ANALOG COMPUTER MODELS OF FISH POPULATIONS

By RALPH P. SILLIMAN, Fishery Biologist (Research)

BUREAU OF COMMERCIAL FISHERIES BIOLOGICAL LABORATORY, SEATTLE, WASH. 98102

ABSTRACT

A modern analo~ computer. tol1ether with an X-Yplotter, I provides a means of constructinl1 useful modelsof commercially exploited fish populations. The modelcombined conventional exponential fishin~and naturalmortality rates with a Gompertz curve of ~owth inwei~ht. By combination of these rates and curve intoa sinl1le differential equation. survival curves for suc­cessive year classes of fish were ~enerated by the com­puter and plotted by the plotter. Wei~hts for all yearclasses present in each season were summed ~raphi­

cally. Recruitment was determined from a stock­recruitment curve, set into a function ~eneratorof thecomputer. Yields for each season were calculated bymultiplyin~ stock weil1ht by rate of exploitation andwere compared with actual yields to test the validityof the models.

Fishery biologists have devoted much effort indetermining changes in fish populations as theyrespond to varying degrees of fishing intensity.Because stocks usually cannot be observed andmeasured directly, it has been necessary to use dataof catch and fishing effort and biological data onrelatively small samples of the stocks. These rec­ords, plus associated data on the environment, havecomposed most of the working materi~ls of the fish­ery biologist. Limited to such materials, the fisherybiologist has been forced to proceed largely by infer­ence. There has been no alternative. As work be­came quantitative, inference came to mean statisticalor biometric inference, or a combination of both.

One method of quantitative inference is that ofsimulation or modeling. Using the best empirical

NOTE.-Appro,'ed for publication Nov. 23. 1911.'i.I Trade names referred to in this publication do not imply endorsemen t

of commercial products.

FISHERY BULLETIN: VOLUME 66, NO. 1

The technique was demonstrated by use of empiricaland hypothetical data for the Atlantic cod (Gadusmorhua). It is ~enerally applicable to fisheries forwhich l100d measures of total catch. I1rowth rate,natural and fishinl1 mortality rates, and stock-recruit­ment relation are available.

It is important to use reasonable values of biolo~ical

variables in constructin~ the models because it wasnot possible to demonstrate beyond doubt that the setof parameters for any ~iven model provides a uniquesolution.

Compared with other techniques, the analo~-~aphic

approach offers low cost of equipment, moderate com­putation speed. ready accessibility of equipment, and~ood visibility of results durin~ computation. It islimited in accuracy (two or three di~its) and in scalin~

requirements.

data and biological judgment available, the biologisterects hypotheses concerning the additive and sub­tractive processes affecting the stocks. The formerinclude recruitment, growth, and immigration; thelatter, fishing mortality, natural mortality, and emi­gration. To test the validity of the hypotheses,characteristics of the models based on the hypothesescan be compared with what is known of the realpopulations. Population models are used for thesame purpose as models in ship or power-dam design :experiments are easier, quicker, and cheaper with themodel than with the full-scale object. Any tool thatwill help the biologist in these processes should bevaluable. This report describes such a tool in theform of an analog computer technique.

Application of the analog computer as describedherein has not, to the best of my knowledge, beenmade previously. Originality is not claimed, how­ever, for the technique of simulation in general or in

31

the field of fisheries. Familiar applications outsidethe field of biology include aircraft and spacecraftdesign, ballistic studies for military services, andmanagement of systems such as power-generatingpools and nuclear reactors. Among biologists, theecologists especially have made use of simulation.Notable examples inelude the simulation of ecologi­cal systems by digital computer (Garfinkel and Sack,1964) and by analog devices (Odum, 1960 and Mar­galef, 19112). In fisheries there have been threegeneral types of simulation: analytical, digital, andanalog.

