,\..

Some Practical Extensions to Beverton and Holt'sRelative Yield-Per-Recruit Model*

DANIEL PAULYMINA L. SORIANO

International Center for Living Aquatic.Resources Management

MCP.O. Box 1501Makati, Metro Manila

Philippines

(Y/R) are perceived as yield predictions (Y), withoutregard to recruitment. Also the decline in catch per effortconcomitant with increased yield per recruit is often notconsidered.

In this paper, a number of concepts and equations arepresented which help in interpreting results obtained usingBeverton and Holt's length-structured yield-per-recruitmodel. These concepts and equations parallel thosedeveloped for use with the age-structured version(s) ofyield-per-recruit models such as

-Zr -Kr -(Z + K)r1 -e 3 3e 1 (1 -e 3,

PAULY. D. and ML. SORIANO. 1986. Some practical extensions toBevenon and Holt's relative yield-per-recruit model, p. 491-495.In J.L. Maclean, L.B. Dizon and L. V. Hosillos (cds.) The FirstAsian Fisheries Forum. Asian FISheries Society, Manila,

Philippines.

-Mr2w~.y

-=F.eR z Z+K

Abstract

The relative yield-per-recniit (Y' /R) model of Beverton and Holtestimates Y' /R values based on few inputs, i.e., c = mean length at firstcapture/asymptotic length, ratio of natural mortality to growth (M/K) andexploitation rate (E = FfZ). However, when used in conjunction with highM/K values and/or a wide selectioo range, which frequently occur insmall, short-lived tropical fishes and invertebrates, the model fails toestimate the optimum level of E for a given fishery. A number ofapproaches are presented to correct for this and related deficiencies.

where Z = F + M. t:l = tc-to. f2 = tr-to and f3 = tmax-tc.

The model assumes isometric von Bertalanffygrowth, natural, fishing and total mortality rates (M, F andZ, respectively) expressed by negative exponential curvesand also assumes that all fish of a given cohort enter theqshing ground or become catchable by the gear or leavethe fishery at the same ages through "knife-edge"recruitment (tv, selection (tc) and "derecruitment" (tmax),respectively. These latter assumptions are reasonable withlong-lived fishes, in which the biomass above tr or tcforms the overwhelming part of stock biomass, but notnecessarily with small animals, in which,' e.g., theselection range may span the entire size distribution (seebelow). De-recruitment, on the other hand, can usually beignored (especially when Z is high), by setting tmax = ~nd simplifying equation (1) to

Introduction

-Kr,3e

Z+K

-2Kr13e

Z+2K

-3Kre 1-Z+3K

~~

2)

Following the development of an age-structuredtheory of fishing by Beverton and Holt (1957), theseauthors subsequently developed (Beverton and Holt 1964)a length-structured version of their yield-per-recruit modelideally suited for use in data-sparse, tropical setups.

The length-structured yield-per-recruit model ofBeverton and Holt (1964) had only three variables (c, M/Kand E: see below for definitions), against seven in the1957 model (W~, K, to, tc, tmax, M, F; see below), the gapbetween the two models being bridged by a set ofassumptions that seemed reasonable enough, and lots ofserious algebra.

Since its publication, the length-structured, "relative"yield-per-rectuit model has been applied widely, notablyin the tropics, and formed the base for a number of usefulgeneralizations in fishery management (see e.g., Gulland1971; Sinoda et at. 1979; Pauly 1984).

Sometimes, unwary users have interpreted results inways not intended by the authors, the most commonmisinterpretation being that predicted yields per recruit (Jones 1957)

491

492

From this, assuming an isometric icngth-weightrelationship, Beverton and Holt (1964) derived the length-structured, "relative" yield-per-recruit model.

which has the useful property of being equal to unitywhen E = 0 and equal to 0 when E = 1. Conversion of thisbiomass index to absolute biomass per recruit can beperformed, in analogy to equation (4) by the relationship

3 (1 -cl2

2 (1 -EI

(M/K)

3 (1 -cl

(1- EI1 +

(M/K)

(1 -c)3..3(1 -E)

1 + :'

(M/K)

v'R = E(1 -c)M/K . 1- +BfA = (B'fA) .Woo oexp-(M(t-tll r 0",'

1 +...6)

Note that MSY is generated when relative biomass is50% or 37% of virgin stock, at least in terms of theSchaefer (1957) and Fox (1970) models, respectively.

