The cod stock-recruitment relationships in NAFO Div. 3NO were studied by the basic Ricker model and modified Ricker model. Although the models do not adjust to the empirical data, inclusion of the American plaice spawning biomass improves the fit. Adding the mackerel adult biomass also improves the model. Including these two variables in the Ricker model explains almost all variation of the cod stock-recruitment relationship in the area. The confidence level of the parameters in the proposed equation were determined by the bootstrap technique, and its accuracy tested. Was realized a simulation study was performed. The response surfaces show the relations between the four variables studied

Larrañeta, M.G. (Manuel Gómez)

Paz, X. (Xabier)

English

Paz, Xabier

Larrañeta, Manuel Gómez

Digital.CSIC

Cod Stock-Recruitment Relationship in NAFO Divisions 3NO

Repositorio Institucional Digital del IEO

Serial No. N2072  Northwest Atlantic NOT TO BE CITED WITHOUT PRIOR REFERENCE TO THE AUTHOR(S) Fisheries Organization NAFO SCR Doc. 92/25       SCIENTIFIC COUNCIL MEETING - JUNE 1992  Cod Stock-Recruitment Relationship in NAFO Divisions 3NO by J. Paz and M. G. Larraneta Institute de Investigaciones Marinas Eduardo Cabello 6, 36208 Vigo, Spain ABSTRACT The cod stock-recruitment relationships in NAFO Div. 3NO were studied by the basic Ricker model and modified Ricker model. Although the models do not adjust to the empirical data, inclusion of the American plaice spawning biomass improves the fit. Adding the mackerel adult biomass also improves the model. Including these two variables in the Ricker model explains almost all variation of the cod stock-recruitment relationship in the area. The confidence level of the parameters in the proposed equation were determined by the bootstrap technique, and its accuracy tested. Was realized a simulation study was performed. The response surfaces show the relations between the four variables studied. INTRODUCTION Stock recruitment theory values attempts to predict the numbers of recruits generated by a certain population level, and has two clasic orientations developed by Ricker (1954) and Beverton and Holt (1957). Recently other simple functions have been proposed but adding parameters (Shepherd, 1982). More complex functions with additional parameters which can represent environment influences have also beeb used. (Hightower and Grossman, 1985; Lovejoy, 1986; Swarzman et al. 1987). The recent proposals to study recruitment processes are more complicated functions, and more terms and more parameters are caculated. There are few reasons for optimism in identifying accurate models of stock-recruitment relationships, and for two kinds of reasons. On the one hand it is very difficult to identify a model from the empirical data, and on the other, even if this can be achieved, the system rarely remains in a similar situation for a long time. In a previous paper (Paz and Larrafieta, 1989) we obtained a significant and positive correlation between cod year-class size in NAM Div. 3NO and American plaice spawning biomass in Div. 3LNO, (r= 0.857, pc 0.001 n= 20). That was interpreted by supposing a strong predation of 0-group cod on 0-group American plaice. A frther result was a negative correlation between cod recruitment and mackerel adult biomass (r= -0.438, p= 0.041, n= 22), which suggests a competitive relation between adult mackerel and prerecruit cod. To take these assumptions into account, the cod stock-recruitment relationship in Nafo Div 3NO by a modified Ricker model was studied. MATERIAL AND METHODS Data on cod recruitment (N3) and spawning biomass (B6+) in Div. 3NO (Table 1) have been taken from Bishop et.al (1990), American plaice spawning biomass (B11+) in Div. 3LNO from Brodie (1989), and mackerel biomass (B1+) referred to Labrador North Carolina area from figure 19.1 in Anon. (1986). The recruitment calculated for recent years by VPA normaly gives underestimates and the data (N3) corresponding to the years 1986-1989 has been rejected. The