Analytical simulation is the oldest and best known.It is so well known that an extensive review is super­fluous here. In it the biologist analytically deter­mines the nature of the mathematical relationscontrolling population structure and populationresponse to exploitation. He then constructs mathe­matical models which predict what will occur undercertain assumed conditions of the fishery and theenvironment. Most of the elassical contributionsto fishery dynamics are of this type. They includethe work of such authors as Baranov (1918), Russell(1931), Thompson and Bell (1934), Graham (1935),Schaefer (1954), Beverton and Holt (1957), andRicker (1958). During the history of the analyticaltechnique the complexity and sophistication of theformulations have generally increased; the work ofBeverton and Holt unquestionably represents thehighest development in this respect to date.

Some of the above authors have also used digitalsimulation. Thompson and Bell (1934) arithmeti­cally constructed tables to simulate the catch andcatch per unit of effort of the Pacific halibut underconstant recruitment and certain assumed rates ofgrowth and mortality. They demonstrated aremarkable correspondence between the values pre­dicted by this method and the observed values forcertain periods and areas. Ricker (1958) madesomewhat similar calculations based on analyticalfunctions and introduced the additional eoncept ofrelation between stock size and rate of recruitment.He expressed catches in terms of "equilibriumyields" which would be obtained when the additiveprocesses affecting the populations just balanced thesubtractive ones. . An outstanding example of digitalsimulation of a fish stock is found in Larkin andHourston (1964).

To my knowledge, only Doi (1957, 196:3) has pro­posed application of the analog computer to fisherydynamics. He set forth mathematical formulations

32

and computer block diagrams and made applicationsto Japanese fisheries. His formulas are similar tothose used here, in general, but differ in detail. Healso adapted the Volterra equations to analog treat­ment of predatory and competitive relations amongfish populations.

With some notable exceptions (Ricker, 1958;Larkin and Hourston, 1964; Larkin and Ricker,1964; International North Pacific Fisheries Com­mission, 1962), fishery-simulation attempts to datehave dealt with equilibrium population conditions.This restriction is understandable in view of themathematical complexities introduced by varyingrates. Such an approach does, however, lead tomodels which are somewhat unreal compared withtheir counterparts in nature. For instance thenumber of recruits annually entering the stocksvaries widely. This problem has been treated byconsidering recruitment constant for short periods(apparently close to the truth for some Pacific hali­but stocks during various 8- to lO-year periods be­tween 1918 and 1933) or circumvented by expressingresults in tel"lllS of "yield per recruit." Likewise,fishing mortality varies with fishing intensity. Anyrealistic simulation of fished populations over con­siderable periods requires some provision for changesin recruitment and other vital rates, as ean be accom­plished on the analog computer. The importanceof this problem was recognized by Schaefer andBeverton (1963) who wrote:

".... the characteristics of the recruitment to marinefish populat.ions-its degree of fluctuation, its relation tostock size and the influence on it of changing environ­mental conditions-are the key to the interpret.at.ion andprediction of the long-term dynamics of a fishery ..."

and:.•Actual fisheries are, however, seldom in st.eadyst.ates .. ."

The remainder of this report is devoted to adescription of the plan of attack (Platt, 1964-stronginferenee) used in the analog computer approach.Briefly outlined, this plan is as follows:

1. Formulation of vital rates in a manner suit­able for analog solution, thus setting up ahypothesis.

2. Simulation of populations and yields over theperiod for which observational data areavailable. .

3. Compal'ison of simulated and observed yields,testing the hypothesis..

U.S. FISH AND WILDLIFE SERVICE

BASIC FORMULATIONS

In addition to the above, t.he following symbolshave been adopted for the formulations here:

(1)

To take account of the growth of individual fish,and to obtain yields in weight. for comparison withcommercial catches, it is necessary to introduce aformula for weight-at.-age. Beverton and Holt em­ployed the von Bertalanffy equation for length-at­age, converting·t.o weight-at-age by means of a cubiclength-weight relation. Use of the cubic relationhas been shown to lead to considerable error whent.he real relation between length and weight involvesa power of length other than 3 (Paulik and Gales,1964). Although this difficulty can be overcome byuse of the Incomplete Beta Function (Wilimovskyand Wicklund, 1963) in t.he yield equat.ion, theformulation still is not well adapted' to analogcomputation.

