A multiple regression model of oyster growth

Fisheries Research.

2 (1984) 167-175 Elsevier Science Publishers B.V., Amsterdam -Printed

A MULTIPLE

REGRESSION

167

in The Netherlands

MODEL OF OYSTER

GROWTH

OTEVEN HALL Technological

Economics Research Unit, Department FK9 4LA (Gt. Eritain)

sf

Monagemenr

Science,

Unbrrsity

of Stirling, Stirling

(Accepted 14 July 1983)

ABSTRACT Dali, S., 1984.

A multiple regression

model of oyster growth. Fish. Res.,

2: 167-175.

A deterministic model of the growth in the sea of Ostreo edulis and Crassostreagigos has been developed from published information. The model is a multiple regression

function in which oyster tize and seawater temperature are treated as determinants of instantaneous growth rate. Simulation of commercisl culti
INTRODUCTICN In the United

Kingdom

ly; the

indigenous

cupped

oyster,

the cultivation

native

two

Crassostrea of both

species

flat oyster, gigas.

species

of

oyster

Ostrea edulk

In recent

are exploited

commercial-

and the introduced

Pacific

years there has been an increase

and a revitahsation

of fisheries

in

for the native

flat oyster.

The main growing tober.

Winter

growth

season for oysters begins in has

been

recorded

for

April

and extends

to Oc-

Crussostreu gigas only.

The growth rate of C. gigas is greater than that of 0. edulis until June-July; thereafter, the rates for both species are similar (Walne and Mann, 1975). Wahre (1975), recorded the average, annual growth increment of juvenile 0. edulis as 18 g. By comparison, the growth rate of C. gigas is much faster reaching 7-10 g per month by the same stage of development (Walne and Spencer, 1371). At most sites C. g&es attains a marketable size in 18-24 months, whereas 0. sdulis is marketed after 3-5 years growth in the sea. The temperature of the sea governs both the beginning and end of the growing season, but the rate of growth within these limits also depends on other factors. The early seasonal growth of C. g&s may be in response to food availability which peaks earlier than seawater temperature (Askew, 1976). Phytoplankton species have differing nutritional vahres and it has been demonstrated that diets of mixed phytoplankton produce the highest oyster growth rates (Helm, 197’7; Epifanio, 1979). At present, however, little is

168

known of the specific nutritional requirements of oysters (Chu et al., 1982). In general, annual mortality rates of between 10 and 15% may be expected for Juvenile and adult oysters held in the sea in trays (Wabre, 1961; Parsens, 1972). ;%Iortalities suffered by oyster spat can be expec+*d to be much higher. There have been previous attempts to derive a general model to predict oyster growth and mortality, notably Askew (1976). From field observations, Askew derived a generalised model based on the relations between monthly instantaneous growth rate (C,,) and in4antaneous mortality rate (Z,,) agamst size. Where (1)

Gjo = log, wt I IV0 and

(2)

.ZxO= loge NJ?:+, W, and

weight over time t (30 days). N, and Nt are the initial and final number surviving over the period. In this model the monthly growth increment of any sixe of oy.rter is predicted from standard growth curves and a value for the annual growth rate. Growth rates compatible with field observations are predicted for the fastgrowing C. gigas, but the seasonal growth pattern used for 0. ~?dnIis overestimates the growth rate of this species. The model described in this paper is a central element of a comprehensive appraisal of the commercial cultivation of oysters (Hall, 1982). The accurate prediction of the growth rate of both species of oyster was the prime consideration in developing an original model. Given the current selling prices of oysters, the model can be used to predict the size and timing of revenue flows at both the investment appraisal and production stages of oyster form operations. Sensitivity analysis can be employed to investigate the effect on commercial viabilitv of changes in the temperature regime, spat size or mortality rate. THE

W, are the initial and final oyster

MULTIPLE

REGRESSION

MODEL

The multiple regression technique has been used to derive a deterministic model which employs oyster size and seawater temperature aa determinants of instantaneous growth rate (G30). The model may be represented by the general formula Y=A,