3)

Model Extension ll: EO.I Concept Analogous to FO.Iin which Y' /R is the relative yield per recruit, E =

F{Z. c = Lc/L~. where Lc is the length corresponding to tc.where tr is set at zero. and where L.. is the asymptoticlength. corresponding to W ~ in equation (1) and (2). Therelationship between Y /R as expressed by equation (2) andrelative yield per recruit as expressed by equation (3) is

given by

Identification, in the context of rapid assessment, ofthe values of F generating Maximum Sustainable Yield("Fopt") is not easy, and practitioners have thereforedeveloped a number of rules of thumb, the most used ofwhich is Fopt = 0.5. Another definition is Fopt = FO.l,the latter term being defmed as the fishing mortality atwhich the marginal increase of yield per recruit is 1/10 ofits value at F = 0 (Gulland and Boerema 1973). Since thefirst of these rules of thumb has been shown tooverestimate Fopt (Reddington and Cooke 1983), anequation is given below which allows estimation of EO.l,the exploitation rate at which the marginal increase ofrelative yield per recruit is 1/10 of its value at E = O.

The first derivative of equation (3) is

.ex p -1M It -t ))r 0

VIA = (V'/AI .W~

4)

Model Extension I: Derivation of a Biomass PerRecruit Index

Because unwary model users equate yield per recruitand yield, and also in order to predict relative catch pereffort, biomass-per-recruit curves are often drawn as afunction of F along with yield-per-recruit curves. Suchbiomass curves can be derived by dividing equation (2) byF

d (V'/A)

dE= [(1 -C)M/K .(V'lA)

3E (1 -c) 1 + M/K

+ .ATables with computed relative equivalents of B/R(i.e., B'/R)~ given in Beverton and Holt (1964). Sincethe equation .they used was not given explicitly, anappropriate equation is presented here, i.e.,

M/K 71

where

2(1 -EI

M/K

-2 2...\A=2(1-cl . -(1-

1

+

8'R-

3(1 -E)

M/K

1~E

M/K

-2 -2

1

+ +

5) 8)

493

which can be solved for any value of E, including E= 0 and E = 1. Using equation (7) it is simple matter toidentify using an appropriate search algorithm, for a givenpair of M/K and c values, the value of E generating a valueof d(Y' /R)/dE equal to 1/10 of the value of d(Y' /R)/dE atE = O. Equation (7), obviously, can also be used toestimate the value of E at which yield per recruit ismaximized, i.e., the value of E at which d(Y' /R)/dE isequal to zero.

Model Extension IV: Compensating for the Effects of aWide Selection Range

Model Extension ill: Relationship BetweenExploitation Rate and Mean Length of the Fish in the

stock

Beverton and Holt (1956) showed that, given thesame assumptions as those used in the derivation of theyield-per-recruit model,

--IZ/K = ( Loo- L ) I (L -L

8)

In large, long-lived fish such as cod or plaice, theselection process usually takes place over a relativelynarrow range of sizes, such that the assumption of "knife-edge" selection is acceptable. In some small animals suchas shrimps caught by trawls, the selection range may covermost size classes represented in the population. In suchcases, yield-per-recruit computations involving theassumption of knife-edge selection may involve a largebias. A simple method is developed here to show theextent of and to help overcome this bias.

Selection curves provide a probability of capture (Pi)for catch-length class (i) between Lmin, the smallest, andLmax, the maximum length represented in the availablecatch samples. In the unmodified model, it is assumed thatPi = 0 when L < Lc and Pi = 1 when L > Lc (hence also L'= Lc). The implicit assumption here is that, if selection isnot knife-edged, the yield from the fish caught below Lcwill compensate for the yield losses due to the fact that notall fish larger than Lc are caught

Although some compensation may occur, theassumption of knife-edge selection does generate a largebias, especially for high values of E, as can be shown byreformulating Beverton and Holt's method forcomputation of yield per-recruit for different E values overthe lifespan of a fISh (section "f' in Bevenon and Holt1964) such that E is assumed constant, but P variable. Thisgives

where [is the mean length computed from L'upward, the latter being a length "not smaller than thesmallest length of fish fully represented in catch samples."Note that Lc < L', except in the case of knife-edgeselection, where Lc = L '.