As an alternative to the von Bertalanffy equation,I investigated the' characteristics of the equationdeveloped by Benjamin Gompertz. He applied itas an expression of human mortality rates, butvarious forms of it have since been used as growthcurves for both length and weight of animals. Itsapplicability in this respect was thoroughly dis­cussed by Winsor (1932). In the form used byWeymouth and McMillin (1931), it is seen to be anexponential curve in which the slope declines expo­nentially. They point out that the relative (asopposed to absolute) growth of an animal declineswit.h age because of an increasing proportion of inac­tive material, and other causes. The declining slopeof the Gompertz curve is in accord with this phe­nomenon. Also, it provided a· good fit to theempirical data of weight-at-age for several fishes.

Beverton and Holt (1957) rejected the Gompertzcurve on the basis that it deals with growth as anadditive process only, i~oring the breakdown ofprotoplasm. The net effect, howevel'; of anabolismand catabolism may well be the kind of decliningrelative growth described by the Gompertz curve.This. curve thus did riot appear to be rejected on

Because interest in this study is centered on thecommercial catch, the model is.limited to the fishablesizes and ages of fish. For a year class of fish passingthrough t.he fishable stock, numbers of fish survivingmay be expressed according to the declining expo­nential formula, as set forth in Beverton and Holt(1957) :

R", = Initial weight of recruits to fishable stock for a singleyear class.

G,g = Constants of Gompertz growth curve.

= Number of recruits surviving at time t.= Initial number of recruits to fishable stock for a

single year class.= Fishing effort.= Instantaneous rate of fishing mortalit.y.= FI/.= Instantaneous rate of natural mortality.=F+M.= Age of fish in years.= Age of fish at recruitment to fishable stock.= Age of fish when first vulnerable to capture by gear

in use.= Weight of aU fish of a given year class surviving at

t.ime t.= Weight of an fish present at beginning of season.= Weight of individual fish at time t.= Upper asymptotic limit of w,.= Weight of individual recruit at time Ir•

= Estimated yield of fishable stock in weight, per year.= Actual yield of fishable stock in weight. per year,

from official statistics.

= Rate of exploitation = _F__(l - e-(F + Ml), F+M .

= Subscript referring to individual year classes.

Pt.

E

N,R

i

P,

4. Acceptance or rejection of the hypothesis. Ifthe agreement of simulated with observedyields is good, the hypothesis is accepted. Ifit is rejected, a new hypothesis is erected byadjustment of parameters in the analog model,and steps 2 to 4 are repeated until satisfactoryfit of simulated to actual yields is either at­tained or found unattainable.

In practice, the process was never repeated morethan a few times, since improvement fell off rapidly.Also, indefinite r~petition would be out of keepingwith t.he scient.ific met.hod.

For the initial trials of the analog t.echnique, Iadopt.ed what seemed the simplest useful model ofa fish population. This model includes rate ofgrowth, rates of fishing and natural mortality, and arecruitment-stock relation. It does not take ac­count of immigration, emigration, or environmentaleffects. Symbols used have been adapted fromHolt, Gulland, Taylor and Kurita (1959) in further­ance of their admirable attempt to secure uniformityin the terminology of fishery dynamics. Definitionsare as follows:

fFqMZttr

t.

ANALOG COMPUTER MODELS OF FISH POPULATIONS 33

and e-Ge-O«-t,.) ----.07 1.00

FIGURE I.-Analog computer and plotter.