+A,X,+A,X,

where Y is the dependent variable and Xi and X2 are independent variables. Al and A2 ue partial regression coefficients and A0 is a constant. Values for oyster size, seawater temperature and the corresponding monthly instantaneous growth rate were extracted from several sources (Walne and Spencer, 19’71; Walne and Mann, 19’75; Askew, 1976; Spencer and Gough, 1978; Spencer et al., 1978). Previous studies describe the relationship between size and growth rate as non-linear. Over the size range of

169

oysters cultivated in the sea, the effect of temperature on growth rate has also been established as non-linear. To calculate the regression for each species, all three vsriabies have been transfcrmed tc the Iogarithmic form (base 10). This permits the introduction of linearity and a homogeneity not present in the raw data (Ricker, 1973). Two growth rate functions have been derived Crassastrea gigas Log,&,

= 1.6767 Log,, Temp -0.329 (0.1804)*

Log,,WO - 2.1283

(3)

(0.0304)

Number of observations = 51 Coefficient of determination (R 2 : = !?.E? Ostrea edulis LogloGs

= 5.I699

Log,,

Temp - 0.5574 Log1, IV, - 6.4489

(4)

(9.1934)

(0.5025)

Number of observations = 54 Coefficient of determination (I?‘) = 0.707. W. = Oyster size (g) at start of month; Temp = monthly seawater temperature (“C); * = standard error of the psrtisI regression coefficient. The yield of oysters dapends on the balance between the mtc of growth and tlhe mortality suffered. In this model, mortality is described by the relationship between initial weight (IV,) and monthly instantaneous mortality rate as given by Askew (1976). The values of W, from which the mortaIi,y coefficient (Z,,) is calculated are those generated by the multiple regression equation. Three levels of mortality are considered: low, medium and high, corresponding to Askew’s derived mortality function and its 50% confidence limits [eqs. 5-7). Low mortality

rate

zss = 0.00993

w’,-0.‘44G

Medium mortality &

= 0.0264

(5)

R* = 0.8571

(6)

RZ = 0.860

(7)

rate

W,-“.4’54

High mortality

rate

z,,

w 0 -s.‘WY

= 0.05688

R2 = 0.7673

An attempt was made to improve on the mortality function by the inclusion of the temperature-dependent mortality relation described for C. edulis

170

by

Spencer

and Gcugh

(1973).

The effect on the model was f:ound to be

negligible because of the sma3 size range over which corded and these data have been omitted. Statistical

+he relation

was re-

validrrtion

The coefficients of determination (R*) go‘ve a measure of causal association between the independent and the dependent variables. In each case the fitted equations explain a statistically significant degree of variation in the dependent variable. Flowever, the values of R’ for the multiple regression functions do not indicate whether each partial regression coefficient makes a significant contribution to the explanation of the total variation. Each growth rate function indicates that an increase in size is associated with a decrease in the ln&urtaneous growth rate. The growth rate of 0. edulis is predicted to be less than that of C. gi@a under the !same conditions. The growth patterns predicted by the model show good agreement with field observations, The analysis of variance results in Table I indicate that the partial regression coefficients are statistically acceptable at the 1% level with the excep tion of the size variable in the 0. edulis equation which can be accepted at the 5% level. It may be concluded that the inclusion of esrh partial regression coefficient significantiy increases the total amount cf variation in th.e dependent variable explained by the model. The standard errors of the partial regression coefficients in eqns. 3 and 4 are an indication of the presence of multicol!ineerity in the data: the z&tence of an imperfect linear relationship between independent variables (Mayer and Mayer, 1976). The computer correlation coefficients between the two independent variables are 0.1228 for 0. edulis and -0.3045 for C. gigas. In both cases, the values are small and whilst indicating some multicuilineeiity, in neither function is the relation important. This effect is not unexpected since both independent variables attempt to explain variation TABLE Analysis Species

I of variance results for the regression

functions

Source of variance

Degre..- of freedom

F ratio

comment

gigas

Regression W, coefficient TEMP coefficient

2, 46 1, 49 1, 49

146.16 75.07 51.66

**p < 0.01 **p < 0.01 **p < 0.01

0. edulis

Reyression W, coefficient TEMP coefficient

2, 1, 1,

C.

