Estimating the parameters of the stock-recruitment model of Ricker from a yield-per-recruit model in data-limited situations

ELSEV 1ER

Fisheries Research 20 (1994) 229-242

Estimating the parameters of the stock-recruitment model of Ricker from a yield-per-recruit model in data-limited situations Luis Cubillos S. lnstituto de Investigacion Pesquera Octava Region S.A., Casilla 350, Talcahuano, Chile (Accepted 25 January 1994)

Abstract A simple method to estimate the parameters of the Ricker stock-recruitment model is described. The method is based on the Beverton and Holt yield per recruit model, as a function of fishing mortality, which can be used to describe the stock-recruitment relationship of an exploited stock even when the data are limited to 1 year of estimates for average biomass and fishing mortality, or, to describe virgin biomass (educated guess). Once the parameters are estimated, the surplus production and maximum sustainable yield can be established. The method is applied to Merluccius gayi of the central zone off Chile and Sardinops sagax off northern Chile. The application of the method is discussed.

1. Introduction

The establishment of stock-recruitment (S-R) relationships and the estimation of recruitment is extremely difficult for many fisheries. This is true particularly in data-limited situations where no recruitment and spawning biomass time series exist. In fact, the traditional establishment of a S-R curve requires the plot of the number of recruits (or some index of recruitment ) on parental biomass. Thus, classical S-R models, developed by Ricker ( 1954; 1975 ) and Beverton and Holt (1957), provide us with curves that describe the data. No analytical theory is available yet by which curves could be constructed in terms of parameters that are independently derived (Cushing 1977 ). In this paper a simple method is proposed to estimate the parameters of the Ricker S-R model in data-limited situations of exploited stocks. The method is based on the Beverton and Holt ( 1957 ) yield per recruit model and requires only 0165-7836/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0165-7836 (94) 00283-3

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estimates of total annual average biomass and fishing mortality during 1 year as input. 1.1. The stock-recruitment model of Ricker

In the model proposed by Ricker (1954; 1975 ), the relationship between the number of recruits and the parental stock size is given by R = aBe - ps

( 1)

where R is the number of recruits, B is the size of parental stock, a is an index of stock-independent mortality, and fl is an index of stock-dependent mortality. When a and fl are estimated, m a x i m u m recruitment (Rm) is obtained by (2)

Rm = a / ~ e

Also, the parental stock at m a x i m u m recruitment (Bin) can be estimated by

(3)

Bm=l/fl

On the other hand, Eq. ( 1 ) can be rewritten as (4)

B/R=ePB/a

which expresses spawning biomass per recruit. Therefore, the yield per recruit model of Beverton and Holt (1957) can be used to derive some relationships with the S-R model of Ricker (see below). 1.2. The yield per recruit model of Beverton and Holt

The yield per recruit model (Y/R) of Beverton and Holt is a steady-state model, i.e. a model describing the state of the stock in an equilibrium condition on the long-term. The equation used here is in the form suggested by Gulland ( 1969 ), which is expressed by Y / R =F[e-M~tc-tr) W ~ ( 1 / Z - 3 S / Z + K +

3S2/Z+2K-S3/Z+ 3K) ] (5)

where S=e-A:(t~-,o); K, to and Woo are Von Bertalanffy growth parameters, tc is the age at first capture, tr is the age at recruitment, F is the fishing mortality, M is the natural mortality, and Z = F + M is the total mortality. An interesting property of the Y / R of Beverton and Holt is that it can be used to express the annual average biomass per recruit as a function of fishing mortality. In fact, it can be shown that the yield per year is Y=FB

where B is the average biomass during a year. The averaging factor is simply B = Bo [ 1 - e (~-z) ] / ( Z - G), where Bo is the biomass at the beginning of the year and G is an exponential rate of growth in weight (see Ricker, 1975 ), which is a gen-

L. Cubillos S. / Fisheries Research 20 (1994) 229-242

231

eralization of the more usual expression N = N o [ 1 - e - z ] / Z which applies to numbers rather than biomasses. It follows that:

B/R = ( Y / R ) ( 1/F)

(5. l )

Thus, it is easy to go from Y/R to B/R. However, if F - - 0 then the following equation must be used

B/Rv=e-M(~c-tr)Wov(I/Z-3S/Z+K+3S2/Z+2K-S3/Z+3K)

(6)

where B/Rv is the virgin biomass per recruit at F = 0 (Sparre et al., 1989). The B/R values obtained from the model of Beverton and Holt, can be used to obtain absolute biomass in Eq. (4) when the ot and fl parameters of S-R model are known (Shepherd, 1982 ), i.e.