THE COMPUTER AND PLOTTER

Since analog machines probably are not familiarto most fishery biologists, a brief description seemsin order. 10dem analog computers (fig. 1) areelectronic and perform operation on voltages. Thevoltage is made numerically equal (analogous) tovariables in the problem (e.g., 1 volt=age of fish of2 years), and component building blocks on thecomputer perform mathematical operations. Vari­ous blocks perform: (1) algebraic summation, (2)multiplication or division by a constant, (3) multi­plication and division of two variables, (4) integra­tion, and (5) generation of nonlinear functions. Thislast building block makes it possible to producefunctions if a graph of the function is available eventhough the equations describing the graph areunknown.

The analog computer is primarily a device forsolving differential equations with time as the inde­pendent variable. It, therefore, becomes evidentthat if a biological process can be expressed as a dif­ferential equation, the equation can be mechanizedby interconnecting analog computer componentscorresponding to the mathematical operations.

(3)

(2a)

(2b)

G - G.-O(t-'T)wt=wTe e

G-Ge-O(l-IT)Wt=wTe

P, = RwTe[G-Ge -O(I-tT)]_(F+M) ('-'T)

Because I dealt with weight rather than numberof recruits, I set

and obtained as my working equation:

as t ----.07 00, e-O«-'T) ----.07 0

This formula can be combined with formula (1)to express total weight of survivors at any time. Itis of interest, also, that it has an upper asymptoteWoo similar to the "Loo" of the von Bertalanffyequation. Thus:

For convenience, clarity, and ready comparability\>"ith other work, I have dealt with all relationshipsso far in algebraic form. Although the computerrequires differential equations, the differentiationcan be performed on the final equation, (4) above.

Substituting in (3) for WI and N, their equivalentsin (1) and (2b):

biological grounds, and since it was practical foranalog computation, I employed it.

It may be noted that growth rates of fish typicallydecline throughout life and can be represented mostsimply by a positive instantaneous rate which de­creases exponentially with time, as in the Gompertzcurve. The relations can be expressed in the follow­ing formulas (where G represents the initial exponen­tial growth and 9 governs the exponential rate ofdecline) :

so that w, ----.07 wTe, G as t-----7 co from equation(2a) above. The limiting value of this expression isWoo' If extended from w, = 0 to w,""'Woo, the Gompertzcurve has a point of inflection lacking in the vonBertalanffy curve. This inflection is found in age­weight curves of many fishes. The total weight ofsurvivors from a single year class may be expressed:

34 U.S. FISH AND WILDLIFE SERVICE

Still disregarding i. C., we have:

/;t-ZRe-Zt

-I Z

so we simply

COMPUTING ELEMENT SYMBOL BLOCK DIAGRAMEQUIVALENT

i.e.

Integrating network wilh D>-~Summing Junction

A

~Potentiometer --0-Summing amplifier -{>-~

dNt =-RZe-Ztdt

The next step is to constmct a circuit diagramshowing the actual computing elements and takingaccount of "initial conditions" and any sign changesthat may occur. For this step we use additionalsymbols:

k dN, ZBut we "now t.hat ~= -R e-ZI,

assemble the derivative:

dNt-- = -RZe-Ztdt

OPERATION PERFORMED SYMBOL EQUATIONBY COMPUTER

i.e.Y=.lI' dl + i.c.Integ/otion wilh /esp~cl 10 lime

.....@Yi.c.= Y]'.o

Inve/sion (Change of sign I xl-G----y y =-xI

Multiplication by a conslont xl-G"y y = Ax,

X

lflSummolion x2 L. y y = xI + x2+ x3 ...x3

dN, = -RZe-ZI

dt

As in digital comput.ation, we start with a "blockdiagram" (ordinarily this step would be omitted insuch a simple circuit, but it is shown here to illustratethe process). A few symbols aloe needed (arrowsindicate direction of information flow):

Remembering- that the analog computer requiresdifferential expressions, we differentiate the aboveto find:

Analog computer programming is based essentiallyon the electrical principle of the feedbaek loop. Fora simple illustration, let us return to the decliningexponential curve, as expressed in equation (1).This expression may be further simplified by settingF+M =Z, as in international notation and assumingthat our measurement of time begins at tr so thattr=O and (t-tr)=t. Expression (1) then becomes:

Using the first three of these symbols, we can nowconstmct a block diagram for our differential expres­sion. We start by assuming that a voltage propor­tional to dN,/dt exists:

dNt

dt

i.e.