51 52 52

61.61 5.74 60.98

+*P < 0.01 *P < 0.05 **p < 0.01

171

due to sources not included in the model. The addition of CIOX data or :/a& ables may reduce this effect, but even with the present model the degree of correlation between oyster size and seawater temperature is not significant and should not und,uly impair the utility of the functions for predicting growth rates. Application The derived growth and mortality equation can be used to predict the yield and prepare production schedules for batches of oysters at any suitable site. A temperature regim.! made up of average monthly seawater temperatures is required for each site investigated. The model uses the initial spat size (W,) and the initial mean monthly seawater temperature to compute, using the appropriate growth equation, the instantaneous growth rate for the month (G,,). This value is employed in eqn. ‘I to calculate the weight attained at the end of the month (Wt). This weigtt forms the initial size for the next time period, and using the correspor>.:ing monthly seawater temperature, the procedure is repeated. The reiterative process continues using discrete time intervals of one month until a predef.ermined maximum size is at tamed. The predicted weights are taken to be the average size of all oysters in the batch. The instantaneous mortality coeff.ci?nt (Z,..) is calculated from the vame of W,, each month. Tne immi~r of oysters surviving each month csn then be &rived using the initial number (No) zind eqn. 2. Predicted growth patterns for both species of oyster have been derived for the two temperature regimes given in ‘lable 11. The seawater temperatures employed are average values calculated for each nonth at each site. In each exampie the mode! simulates the growth of O.l-;f batches of spat laid out in the sea in trays in April, May and dune. The predicted growth curves are shown in Figs. 1 and 2. The derived growth patterns reflect the marked difference in the growth rates of the two species under the same conditions. At all stages of development the growth rate of C. gigas is greater than that of 0. eduU.s. The step-like fomi of the growtll patterns mirrors the seasonal nature of growth. The horizontal phase represents winter when little or no growth occurs. The presence of winter growth is evident in the C. gigus plots. The model predicts a significant difference in the ongrcting time between the two temperature regimes employed. A relatively small increase in monthly seawater temperatures will increase the growth rate significantly. The difference for C. g@s is a few months, but for 0. edulis the choice of site can result in nearly two years difference in ongrowing time. The results indicate that there are advantages to be gained by spreading the planting out of spat over several months both in the use of equipment and maintenance of a supply of mature oysters. DISCUSSION The model described

in this paper employs field observations

in an origin-

II

Hnrhour*

(“C)

*Askew (1976). **itiinistry of Agriculture,

Menai Straits** temperature 7.0

5.3

Feb.

8.2

6.3

Mar.

9.2

10.5

Apr.

12.5

15.0

May

15.0

16.0

JUW

regimea at two established oyster cultivation

Fisheries and Food.

7.0

6.0

Jllll.

Month

seawater tzmpemture

temperature(“12)

Emsworth

~_~

Mean monthly

TABLE

17.2

18.8

July

17.0

18.5

Aug.

sibs: Emsworth

15.2

16.6

Sept.

13.1

14.3

Oct.

8.‘)

11.0

NOV.

Harbour and Menai Straits

7.4

7.5

Dec.

Fig. 1. tGrowth curves predicted from the Emsworth iii& laid our Lr ‘.iii ran irj Aprii, May and June.

temperature

Fig. 2. Growth curves predicted from the Menai temperature laid out in the sea in April, May and June.

regime

regime

for 0.1-e oyster

for 0.1-g oyster

spat

al way to derive functions which represent the growth in the sea of both 0. edulis and C. gigas. The inclusion of seawater temperature as a determinant of growth gives the model a flexibility not present in earlier studies. ‘I’he statistical analysis confiis that the derived equations give a good fit to the data.