B=ln( a B / R ) /fl

(7)

However, t~ and fl parameters are not available for many fisheries; especially those undeveloped a n d / o r in data-limited situations. If this is the case then the procedure described below can be used to estimate the parameters of the S-R model of Ricker. It must be mentioned that the B's used in Eq. (4) and (5.1) may be inconsistent, with one referring to parental biomass, the other to mean cohort biomass per recruit. Therefore, the mean total cohort biomass per recruit must be referred to adult fish only, by using a mean age at first maturity (tin), i.e. t~ = tr = tm. 2. Method

2.1. Estimating the a and fl parameters The principal assumption of the method described here is that the maximum recruitment (Rm) and the biomass at maximum recruitment (Bin), or Bm/Rm ratio, tend to occur at some level about Fm~x, or Fo.l, in the B/R-curve. These biological reference points (Fm~x and Fo.l ) can reasonably define an interval in which B J R ~ ratio can be estimated. In fact, the stock is "growth overfished" if F exceeds Fmax and also "recruitment overfishing" could occur (see Sissenwine and Shepherd, 1987 ). Furthermore, it is assumed that the observations of annual average biomass (Be) and fishing mortality (Fc) are available for l year only. However, when Bc and F¢ are not available some educated guess about virgin biomass (By) of the stock under consideration can be used. Typical Y/R-curves derived by applying the model of Beverton and Holt are shown in Fig. 1. In Fig. 1 (a) the Y/R-curve has a maximum as compared with Y/R-curve in Fig. l (b), which has no maximum. These two kinds of Y/R-curves are treated as two cases which, however, do not differ in essence.

2.1.1. Case I." when Y/R-curve has a maximum Because of the assumption that the Bm/Rm ratio tends to occur at some level about Fmax value in the B/R-curve, it is necessary to give an interval in which Bm/Rm ratio can be estimated. The only reference points that are available are

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232

a)

Y/R max

=:

i I= >-

FO.1

Fm~

Fishing mortality (1=)

b) Y/R o.1

tc

,i Fishing mortality

(F)

Fig. 1. Typical yield-per-recruit curves as a function of fishing mortality for a given age at first capture, which are considered as two cases in the estimation of the parameters of the stock-recruitment model of Ricker (see Fig. 2).

Fo.~ and Fma x. Thus, I believe that B/R at Fo.l can be the lower limit of the interval and that the upper limit can be beyond Fmax and that it can be determined by drawing a straight line from B/Rv (the virgin biomass per recruit at F = 0 ) that goes through B/R at Fm,x to B/R = 0, whose F value at B/R = 0 is the upper limit of the interval (see Fig. 2 (a)). Once the interval is defined, it is assumed that the average of the B/R values, between Fo. l and F,p (Fig. 2 (a)), represents the B m / R m ratio. Then the ot parameter of the S-R model of Ricker can be estimated by Or= ( B m / R m ) - l e

(8)

which was obtained by combining Eqs. (2) and (3). Subsequently, fl can be estimated using Eq. (7) and the observations of Be and F~, i.e.

fl=ln( aB/Rc) /Bc

(9)

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L. Cubillos S. / Fisheries Research 20 (1994) 229-242

(a) B/Rv FO.I

~

-tc mit of the interval

J

~"

o

=

Fishing mortality (F) Fmax

(b) B/Rv

to

FO.1 Umit of the IntervaJ

Fishing mortality (F)