--+)--IZJ,---;)~-- Nt + i. c.

zFinally we must supply the "initial condition"

voltage, £.e., which is equal to the constant of integra­t.ion. We noted above, in the definition of the

Considering only the rate at which N , changes,and disregarding its absolute value, we can omit theconstant of integration (this constant will be addedin the circuit diagram):

i.e. and i.e. =Y],=O. In our equation "y" is replacedby N /_ Since we set tr = 0, by definition N ,],..0 = R =

i.e. The completed diagram, assuming a computer

ANALOG COMPUTER MODELS OF FISH POPULATIONS 35

reference voltage supply of -10 volts. becomes:

-10

z

The plotter has a pen actuated by two servo­motors. One moves it along the "X" axis and theother along the "Y" axis in proportion to an inputvoltage supplied by the computer. Thus the penmoves to any point X. Y in a system of rectangularcoordinates on the plotting surface, corresponding toinput voltages V r and V u' Since the computer in­tegrates with respect to time, a voltage directlyproportional to time is usually fed into "X." Theinput for "Y" can be taken from any point on thecomputer circuit to plot the variable(s) desiredagainst time.

As an alternative method of output display, anoscilloscope can be used. This combination requiresthat the computer be modified for "repetitive opera­tion." Problem solutions are repeated 10 to 100times per second, so that they appear as curves onthe oscilloscope screen. For a permanent recordthe screen can be photographed as mentioned byDoi (1962).

Tilis oscilloscope display is particularly valuablefor curve fitting, since points can be plotted on theface of the tube. In this manner the effect of thepote!ltiometer adjustment in improving the fit isinstantly seen.

The work described below was performed on aPace TR-10 analog computer in conjunction withan EAI Variplotter 1110. The TR-10 is one of thesmallest general-purpose analog machines. Sinceboth computer and plotter are fully transistorized,they are small and can be used conveniently atop adesk or small table. The set of units available inthe computer as I used it was as follows:CIRCUIT DIAGRAMBLOCK 01 AGRAM

In the operation of the integrating network, i.e.only determines the value of N , (condenser charge)at the beginning of the computation. It is cut offthe instant computation begins, and current flow isthen confined to the feedback loop. That is whyi.e. does not appear as a variable in the block dia­gram. The figure "1" by the integrator input 'refersto the input resistor. Operation of the integratoris such that the input is multiplied by a factor1oo/R, when R,=input resistance in kilohms.

With the knowledge now at hand we can proceedto set up a circuit in which Z is divided into itscomponents F and M. The equation becomesN ,= Re.- IF+J\O I and the dial!:rams are:

GENERATION OF SURVIVAL CURVES

The differential equation needed for generatingthe survival curve of a given weight of recruits, RID,is obtained by differentiating expression (4):

~f..!= RlDe[a-a. -q(t-ITl]-CF+J\[) (I-Ir)[gGe-OII-lr) - F-lIf]

(5)

NumberU"itAmplifier (used with integrat.or, Illultiplier, etc.) _____ 10Coefficient potentiometer (used as above) ___________ 18Null potentiometer (used to Ret coefficient

potentiometers) ________________________________ 1Integrator (uscd as above) ________________________ 4Multiplier (used as aboveL _________ __ ____ ____ 1

Diode function generator (use described laterin text.) _______________________________________ 1

c.omp:tmtor (use described later in text) ____________ 1

-10

Study of the circuit used here (fig. 2) reveals it tobe essentially an elaboration of the basic feedbackloop. A multiplier has been added to take care ofthe nonlinear Gompertz growth curve, and otherelements have been added as described in the text.The "time-base" (t) is generated by a simple inte­grator output. Biologists interested in further in­formation on analog computer programming willfind an excellent brief introduction in Strong andHannauer (1962) or a more extended treatment inAshley (1963).