last most recent estimate accepted was N3 in 1985, which corresponds to the 1982 year- class. Table 1 shows recruitment values corresponding to the 1956-1983 year-classes. Taking as recruit the numbers at 3 years age (N3), each recruitment value corresponds to the year-class 3 years previously. The same table gives the other three variables considered: cod spawning biomass, American plaice spawning biomass and mackerel adult biomass. The Ricker model. Ricker (1954) proposed a model applied frequently. It can be writte as: R = AS ebS when: R is recruitment ; S is spawning population size; been A and b are the density-independent and density-dependent parameters respectively. That is to say, A is a survival rate of recruit numbers by biomass unit, and b is a density-dependent coefficient that describes how recruit numbers decline with increasing biomass. The outstanding general property of this model more is that it can be dome-shapep, so that recruitment decreases at high levels of biomass . Biological asumptions of the Ricker model. In the Ricker model, the underlying hypothesis is that egg and juvenile mortality rates are more dependent on the intiial stock than the present population density, and recruitment increases with biomass until certain levels. Parameter estimation. For calculating the parameter values we start from the Ricker equation: R= ASebS , or 12/S= Aeb .S. Taking logarihtms In(R/S)= In (AebS), or: In(R/S) = In(A)+(-bS) --> InA-bS. In this way the expressions are transformed into linear expessions, and ,b can be calculated easily, since the regression slope represents In(R/S) on S. Then In A is the origin ordinate value (when S= 0). Several authors consider that finding a significant relation between In(R/S) and S demostrates a stock- recruitment relationship, but it is not true. It can be a mistaken result due the presence of the variable S in the two terms of the expression. The stock-recruitment relationship validation must be done by statistic comparison between empirical recruitment values and the predicted values from the model. Inclusion of environmental variables in the Ricker model. According the results obtained with the correlations (Paz and Larrafieta, 1989) it is reasonable to think that American plaice spawning products can have an effect on the cod stock-recruitment relationship. So, its inclusion in the model should improve its correspondence with the empirical data of cod recruitment NA FO Divisions 3NO. To begin, we used the following the transformation A = ea, that is to say a=lnA, and obtained the expression: R= ease-bS,  equal to: se a' laS . In this way, including as exponents the density-dependent and density-independent parameters, it is easy to add to the model the factor effects, fi: R= so' -bS cfi In the linear form we obtain the expression: In(R/S)= acbS+cli E, where a is the residual or total error. Stocker et al. (1985), applied the Ricker model to mackerel data from Georgia Strait and included several environmental variables. In the same way, one can add interactions between species such the competition, and predation (Walters et al., 1985). The correlation between American plaice spawning biomass and cod recruitment has a positive sign, and this will be the sign in the equation proposed. Writing plaice spawning biomass as the the resulting equation is: R= sea -bS+cSpiat, or in linear form: In(R/S)= a -bS+cS p iat Where a, b and c are the parameters to be calculated; a and b express mean environmental conditions, density-independents and density-dependents respectively, whereas c expresses the magnitude of the American plaice biomass effect on cod recruitment. The same criterion was used to include the mackerel adult biomass. The comparison between the calulated or predicted values from the model and the empirical values was done to obtain a e value, which represents the residual unexplained due to factors not considered, and also measured errors. RESULTS AND DISCUSSION The calculations made the grafics which appear