Fig. 2. Method to define an intervalin whichBm/Rmratio could be estimated fromB/R-curve.The intervalis defined between Fo.l (lowerlimit) and F,p (upperlimit). In (a) the straight lineis passing through B/R at Fm~,whilein (b) the straight line should pass through B/R at Fo.~,the onlyreference point that is available (see text). where B/Rc is the B/R value at Ft. However, when Bc and Fc are unavailable an educated guess of virgin biomass (By) can be used to estimate fl, i.e. fl=ln(otB/Rv)/B~

(10)

2.1.2. Case II: when Y/R-curve has no a maximum

When the Y/R-curve has no maximum, the Fm~xpoint cannot be determined. If this is the case, the only reference point that is available is Fo.~. Thus, to define the upper limit of the interval the straight line should pass through B/R at F0.~ to B/R=O (see Fig 2 ( b ) ) . Once the interval is defined hence Bm/R m ratio obtained, the parameters of the S-R model of Ricker are estimated by using Eqs. (8) and (9), or (10).

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2.2. Estimating the surplus production curve According to Shepherd ( 1982 ) a surplus production curve can be constructed by transforming the B/R values of Beverton and Holt to absolute biomass through Eq. (7). Subsequently, the absolute biomass is multiplied by the fishing mortality that produces the B/R value to obtain yield as a function of fishing mortality, It must be mentioned that the surplus production curve is an alternative to validate the parameters of the S-R model of Ricker estimated by the method above proposed by comparison with catch data.

2.2.1. Application of the method and validation of the estimates To illustrate the method and to validate the estimates, it is applied to species for which recruitment and spawning biomass time series exist: the hake of the central zone off Chile and the sardine off northern Chile, which represent demersal and pelagic species, respectively. The method is applied by assuming that no recruitment and parental biomass time series exist and by using a single estimate of average biomass and fishing mortality, which has been estimated through independent methods.

3. I. Case I: the hake of the central zone of Chile The hake (Merluccius gayi) resource has been exploited by trawling and artisanal boats in the Los Vilos-Corral ( 31 ° 55'S-39 ° 50'S) area off Chile since 1940 (Bustos et al., 1990). The parameter values used for derivation of the Y/R and B/R curves are given in Table 1. The curves are plotted as a function of fishing mortality in Fig. 3 and, as might be seen, the Y/R-curve has a maximum. Therefore Case I of the method has been applied. Table 1 Parameters used to derive the Y/R-curve of Beverton and Holt ( 1957 ) for the Chilean hake (Merluccius gayi) and sardine ( Sardinops sagax) Parameter

Definition and unit

M. gayi

S. sagax

W~ K to M tcd trd F

Asymptotic weight (g) Growth constant (year- ~) "Age" at curve origin (year) Natural mortality rate (year-J ) Mean age at first capture (year) Mean age at recruitment (year) Fishing mortality rate (year- t )

3196.3 a 0.139 a -0.923 a 0.35 b 3.0 3.0 variable

612.2 a 0.205 a -0.917 ~ 0.4 c 5.0 5.0 variable

aChilean hake von Bertalanffy growth parameters given by Aguayo and Ojeda (1987) for both females and males. Growth parameters for the Chilean sardine given by Aguayo et al. (1983). bAssumed between 0.3 for females and 0.45 for males (Bustos et al., 1990). CAccording to Pauly ( 1980). aAge at first capture and age at recruitment should be identical to age at first maturity.

L. Cubillos S. / Fisheries Research 20 (1994) 229-242

235

Fmax

B/Rv 2000 \

+

3OO

FO.1

..= =

,I=

1000

"o

Fup

"3

-

i

0

i

0.5

J

i

i

1 Fishing mortality (F)

i

1.5

i

2

Fig. 3. Yield-per-recruit and biomass-per-recruit as a function of fishing mortality for the Chilean hake (Merluccius gayi) for the parameters given in Table I. Fo.~and F.p are the limits of the interval in which Bm/Rm ratio is obtained ( • ) to estimate the parameters of the stock-recruitment model of Ricker (see Table 2).