Equation (8) can be solved for L (J. Hoenig; pers.comm.), in which case we have

LoaVilA = ~ p. «(V'IR). .GI. I -11-((Y'/Rli+1 .Gjl[ = (l.oo + (L' .Z/K)) / ((Z/K) + 1 = Lmin

11 )9)

Now sinceE =F/(F+M) andE = 1- (M/K)/(Z/K), wealso have

in which (Y' /R)i and (Y' /R)i+ 1 refer to relative yieldper recruit as computed from the lower limit of lengthclass (i), Picrefers to the probability of capture between Liand Li+l (see Table I), while Gl is defined by

10)G. ~

I nr.J

a1for 0 ~ E < 1. Thus, one can, when performing arelative yield-per-recruit assessment, also assessstraightforwardly the reduction of mean length broughtabout by an increase of E.

...12)

and where ri is a factor expressing the proportion ofrecruits of length Li which survive, grow and reach lengthLi+l, which is computed, for 0<£ < 1, from

(1 -c,l(M/K) ( E/(1 -E) IPj

131

494

Model Extension VI: An Alternative for Plotting YieldIsopleth Diagrams for High Values ofM/K

In the tropics, and/or when dealing with smallanimals, use of the relative yield-per-recruit model oftenimplies use of high M/K values «2 see above).

In such cases, however, yield per recruit havemaxima occurring at values of E (> 0.5) and oftencorresponding to extremely high values of fishingmortality. When Lc/L~ is near 0.5, yield isopleth diagramsbased on equation (3) then usually consist of 4 quadrants(A-D) with properties as given in Table 2.

which is analogous to Beverton and Holt's"reduction factor", but considers P as a variable and hasthe exponent (M/K)(Ej(I-E» instead ofF/K.

Replacing the (Y' /R) tenDS in equation (11) by(B' /R) as given by equation (5) is straightforward and willlead to estimates of biomass per recruit independent of the

knife-edge assumption.Table 1 gives probabilities of capture for selection

gives with increasingly large ranges, from knife-edgeselection in case 1 to a selection range spanning most ofthe range between 0 and Looin case 3. In Fig. lA, departurefrom knife-edge selection has a profound impact on yield-per-recruit estimation, particularly at high values of E;similar results are obtained for relative biomass per recruit

(Fig. IB). Discussion

Model Extension V: and Empirical Equation to PredictM/K

Beverton and Holt (1964) presented their relativeyield-per-recruit model in the form of yield tables at a timewhen microcomputers and programmable calculators didnot exist. That they were able to include in their table arealistic range of M/K values is due to a previous reviewof the growth and mortality of fish (Beverton and Holt1959), in which they showed that M/K varies less betweenstocks than either K or M alone. Pauly (1980) used theirdata and a number of other data sets to demonstrate theexistence in fish of strong partial correlations between Mon the one hand and L ~ and mean environmentaltemperature on the other. Here, the data compiled in Pauly(1980) (see Pauly 1985 for correction of 4 outliers), wereused to show that M/K in fishes is affected by thetemperatures of their habitat, i.e.,

The various extensions of the length structured yield-per-recruit model presented here should help make itsapplications to tropical fish and invertebrates considerablyeasier, and lead to results that will be less biased andstraightforward to interpret These extensions alsoillustrate how "classical models" rather than being rejectedout of hand can be adapted to better fit situations for whichthey may not have been originally intended.

The listing of a BASIC program implementing theequations presented in this paper is available on requestfrom the authors. .

References

loge (M/K) = -0.22 + 0.30 loge T

14)

where T is the mean water temperature, in oC (r =0.308, 173 d.f., P < 0.001; s.e. of slope = 0.07; range 3-

30OC).Equation (14) implies that the values of M/K

commonly used for stock assessment in temperate waters(say 100) should be used in the tropics (say 300) only aftermultiplication by a factor of about 1.4.

Beddington,J.R. andJ.G. Cooke. 1983. The potential yield offish stocks.FAO Fish. Tech. Pap. 242 47 p.

Beverton, RJ.H. and S.J. HolL 1956. A review of methods for estimatingmortality rates in fish populations, with special references tosources of bias in catch sampling. Raw. P. -Y. Reun. ClEM140:67-83.

Beverton, R.J.H. and S.J. HolL 1957. On the dynamics of exploited fishpopulations. Fish. InvesL MinisL Agric. Fish. Food G.B. Ser. n19.533 p.

Beverton, RJ.H. and S.J. HolL 1959. A review of the life-spans andmortality rates of fish in nature and their relation to growth andother physiological characteristics. Ciba Found. Colloq. Ageing5:142-180.

Beverton, R.J.H. and SJ. HolL 1964. Manual of methods for fish stockassessmenL Part 2 Tables of yield functions. FAO Fish. Tech.Pap. 38 (Rev. 1). 67 p.