in figure I. It are shown: a) The relationship between empirical values of cod spawning biomass and subsequent recruitment . b: The fit to basic Ricker model from the data. c: The fit to the same model when it the American plaice spawning biomass is included . d: The fit when the mackerel adult biomass (B1+) is also added. In the figure I we see that model Ricker basic does not fit to the data. According to the model there is no cod stock-recruitment relationship, but when American plaice spawning biomass is included, denoted by the correlations (Paz and Larraiieta, 1989) as a variable related with cod recruitment, the fit improves. Finally, when the mackerel adult biomass is also added as depresing element (predator), the fit improves a little more, as can be seen in figures 1 and 2. The resultant equation was: In(R/S)= 0.25596 + 0.00089 S, and the parameter values were A= e0 .25596= 1.29170; b= 0.00089. The correlation coefficient value result low: R 2= 0.364. (Table 2) By introducing American plaice spawning biomass we obtained the equation: In(R/S)= -0.1980 +0.00110 S+0.00078 S piat Parameter values were: A=0.82037 and b=0.0011. The fit improves appreciably: R 2= 0.647. The result is shown in figure 2b. and Table 3 Adding mackerel adult biomass, with minus sign, the fit improves again as shown by the R2 value (R 2=0.688) and by the residual values decreasing, from 2.142 to 1.89. (Figure 2c and Table 4) The equation is: In(R/S)= 0.44 M.00115 S + 0.00079 Spat-0.00023 Sur  (1) In this case the parameter values A and b were 13527 and - 0.00115 respectively, and e= 1.89. Determination of the confidence levels of the parameters values calculated. The equation with the best fit (/) has a high significance level (p< 0.001), but the confidence levels of the values obtained for each coefficient are unknown . There are several procedures for calculating the statistical variability by resamplini from real data. Here we applied the booststrap method. The bootstrap was proposed by Efron (1979a, 1981). It consists of generating new random sanples with repetition the original serie, and realizating after the calculation with each its. The values distribution of each coefficient was trated as a distribution make with real values (Stine, 1985). This method involves no assumption of normality in the data. This technique was used in the cacth age analysis (Deriso et al.', 1985). Were calculated the residuals of each datum in the adjusted curve; random it with repetition this residuals; was make new serie of recruitment values adding the randomly residuals to recruitments predicted, calculated by the adjust; was adjusted the curve from new data serie resulting a new equation. By a iterative process was obtained a coefficients values serie. It will has a normal distribution, can estimated their variability or confidence level. We did 100 repetitions. The coefficient value distributions are show in figure 3. Table 5 gives the coefficient values calculated from equation (I); the mean value of 100 replicate, and the interval 2S values. The parameters values calculated for the adjusted equation proposed lie within the 99% confidence interval with regard to the normal distribution. As a final synthesis, we show the data and the three fitting attempts in chronological order (Fig. 4). The figure shows the progession of the fit in relation to the data. Simulation models with the modified Ricker equation. Afar testing the fit to the Ricker model with American plaice (B11+) and mackerel (B1+) included, we did a simulation to test the performance of the four variables studied. The simulation was made in a range close to the empirical range. The coefficients obtained in the best fit (equation /), corresponding to the Ricker model including American plaice (B11+) and mackerel (B1+) were used. Values of cod spawning biomass, and also a. plaice spawning biomass in increasing order were taken. The mackerel adult biomass was considered in three levels: high, mean and low, as a state variable, as well