36 U.S. FISH AND WILDLIFE SERVICE

+10

P,. setE

P,.(Y plot)

)----------Yw (Yplot)

1-----------------I -10

, COMPARATOR

(X plot)

-~

t /chart unitI

;-10-10

.,-' - - - - - - - - - - - - - - ---- --~

FUNCTIONGENER'ATOR I

~ II I

___________ J

FIGURE 2.-Analog computer circuit for generation of P, and Y,•. Standard symbols for computing elements, scaling excluded.

By integrating this expression, the computer gen­erates P, as a function of (t-t r) for any given valuesof R w, g, G, F, and AI. The circuit diagram, whichincludes conventional symbols for the computingelements, is shown within the broken line enclosuresof figure 2. The elements outside the broken linesare used in the combination of analog and graphicalcomputation employed to sum the weight.s of yearclasses for each season and to det.ermine subsequentrecruitment. Their nature and use are describedlater.

As examples of analog computation, growth andsurvival curves from the plotter are shown in figures3 and 4. The data are hypothet.ical except for t.hegrowth constants, which have been derived fromdata for t.he California sardine.

Data of growth in weight were obtained by com­bining a curve of growth in lengt.h wit.h a weight­length relation. A table of length-at-age was givenin Phillips (1948) and a weight-length relation inClark (1928). Fitting of the Gompertz relation to

these data (fig. 3) was readily accomplished by suc­cessive trials, with appropriate adjustment of thepotentiometers for G and g. The fitted curve fol­lowed expression (2b), with 10r =93 g., 0=0.825,g= 0.445, and tr = 2 years.

St.arting with a hypothetical 1,000 fish, Rw=93 kg.when 1Or =93 g. The upper curve (fig. 4) shows how,wit.h no fishing mortality and low natural mortality,P I may increase for a year or two before mort.alityovercomes growth. In the t.wo lower curves theeffect of adding a·substantial fishing mortality maybe seen. Application of an increase in fishing mor­tality at t- i r = 2 resulted in the lower branchedcurve. This change is readily made on the analogcomputer by placing the machine in the "hold"mode. The potentiometer for"F" is then reset, themachine returned to "operate" mode and the com­putation resumed. The ability to change quicklyt.he vit.al rat.es during a computation is one of theadvantages of the analog machine. It may be doneeven more conveniently by presetting a number of

ANALOG COMPUTER MODELS OF FISH POPULATIONS 37

22201812 14 16

(YEARS)10AGE

864

l,---y/

//

,o

2

25

50

75

225

200

_150en::!:«0::: 125(,!l

175

....= 100~

FIGURE 3.-The Gompertz curve fitted to weight-at-age data for the Pacific sardine following expression(2b) in text; 11'. = 93 g., G = 0.825, g = 0.445, and I. = 2 years.

----V """t = . OOM =.20

1\ 'i. .-'

\- 'F= ~~lM =:20

\

~~,

\ ~~

F=.50}_~ -----M=.20 -......... -

120

en 100::2:«~ 80o....J

::.:: 60

40

20

oo 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

t - t r

FIGURE 4.-Computer curves of PI as a function of (I-I,); growth data from the California sardine, accordingto expression (4) in text. Values of constants are: R = 1,000 fish, 11', = 93 g., RID = 93 kg., G = 0.825,g = 0.445.

38 U.S. FISH AND WILDLIFE SERVICE

potentiometers at needed values of "F" and shiftingfrom one to the other.

PROCEDURE OF SIMULATION

Simulation of populations and yields by a com­bination analog-graphic approach required develop­ment of a standardized procedure. A cllart wasprepared for the plotter, with appropriately scaledcoordinates for time (X-axis) and stock (Y-axis).Vertical lines on this chart marked the points forchanges in mortality rates or recruitment relation.

