for long lived species where long series of ageing are at hfu"J.d, :"l'or the North Sea mackerel there are on.ly three yoaro of data available in the Norwegian material. But i t is evidez,-t that these three years

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show reaDonable correspoildence ia tlle ourvival rates calculated by the new method. ?or this 1:i::.,-c1 of data, botb. e:le If logari tl-u:lic 1;}etl1od"

and Lea's ll1et:::-lOct brea1-: dow:;.:.., at least i21 t':wir origi:1al forma.

Tl-:..e higIl survival l~a te i'ov . .nd f'oI' tl:,e Elacl:erel, is rat?c.er m.:I'- prising. Accepti:a_{b ,} hOHever, the evidence, i t ""ill l:~ea~:: t~_2t; t~J.8

mackerel reach a high

age,

say up to

24 -25

years. I t l'airly obvious, that 1rrhel1.

tJ:w

ring TJ.et f'is:':l'ary for r:mckerel stal~';od in the

IT orth Sea area, there i'las a;'l accumulated DtoC:'::.

survival rate, this stock should be expected to be large. That thiD is so, is bor::w out by t~.le fact tJ:!.at acoording to the of'ficial

Norwegi2~1 :fis:lery statiotic8, about

2.5

r,lillion to:'.18 Lave been landed in the period

1965 - 68.

m.ercial catches 1:1£;.en the inte:;:lSive fishery otartled , should J:.l0ro or loss show t:G.e natural survival rate.

But i t 'VIill be realized, tlla t t::'"-e saue would be tb.e caDe if' there vIas no recruitment to the stocl::, irreopective of' tIw stool:

level. An ir18pection of ':Cable J sJ:wvV/J tLat t1-::.e laot strong year .. ; class to enter the adult stock vIas cne ^{J •}

1962

year class.

I may be concluded then, that t7:18 calculated average survival ra te. 0.78[1·, is approachil1.g' t::l.G natural sU1'vi val 1'a to for the IT orth Sea Ii!ackerel becauGe the eJ':f'ects o:f the :-:eavy :fishing is cov.l1.teraoted in the formula by :fai1.il1.g recruitment • This Imow'ledge may be useful at a later stage, vihen the recruitrael1.t is 1In ornalizec1", or if', :f.i.

tagging data will provide reliable estimates of' the exploitation rate.

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Heferel1.ces

Lasen, D.

19.53

Lea,

1929

Hollefsen,

G. 1933

( )

- 0 -

"A hetl:od for theoretical Calculation of

the numerical stock-stre!lgth in Fish_

:-::0i)ulatio"-1.s subjected to seasonal i<'isheries".

Happ. Proc~\Terb. Vol. 136.1952~o

"Hortali ty in the TribeofNor-:J:~· :-::Lan I-Ierrh'lg".

Rapp. Proc-Verb. Vol,

6.5, 19.30

"The otolith of' the 0oc1",

Eep. ITor~v. Fish. Lar. Inv. Vol. L!_. Ho.],

1933.

liThe Age-,Jompositio1'l of the Norwegian iJpaun

herring obsorved during

36

years",

Al"J.nales Biologiqueo, Vol. 1,

19[1-3.

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Tal: le 1 Average age frequency distribution ( 0/00 ) of the Horwegian Winter Herring in the years

1907 - 1)28

0 'Year! Age

t.;907i __ 3 i 4! 5 I 6 7 i 8 ' 9 i 10! 11 I 12 13 i 14 ! 15 ' ,61,7 L'_8 L:~[j\1 121_hI23_f'~al

1-28:!!

!! I

I

I 1 . I '1-=-1.1 I '. . ',1 "". I . . l.~ 14,1.l?_1,4 l5 0 ,2\148,8 i 12.8,6 .114,3! 102'4176,0149'2139,8~33'2 26,7! 20,3 15,1\9,915,912,8, 0,61 0,6J~210'7!

Table 2 Average distribution ( 0/00 ) for the spawnL1.g groups of Lofoten cod (~) in the years

1932 - 1952.

" ---,----.---'--Spa\'lning group s

[~~~~! - ~"l--'-r;--'-l

IlL

1 1932

I-j --

- 52 ! 473,&! 277",0

VI

131,4 52,4 28,J 15,9

VII

9,9! 5,5

IX

2,5 x I

XI XII XIII_-t-___ .

.! 1, °

0 ,

4

0,2

1000,3

Total 2,2 Table

3

Age groups ( %0 ) North Sea Mackerel. (No age reading: U 2

>

.38cm).

u I:

38cill; 1

r

^y~ar[O-1

1

T' -2'---i-

3

-r 4 ---j .-- 5---'1- 6 ---1--7-. -I-'S-I u 1-'-T-'-9-I2

o--r'11i-12-~'~13·r -1-4-'l~5T-~6:

.

~7 j

u

2 rro~~~-T

196;L! 7,7 1 78,5 15 0 ,6120,8 i 24J~43'81 Jl'5!,39'2i154'6i35~4i16'9120,0124~6i 2r31 ,),8\-:-1- ,- 230~11000,c .-,-, I! I !

I ., i • 'r-'-'

; ; *

i

I

i

I : I

i I I

I

I 1

l~968i -!61 ,2;302,°1126,5,42,9

I

14,3 11 20 ,41 16,3; 16,3 J144,9 18,4j38,8)10,2118,4 6?1\ 2,012,OJ - 2,0 57,1; ! ! i ill

I

i ,

I ! i

i , I

I _ I

i I

i

1 !6

9 ,-: - I 7,7!15 1 ,3;17 4 ,4 i 59,0

i

15,4J169,2!46,2\120,5j10,2i2J,1 12,817,7 15,41 1 ,J,4f-l- - 171,8 !1000,1

1 I ! 1 i ! l I :! 1 j i i

IMeanl- 23,01129,41109,3; 79,41105,5r59,9i72,J!33,9i140,Oi21,J/26,3 14,3 16,9 7,91 5 ,11°,7 -1°,7 1.53,21 1000

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lnf'

---;...,

t---·-·-,---·T4~---

2.0 - - & . - - - - L . , _ , ___ ~, "'----~-.,__t .. - 2Git~~

7

⁸

9 10

11 12

1:1 14 15

16

17

A g

e

Fig. ₁ Logarlthm.ic method.

⁰ ^Natural

logarithms

to

^t!.':o

t'l!')quenoies in

Table 1 •

z

^:-;

O",;:CG , ^x

^a

^{O. '769}

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o

- 100

\ _•

o ^0,1 0.2 0., 0.4 0.5 0.6 0.7 0.8 0.9 1.0 x

Fig. 2 Power eerie s method.

,"

• Caloulated values ort r(x)

_III

'45 x" - 615 x' + 270 r(x)

^la

0 for x

^=-:

0.78 tt (x)= 0 for

x et

0.915

~(x)=

0 for.

x ::

0075'

(12)

In

f

3.0 ..

Fig.

~

Logar:tthtllio method.

• Natura.l loga.rithms to

th~)

frequenoies in Table 2 • z = ^{OQ703 ,} ^x = ^OQ495

+--_~

""'

^-.L^{_ _}....;,..ji--_ _ _ • J..L ______ ... , _ . _---'1--_ _ _ _ _ _ _ _ _ _ . ____ ^M"

I

II

I I I

IV

V

VI VII VIII

Spawning' classes