as in a range defined by the empirical values. The response surfaces appears in figure 5. The three grafics shown the progressive pointing of the surface when increqasing American plaice biomass value, even with cod biomass equal values. The cod recruitment increases with increasing American plaice biomass . in the three cases. Also, observe the inhibitory effect of adult mackerel on cod recruitment. So, for identical levels of American plaice and cod biomass, the highest recruitments were obtained with the lower level of mackerel, and the smaller values with the maximum mackerel level. The response surface results accord with the hypothesis that American plaice spawning products are food for 0-group cod and increase its survival, and that mackerel, during northward spring migrations, feed on cod larvae and fingerlings and have a negative influence on cod recruitment. ' Summarizing, in the period studied by the modified Ricker model, with combined effects of the three variables: cod spawning biomass, American plaice spawning biomass and mackerel adult biomass, we can explain the cod stock-recruitment relationship variations in NAFO div. 3NO. REFERENCES. Anon. 1986. Status of the Fishery Resources Off Northeastern United States for 1986. NOAA. Tech. Memorandum NM FS-FINEC: 130 pp. Beverton R.J.H. and SI Holt, 1957. On the dynamics of exploited fish population. Fish. Invest. London. Ser. 2, Vol 19. Bishop C.A. J.W. Baird and E.F. Murphy, 1990. The assessment of the cod stock in Nafo Div. 3NO. NAFO S.C.R. Doc. 90(73, Serial n° 1795: 35 pp. Brodie, W.B., 1989. An assessment of the American plaice Stock in Div. 3LNO. NAFO S.C.R. Doc. 89/37, Serial n° 89/37: 42 pp. Deriso R.B., T.J.Quin 11 and P.R. Neal, 1985. Catch-age analysis with auxiliary information. Can. J. Fish Aquat. Sci. 42 (4): 815-824. Efron, B., I979a. Bootstrap methods: another look at the jakcnife Ann. Statis., 7 . 1-26. Efron, B. 1981, Non parametric standard errors and confidence intervals. The Canadian Jounal of Statistics, Vol. 9(2):139-172. Hightower J.E. and D.G. Grossman, 1985. Optimal policies for rehabilitation of overexploited fish stock using a deterministic model. Can. J. Fish. Aquat. Sci. 44(4): 803-810. Lovejoy, W.S., 1986. Bounds on the optimal age-at-first-capture for stochastic-age-structured model. Can. J. Fish. Aquat. Sci. 43(1): 101-107. Paz J. and M.G. Ltu-nifieta, 1989. Cod in div. 3NO: Year-Class Variations and Abundance of other Commercial Fish. NAFO S.C.R. Doc. 89/72, Serial n2 1653, 9 pp.. 	Ricker, W.E., 1954. Stock and Recruitment. Bull. Fish. Res. Board. Can., 119, 300 pp. 	: Sepherd, J.G., 1982. A versatile new stock-recruitment relationship for fisheries, and the construction of sustainable yield curves. J. du Conseil 40(1): 67-75. • 	• 	• Stine, R., 1985. Bootstrap Prediction Intervals for Regression. , J. of the American Statistical ' Association, Vol.80, n 2 392: 1026-1030. Stocker M. and D. Fournier, 1985. Environmental variation and recruitment of Pacific herring (Clupea arengus pallasi) in the Strait of Georgia. Can. J. Fish. Aquat. Sci. 42 (Suppl.): 174-180: Swarzman G.L., W.M. Getz and R.C. Francis, 1987. Binational management of Pacific hake (Merlucius productus): a stochastic model approach. Can. J. Fish. Aquat. Sci. 44: 1053-1063. Walters et al., 1985. Bias in the estimation of funtional relationships from time series data. J . Fish Aquat. Set, 42: 147-149. Cod plaice 	Mackerel (811+,tons) 	(B1+,tons) Year 	Recruitment Recruits Biomass 	A. Class 	(N3) 	(000) 	(B6+,tons) 1956 1957 1958 1959 1960 1961 53698 53179 82134 -- -- -- -- 1959 1962 107755 89220 -- 1960 1963 78284 72956 -- 1961 1964 112516 90018 -- 1962 1965 162762 80480 -- 1963 1966 210243 88029 -- 275 1964 1967 183387 112830 -- 311 1965 1968 101714 120660 138197 323 1966 1969 128599 104875 158756 371 1967 1970 80905 93820 157325 623 1968 1971 86121 83341 138837 1198 1969 1972 63163 81217 120238 1533 1970 1973 37594 82481 94378 . 	