Because the commercial stock in any fishing seasonis made up .of survivors from individual year classesof various ages, that stock cannot be started "fullblown," with all year classes present. I thereforestarted with a stock size such that the. rate of exploi­tation (E) estimated to be in effect would produce acatch (f"",) equal to the real catch fO!' the first fishing­season, OJ' the mean catch for the first two fishing­seasons of the study pCJ'iod. This stock was thenbuilt up by starting year classes at times 1, 2, 3, ...n years before the beginning of the period; n repre­sents the numbel' of years for a year class to passthrough the fishery. DlII'ing this "prestudy" period,the mortality rates in effect at the beginning of thestudy period were assumed to be in effect. Eachith curve begins at the value of R,o (assumed thesame for each year class) required to produce the

i=".

specified initial vnlue of total stock P,o = ~ P".iJ:l

Thjs value of R"" can be quickly determined by afew computer tl·ials. P r fOl' each year class is gen­erated until it declines to a small arbitrary valueclose to zero, but only the portion extending into thestudy period is plotted.

The initial and subsequent values of p,. are cal­culated .by graphically summing the heights P" ofthe ith slll'vival clll'ves for each fishing season, bymeans of a pail' of dividers. Once the initial valueof P,o has been obtained, the calculation is self­sustaining. From the recruitment curves, values ofR,Oi corresponding to P 10 tr years before are obtained.Rw, is generated as a function of P,o in the DiodeFunction Generator (fig. 2) by the simple device ofsetting the P,o potentiometer so that the plotter "Yvalue" corresponds to the P,o value for the particularyear in question. R ,OI is then plot,ted by attachingthe Y input to the RIO point in the computer circuit.Any empirical or theoretical curve relating recruit-

ANALOG COMPUTER MODELS OF FISH POPULATIONS

ment to spawning stock can be set into the DiodeFunction Generator, which, with an input XI, pro­duces an output in the form of a clII've 1I = !eXI),composed of 10 stmight segments.

SlII'vival clll'ves as descI'ibed above can be gen­erated for each year class entering the fishery duringthe study period. As in the case of the initial seasonjust described, hei6hts P'I of the individual survivalclll'ves fO!' each year arc summed graphically, and amark made representing the total commercial stock,

i=n

P,o= ~ P". The "P,o set" potentiometer is ad-i=o

justed to bring the "Y-value" of the plotter intoconformance with the total stock value P,o. Thecatch is calculated by setting potentiometer "E"

(Fig. 2) at the vahll~ E= F~MfJ.-e-(FH[)l. The

catch or yield value proceeds from the simple rela­tion i-"10 = EP'0' It is plotted by attaching the "Yinput" of the plotter at the point Y 10 in the computer.After plotting of 1\. the process for R"'l (above) isrepeated, and the cycle recommenced.

In outline, then, the process of simulating popula­tion and yield is as follows:

1. Set the initial value of Y '" at the size of theactual catch for the initial year or two of thestudy period.

i=n

2. Determine initial P,o= ~PII from the rela-i=o

tion P'o= Y",/E.

3. By computel' trial, find value R Wi such thati=,1-

~ P Ii =initial P,.,i=o

4. Generate n curves of PH, where n is the num­ber of years required fOl' P II to decline fromR w, to an arbitrary small value neal' zero.Start at 1, ~, 3, ... n years befO!'e the begin­ning of the study period.

.5. By Diode Function Generator evaluate RWI

for each season from P,o for season tr yearsbefore. Generate curves P'I starting atP Ii = RWI for each fishing season.