174

Temperature data for Loch Sween, Scotland, which were not considered -__A--. +%.--A’. _r ‘, ,..I,. !,.A.,. in the calculation of tbi resIGu.rull lularr;vlrO ..U._ ___,I tested in {:ne nlod*3 “d,Jd abse~~atticns obtained. and predicted growth rates consistent with :.,. There is a constraint on the number of independent variab!ef whi:h can bc included in a model of this type. The c~&~th~n of data in iit+ fis’4 will impose a practical (but not theoretical) limit of three or four variilbles. The paucit.y of data describing growtn at high seawater temperatures prohibits extrapolation of the 0. edulis function to temperatures above c:hose employed in its preparation (<19”C). This is only a minor restriction as mean monthly seawater temperatures will rarely exceed this temperature at suitable locations in the United Kingdom. The omission of food availability as a variable does not impair the use of the model provided food levels are not limiting ail year round. Data have been drawn priucipaily from work describing intertidal oyster cultivation, The increasing use of suspended culture techniques may warrant a revision of the model at some future date. The importance to the commercial operator of being able to prepare production schedules and predict stocks at the time of marketing can readily be appreciated. Shellfish investment appraisals in the past have relied too heavily on simplistic models which take no account of the effects of season on growth and the market for shellfish. These factors can exert a considerable influence on the return on investment by controlling the timing of cash flows. The commercial success of sheEfish witure often depwllds on the producer’s ability to optimise the use of resources so as to take advantage of seasonal growth patterns. In this context, the model can contribute valuable information both to the initial evaluation of potential sites and ‘to the management of production. ACKNOWLEDGEMENTS I am indebted to the SRC/SSRC for providing a student&p and to my colleagues at Stirling for their invaluable advice and encouragement.

REFERENCES Askew. CD., 1976. Biolc$ical aesezzrrent QC off-bottom culture of oysters in Britie!l waters, including a consideration of econcmic implications. Ph.D. Thesis, Portsmouth Polytechnic, 296 pp. Chu, F..L.E., Dupuy. J.L. and Webb, K.L., 1982. Polyeaccheride composition of five algal species used as food for larvae of the American oyster Cmssosfrea virginico. Aquacultwe, 29: 241-252. Epifenio, GE., 1979. Growth in bivalve nollusw nutritional effects of two or more species of algae in diets fed to the American oyster Crassostrea uirginica (Gmelin) and the hard clam Mercenoria mercenario (L.). Aquaculture, 18: l-12. Hsll, S.. 1982. An appraisal of commercial oyster culture in the United Kingdom. Thesis, University of Stirling, 286 pp. Helm, M.M.. 1977. Mixed algal feeding of Ostreo edulis larvae witn IwxhrysiB and Tetroselmis SUPCIC(I. J. iviar.timi. Assoc. U.K., 57: 1019-1031.

Ph.D.

gulbana

175 Mayer, A.C. acd Mayer, D.G., 1576. Introductory Economic Statistic.% Clley, London, 223 PP. Parsons, .I., 1972. Gzowth -zzte and rzwtality of hatrbeg-rzr. r=ri Pecxia n”ste~s emsmstreo gigas in Strangfrrrd Lough, Northern Ireland. Cons Pwm. Int. Explor. Mer. C.M. 19’72/K32, 12 pp. Zidt?r, W.E., 1973. Linw regression in fisheries ressarch. a. Fish. Res. Board Can., 30: 409-434. Spencer, B.E. and Gough, C.J., 1978. The growth and survival of experimental batches of hatchery-reard spat of O~trea adulis (L.) and Cmssostren giga (Thunberg), using different methods of tray cultivation. Aqunculture. 13: 293-312. Spencer, B.E., Key, D.. Milliean, P.F. ar.l Thomas, M.J., 1978. Tine effect of intenidal exposure on the growth and survival of hatchery-reared Pacific oysters Crossos~ea giias (Tbunberg) kept in trays during their first ongrowing season. Aquaculture, 13: lGl203. Walne, P.R., 1961. Observationx on the mortality of Ostreo edulis. J. Mar. Biol. Assoc., U.K., 41: 113-122. Walne, P.R., 1976. The growth of Ostrea edulis in the Menci Straits. Proc. ChaUanger sot., Iv@): 272. W&e, P.R. and Mann, R., 1975. Growth and biochemical composition in Osfres sdulis and Cruw~trea gigas. Proc. 9th Europ. Mar. Biol. Symp., Aberdeen University Press, pp. 587-607. W&e, P.R. and Spencer, B.E., 1971. The introduction of the Pacific oyster Crassostrea &I?? Int” the Shellfish informatioa lea&t No. 21.8 pp.

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