A hake standing stock estimate of 217 000 t was obtained in 1980 by the sweptarea method (Aguayo et al., 1981 ), which with a total catch of 32 000 t in the same year permits estimation of the parameters of the Ricker S-R model for the Chilean hake (see Table 2 ). The graph of the average recruitment on average biomass (Fig. 4 and Table 3 ) suggests that the parameters estimated appear to be reasonable with respect to the recruitment data of the Chilean hake stock. However, is a Ricker type S-R relationship the most appropriate for the Chilean hake?. It can be strengthened if further information is considered. In fact, the model of Ricker can describe the relationship between recruits and parental biomass of the hake stock appropriately due to evidence demonstrating a strong cannibalism of the young by the adults (Melendez 1983; Arancibia 1987 ); furthermore, younger fish ( 17- 25 cm of total length) cannibalize postlarvae (Arancibia 1989 ). On the other hand, the derivation of the surplus production curve for the Chilean hake suggests that today the resource is underexploited (Fig. 5 ), which is in agreement with Bustos et al. (1990).

3.2. Case II: The sardine stock of northern Chile The sardine (Sardinops sagax) is the principal pelagic resource exploited in the north part of Chile (18 ° 20' S-24 ° 00'S). Total landings of sardine increased notably after 1973 from a total catch under 10 000 t before 1973 to an annual

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L. Cubillos S. / Fisheries Research 20 (1994) 229-242

Table 2 Results of the method applied to estimate the a and fl parameters of the Ricker stock-recruitment model for the Chilean hake and Chilean sardine (Case I and II of the method) Species

Input

Results

M. gayi

Bc a = 217000 t Yc ~= 32000 t

Fcb=0.15 year - l

c~e= 6670.215 ff=9.42 10-6

Bm/RcC= 407.5 3 g Bc d = 3780000 t

S. sagax

ycd=2081000 t Fcb=0.55 year - l Bm/RmC=237.27 g

cte= 11456.59 fir=3.09 10-7 flg=3.98 10-7

aBiomass estimated by the swept-area method during 1980 (Aguyao et al., 1981), and total catch obtained in 1980. bAccording to F = Y/B. CAverage of the B/R values in the interval between Fo.~and/7,v (filled square in Figs. 3 and 6 ). dAverage biomass estimated by acoustic survey during 1981 to 1985, and mean catch in the same period. CAccordingto Eq. ( 8 ). fAccording to Eq. (9). gAccording to Eq. (10) for an educated guess of 5.6 × 106 t of virgin biomass and B/Rv = 812.1 g. 300-

25O"

v-

80-

0

o

so

~o Average Biommm (thousand tons)

Fig. 4. Stock-recruitment curve for the Chilean hake (Merluccius gayi) as estimated by the method proposed. The average recruitment is plotted on average biomass to validate the ct and p parameters (see text). t o t a l g r e a t e r t h a n 2 × 10 6 t d u r i n g 1 9 8 3 - 1 9 8 6 . H o w e v e r , i n r e c e n t years t h e b i o m a s s a n d t h e c a t c h e s o f t h e s a r d i n e h a v e t e n d e d to d e c l i n e d u e to o v e r e x p l o i t a t i o n (Y~lfiez a n d B a r b i e r i 1988 ). T a b l e 1 p r e s e n t s t h e p a r a m e t e r v a l u e s u s e d to d e t e r m i n e t h e Y / R a n d B / R

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L. Cubillos S. / Fisheries Research 20 (1994) 229-242

Table 3 Catch, recruitment and biomass time series of the Chilean hake and Chilean sardine, obtained by virtual population analysis Year

1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989

M. gayi a

S. sagax b

Catch t 10 -3

Biomassc t 10 -3

Recruitsa No. 10 -3

Catch t 10 -3

Biomassc t 10 -3

Recruitsd No. 10 -6

32.0 30.0 37.0 34.0 32.0 32.0 33.0 26.0 25.0 34.0 28,0 29,0 30.0 39.0 47.0

208.4 198.2 199.1 204.8 201.0 207.9 209.6 217.3 186.3 191.1 224.0 208.5 210.5 222.2 218,6

210.3 197.3 199.7 214.8 226.5 224.6 214.0 216.1 189.4 199.6 193.0 170.2 184.7 224.0 214.0