1856 1971 1974 41586 90799 81403 1868 1972 1975 26552 81758 62129 1653 1973 1976 33510 61446 52619 1389 1974 1977 52753 60994 50889 1126 1975 1978 61688 27912 45906 970 1976 1979 21747 23903 39864 719 1977 1980 23345 36541 45515 491 1978 1981 33823 42314 48818 467 1979 1982 24673 52004 60846 503 1980 1983 35426 103667 67125 467 1981 1984 47223 157436 53278 479 1982 1985 32581 172654 46852 599 Table 1. - Cod Div. 3NO spawning stock Table 1.- Cod Div. 3N0 spawning stock (B6+) and recruitment (N3) From (Bishop et al. (1990); American plaice spawning biomass (B11+) from Baird and Bishop (1989) las Divisiones. 3N0; Mackerel adult biomass from figure 19.1 anon. (1986). Source 	F.D. 	Sum Squares 	F Test Regression 1 2.206 9.154 Residual 16 3.860 p< 	0.05 Total 17 Table 2. - ANOVA corresponds to fit Ricker model from the empirical data. Source 	F.D. 	Sum Squares 	F Test Regresion 2 3.924 13.739 Residual 15 2.142 p<0.05 Total 17 6.066 Table 3. - ANOVA corresponds to fit to Ricker model modificated with the inclusion of the American plaice spawning biomass. Source 	F.D. 	Sum Squares 	F Test Regression 3 4.176 10.311 Residual 14 1.89 p< 0.05 Total 17 6.066 Table 4.-ANOVA corresponds to adjust to Ricker model modificated also, with the inclusion of the mackerel adult biomass. Cod (86+) A. plaice 	(811+) Mackerel (81+) Coefficient value obtained 	-1.102 10 -3 8.545 10 -4  -2.293 10 -4 Mean bootstrap distribution 	-1.154 10 -3 7.933 10 -4 -2.205 10 -4 Interval 	999(mik.)-1.191 	10 -3 8.119 10 -4 -2.639 10 -4 values 	(max.) 	-1.03 	10 -3 8.971 10 -4 -1.77 10 -4 Table 5.- Coefficient values of the equation result and mean and interval +/-2s parameters bootstrap caculated. a . • b crxre_ora-32--0-0---0_, ' §  §1  §  § §  `  c d 0 	200 400 600 800 1000 1200 1400 1600 1800 2000 Biomass Figure' 1. - Cod stock-recruitment relationship, NAFO divs. 3N0; 1956-84. a) empirical data; b) values obtained with Ricker model; c) modified Ricker model including American plaice spawning biomass; d) Values resulting to adjust the,modified Ricker model with inclusion the • American plaice spawning biomass. and the mackerel adult biomass: C E , I lam, 4 	a 	a 	o 	12 A. plaice 1104 ) Mackerel (104 ) .11 	-1s 30 20 I0 • • Coefficients -6 .4 1400 1200 1000 800 600 400 200 0 1200 .14 1000 800 600 W 400 ry 200 0 1200 1000 800 600 400 200 0  a    b       800 1000 1200 Biomass  1400 1600 0 200 400 600 t BOO 2000 Figure 2. - Cod stock-recruitment in NAFO divisions 3N0: 1956-84. Empirical data compared with a) the result values obtained with the Ricker model; b) the fitted curve by the Ricker model with introduction American plaice spawning biomass c)results values from Ricker model with introduction American plaice spawning biomass and the mackerel adult biomass. • Figure 3. - Coefficient values frequency distribution, auto generated by bootstrap, corresponding the equation results to applied the Ricker model modified with American plaice spawning biomass and mackerel adult biomass. - 10 - Recruitment  1066 Recruitment data 1073 	 1r 	 Ricker Years 	ien 	 A. plaice I 261  A. plaice + Mackerel Figure 4. - Chronological evolution of the recruitment empirical values, and de los three adjusts attempts: Ricker, Ricker with the American plaice (B11+) and Ricker. with the American plaice (311-) and mackerel biomass(B1 4 ). Mackerelbiomas:300 1000 800 600 400 200 0 1550 1350 1150 950 750 	' 550 2600 350 Mackerelbiomass:850 1000 600 400 200 0 1550 1350 1150 950 750 550 2600 350 Mackerelbiomass:150 0 1000 800 600 400 200 0 	1550 	' 1350 	• 1150 950 750 8002200 	550 A. plaice biomass Cod biomass 	2600 350 Figure 5. - Simulation by modified Ricker model including the American plaice (611+), and considering three mackerel biomass (B1+) levels; a) mackerel biomass low le‘kel; b), mean level Mackerel biomass and c)high level mackerel biomass. 1902 1000 800 600 400 200 0 200 Recruitment  000 800 600 400 200 0 200 1000 600 600 400 200 0 200 6  a l 

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Cod Stock-Recruitment Relationship in NAFO Divisions 3NO

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