Fox, W. W. 1970. An exponential yield model for optimizing exploitedfiSh populations. Trans. Am. Fish. Soc. 99:80-88.

Gulland, J.A. 1971. The fish resources of the oceans. Fishing NewsBooks, Ltd., Surrey, England. 255 p.

Gulland, J.A. and L.K. Boerema. 1973. Scientific advice on catch levels.Fish. Bull. 71:325-335.

Jones, R. 1957. A much simplified version of the fiSh yield equation.Doc. p. 21. Paper presented at the Lisbon Joint Meeting ofInternational Commission Northwest Atlantic Fisheries,International Council for the Exploration of the Sea and Food andAgriculture Organization of the United Nations. 8 p.

"."

495

Tobit 2. Proptrtlt. of Oht 4 qu.JranU of ralt,lva iBOplaOh dl~rtm.. whtn M/K > 2, or;olool .Izt raO.,ILo/,-1 -0.5 tnd or;oloal a"plOiOa'lon raoa E .0.5.

PO"ib.'o bonto,..ntoonsFishing

regime"Quad,ont locot;on Assessment

underlioh;ng 'e. eflon inae-..'ong'Vor

cdo noth,"9

A c-0.5'ol

E -0 to O~

larg. lish .,. cought.t low .ffortI.vels

ome" !;", e,e caughtet low elfonlovols

...me..ic fish;ng

developing f;shery

do nothing, butnot. that m..h.;z.. w;II have tobe ;nc,...od a..It"rt ono,

B

0'OtoO.5

E' 0 to 0.5

oflo" must be,'ob;';'"" end

-"tpo",bl,! ,edu.~

C c-OooO.5

£-O.5tol

'..g. fish ...caughtat h;gh .ffo"'oval

eumet,;c fish;ng

developm fish.'Y

ov.,f;sh;ng 'nc, m.'" ,;,..nd doc'..".tlo"

D ..0_005

E-OS_ol

oma" fi,h a'o coughtat high ottonlovei,

Pauly, D. 1980. On the interrelationships between natural mortality,groWth parameters and mean environmental temperature in 175fISh stocks. J. Coos. CIEM 39(3):56-60.

Pauly, D. 1984. Fish population dynamics in tropical waters: a manualfor use with programmable calculators. ICLARM Stud. Rev. 8.325 p. Iotematiooal Center for Living Aquatic ResourcesManagement, Manila, Philippines.

Pauly, D. 1985. Zur Fischereibiologie tropischer Nutztiere: eineBestandsaufnahme voo Konzepten und Methoden(Habilitationschrift). Ber.lost. Meeresk. Christian-Albrechts Univ.Kiel147.155 p.

Schaefer, M.B. 1957. A study of the dynamics of the fishery foryellowfUl tuna in the eastern tropical Pacific Ocean. Bull. 1-A 1TC/Bol. CIA T 2:247 -268.

Sinoda, M., S.M. Tan, Y. Watanabe and Y. Meemeskul. 1979. A methodfor estimating the best cod-end mesh size in the South China Seaarea. Bull. ChoshiMar. Lab. China Univ. 11:65-80. 'In ,.,m. of y;eld per ""N;t only.

bCon...enng the' open...,.e.. fi.her;e. will become o"ercepi,el;zed If no' probebly menagad.cSoca".. the f;ohery i. probably g.ne,.t;ng mexim"m _nom;c yi.,d.

*ICLARM Cootribution No. 291.

Toblo 1. Poobobilitio. of copt..,o fo, 0 .im..iation of off on ,oloti.o yiold po, ,..,..it of inc,...ingtho ..'oction ,onue (fo, L.,.-10. M/K -21.

Longohcl... 1J)-I~ 2.0-2~ 3.0-3~ 4.0-4~ 5.0-5.9 6.0-6.9 7.0-79 8.0-89

1.0.,

0.9

0.8

0.7

iI 0.6a.'" 0.5,,: 0.4)i 0.3

0.2

0.1

8

,

A Cose I( knife-edge

selection)

,Case I (knife-edgeselection)

""Case 2

.5 0.030..uQ)....Q)Co 0.020"0""4i';;'

0.010 3'3

1.0I I I I I

0.2 0.4 0.6 0.8 1.0

Exploitation rate Exploitation rate

Fig. 1. Effect of an increasingly wide selection range on relative yield per recruit (AI and relative biomass perrecruit IB) as assessed through application of equation 111) to the data in Table 1. Note that the knife-edge

assumption leads to overestimates of yields and of optimum effort; based on Table 1.