169.5 134.3 280.3 551.7 692.4 1420.1 1606.0 1425.5 1663.3 2421.0 2289.1 2607.1 2223.0 1880.2 1356.8 -

897.8 1935.1 2890.1 3217.5 3412.4 4138.4 5400.0 5206.5 4905.4 3474.3 2306.6 1938.9 2324.6 1446.5 1096.4 -

3049.5 4932.5 5187.0 5274.0 7592.5 13626.0 14002.0 13273.8 12768.2 9816.9 6075.4 7159.4 9285.9 5256.3 4321.0 -

aAfter Bustos et al. ( 1990); these results have been adjusted to average recruitment and average biomass, both sexes combined. bCubillos (1990). CAverage adult biomass, 3-year-old fishes and older for the Chilean hake and 5-year-old fishes and older for the Chilean sardine. dAverage number of year class at age 3 for the Chilean hake and number of year class at age 5 for the Chilean sardine. curve. Note that Y/R-curve has no maximum (Fig. 6) implying application of C a s e II. A c o u s t i c s u r v e y s c o n d u c t e d in t h e z o n e u n d e r c o n s i d e r a t i o n ( 1 8 ° 2 0 ' S 24°00'S) during 1980-1985 provide an average sardine standing stock estimate o f 3.780 X 10 6 t ( M a r t i n e z et al., 1987 ), w h i c h i n c o n j u n c t i o n w i t h a t o t a l a v e r a g e c a t c h 0 f 2 . 0 8 1 X 106 t p e r m i t s a n e s t i m a t e o f f i s h i n g m o r t a l i t y o f 0.55 y e a r - ~. T h i s i n f o r m a t i o n is u s e d t o e s t i m a t e a a n d fl p a r a m e t e r s o f t h e m o d e l o f R i c k e r ( T a b l e 2 ) a n d , as m i g h t b e s e e n , t h e fl p a r a m e t e r h a s a l s o b e e n e s t i m a t e d u s i n g Eq. ( 1 0 ) for a n e d u c a t e d g u e s s o f 5.6 X 106 t o f v i r g i n b i o m a s s . The S-R curves obtained and the plot of the average recruitment on average b i o m a s s ( s e e T a b l e 3 ) a r e s h o w n in Fig. 7. T h e S - R m o d e l o f R i c k e r a p p e a r s to r e f l e c t a d e q u a t e l y t h e s t o c k - r e c r u i t m e n t r e l a t i o n s h i p o f t h e s a r d i n e stock. H o w e v e r , t h e S - R c u r v e i d e n t i f i e d b y a c o n t i n u e d l i n e is b a d as f a r as t h e fit is c o n c e r n e d ( F i g . 7 ) . T h i s is p r o b a b l y d u e t o t h e a v e r a g e b i o m a s s (5 y e a r o l d fishes and older) of the sardine stock being overestimated by acoustic survey. Thus, the

L. Cubillos S. /Fisheries Research20 (1994) 229-242

238

80

60 C

o

¢: :l 0 .l:

40 75

t ¢J

20

oo

ols

I

' ~ Fishing mortality (F)

1;6

=

Fig. 5. Surplus production curve for the Chilean hake (Merlucciusgayi) as derived by using Eq. (7) and the a and fl parameters estimated in Table 2.

1000 I

I:o

B/Rv 800

i

.i m

FO,1

600.

Fup 200-

o

0

I::i 50

o~5

1 1'.5 Fishing mortality (F)

Fig. 6. Yield-per-recruit and biomass-per-recruit as a function of fishing mortality for the Chilean sardine (Sardinops sagax) for the parameters given in Table 1. Fo.t and F.p are the limits of the interval in which Bm/Rm ratio ( • ) is obtained to estimate the parameters of the stock-recruitment model of Ricker (see Table 2).

S-R curve identified by a dashed line, and obtained with the fl parameter computed from the guessed virgin biomass, appears to be the most adequate. However, the method outlined here can be appropriate for a preliminary rough estimation of the a and p parameters of the S-R model of Ricker. Furthermore, the surplus production curve suggests that the parameter ( a and fl) estimated by the method proposed here can be useful for management advice (Fig 8).

L. Cubillos S. / FisheriesResearch 20 (1994) 229-242

239

16-

79/74

80/76

12

< O X O

.= IIC

0

i

0

r

1 000

~

,

2000

i

i

J

3000

i

i

4000

i

6000

,

i

J

6000

7000

'

8000

Average Biomass (thousand tons) Fig. 7. Stock-recruitment curves for the Chilean sardine (Sardinops sagax) as derived by using the a and fl parameters estimated in Table 2. Mean recruitment is plotted on mean biomass to validate the a and fl parameters (see text).

88

A

.g

o 8OO-

o o

0.5

1

1 ,K

2

2.5

Fishing mortality (F) Fig. 8. Surplus production curves for the Chilean sardine (Sardinops sagax) as derived by using Eq. (7) and the o~ and p parameters estimated in Table 2.

4. Discussion The method described above to estimate the parameters of the S-R model of Ricker, is based on the well-known yield per recruit model of Beverton and Holt (1957). Pauly (1982; 1984) suggested that the yield per recruit model can be used to estimate recruitment by dividing yield by yield per recruit at a given fishing mortality value, especially in fisheries that have stabilized at a given level of

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L. Cubillos S. / Fisheries Research 20 (1994) 229-242

effort and/or those consisting of short-lived fish. The method described in this paper is in agreement with the approach developed by Pauly (1982; 1984), except that the parameters of the S-R relationship type Ricker curve can be estimated directly by applying the method proposed here. It must be mentioned that the performance of rough estimation of a and fl parameters depends on the accuracy of the data used. This is true particularly for the fl estimation, which is mostly associated with the stock size, i.e. was the stock really in equilibrium when the biomass was measured? It is not possible to answer this without long data series. However, when working with the method proposed here, the biomass must be the adult average biomass, and both the fishing mortality and biomass to which is referred should have the same age (or size) structure. Furthermore, if the observations of biomass and fishing mortality are available for various years, an average of its values should be considered as input to the method. The estimation of the B m / R m ratio from an interval of the biomass per recruit curve, is the principal assumption of the method. I believe that the B m / R m ratios tend to occur at some level about the only biological reference points that are available (F0.t or Fm~x), which are targets commonly used. However, the B m / Rm ration may be estimated directly by multiplying the virgin biomass per recruit by a factor of 0.2 to 0.4 and hence estimate the a parameter. When parental biomass and recruitment time series of the stock under analysis are not available, the S-R relationship could be established by using the method proposed here in order to determine the potential yield of the stock and its trajectory under various fishing levels. In fact, often the managers and fishermen are not interested in a imaginary yield-per-recruit; rather, they are interested in a "physical yield" (Pauly, 1984). Thus, the method proposed here could be useful to express the results of the yield per recruit model in terms usable by managers, i.e. surplus production curve and maximum sustainable yield. Fishery scientists should check whether the basic assumptions of the S-R model of Ricker are fulfilled for the stock under analysis (see Ricker, 1975; Cushing, 1977; Bakun and Parrish, 1980). Also, one must bear in mind that the S-R relationship obtained by using a single estimate of biomass and fishing mortality should be regarded as tentative until verified, i.e. as a null hypothesis. However, it must be mentioned that many fish stocks do display S-R relationships of the Ricker type rather than the asymptotic Michaelis-Menten type, in which recruits are independent of stock size over a wide range of stock size (Parrish, 1978; Saville, 1980; Murphy 1982). The method should not be considered as a substitute for the classical recruitment and spawning biomass analysis, but rather as a robust estimator of the parameters of the S-R type Ricker curve in data-limited situations, which also provide an estimation of the maximum sustainable yield. Obviously, the new proposal may offer a better basis for a preliminary management- oriented analysis when parental biomass and recruitment time series are unavailable.

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5. Acknowledgments I am grateful to Drs. D. Pauly and G. Silvestre, of ICLARM, for reviewing a draft of this paper. Any remaining mistakes or inconsistencies are completely my own responsibility.

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