6. For each fishing season, gl'aphically deter-L=n

mine P,o= ~PII' Calculate Yw=EP,o.i=1

39

7. Repeat cycle to end of study period, startingeach cycle with step No.5.

EXAMPLE OF APPLICATION

Since hypothetical data are seldom satisfactoryto demonstrate the application of a technique, thefollowing example of application to the fishery fOI"At.lantic cod (Gadu.~ morhua) is included. It hasbeen used to achieve concreteness, not to make newdiscoveries about the cod. Catch data were summedfor International Commission for the NorthwestAtlantic Fisheries Divisions 5Y and 5Z, and thefollowing parameters were assembled for analogcomputation:

1. The central value of F = 0.35 used in Bever­ton and Hodder (l96~) was assumed to bethe average CF) for the entire study period1932-1958. From this figure, values werecalculated for eight periods, from the relationF = qf, where fwas value of fishing effort fromBeverton and Hodder and q=F/f:

10'='1[. h(Jhlin~menn 7- =0.55

Cor:

other species. From data on age composi­tion in Beverton and Hodder it was estimatedthat tT = 4 years.

5. From data in Bevcrton and Hodder (1962)and Silliman and Wise (19131), it was esti~

mated that if t, = tT before the change in cod­end mesh size from 2rs to 41/6 inches in 1954,then to = tT +0.2.5 year fOl' 1954 and there­after. This change was accomplished by thecomparator (fig. ~), a device which actuatesa switch when the input (t in this application)reaches a predetermined value U, here). Thecomparator delayed application of F for onequarter of a year, for year classes enteringin 195-'1-58.

The initial trial simulation, hased on the parame­ters just listed. produced a poor fit of calculated(l'w) to actual (Yu,) catches (r= -0.17). On thebasis of this trial, the shape of the recruitment curvewas altered somewhat (fig. 5B) for the second trial.All other parameters remained the same. After therecruitment relation was revised, a second trial pro­duced a considerably improved fit (r=0.59). Finally,the value of M was changed to 0.25 and mean F to

2. The central value M = 0.2 given by Bevertonand Hodder was assumed to be in effect dur­ing the entire study period.

3. Lengths-at-age were oU "led from Schroeder(1930) and convert.r . ights-at-age witha length-weight Clio ;. to data given inBigelow and Schroeder (1953). A Gompertzcurve was fitted to the weights-at-age withconstants G= 1.47, g=0.340, and wT =5.9 lb.

4. No empirical data were available as the basisof a stock-recruitment curve. After somepreliminary experimentation, a hypotheticalcurve (fig. 5A), with mode at an arbitraryRIC; = 32,000 metric tons, was adopted for thefirst formal trial. The concavity of the lefthand limb was based only on experience from

2.00 0.~6 0.~1

1.1:1 .20 .182.00 .~6 .~12.77 .50 A~

1.80 .:i2 .2R1.22 .22 .191.80 .~2 .282.24 .40 .35

'8 C. TRIAL 3o.

Ol--..::::::::::L+----+----+-----'t--"""---j..

40

A, TRIAL I<:t+c 20

0:«ILl~

III 0z

B. TRIAL 2°I-

LL.o

00 40 80 120 160

P , 1,000's OF METRIC TONS, YEAR nto

U

0: 20I----I----+-+--~-----l~i_I_--____tI­ILl::iE

FIGURE 5.-H~,pothetical fecruitment curves fOf New Eng­l:md cod. Vertical broken lines indicate mnge of valuesused in comput.ations.

," =0.2 ," =0.21;('I'dal. ('l'rir,1 3)

I and 2)

Ml'an effortthollsnnr!s oC

bont-dnysPeriod

1932-33. • _• •••. _1(J.'l4-36.• _.. _... • • ..• _1~~7-38 .• • •. _••• _19:19--11.. ... •• _. .1942 only . __ •• •. _1943-46.• _. " . __ ••• ._1947-48_... .• • __ .•••• .1949-58_•. _ .• _. .• _•• .•

40 U.S. FISH AND WILDLIFE SERVICE

0.30, so that F+lIf remained 0.55. A further ad­justment was also made in the recruitment curve(fig. 5C). The third trial, in which these adjust.­ments were employed, yielded a further improve­ment in the fit (r=0.69), which is shown in figure 6.A notable feature of the trials was the sensit.ivit.y ofthe fits to changes in the stock-recruitment relation.The greatest improvement in fit occurred betweenthe first. two trials, as the result solely of a moderatechange in the s~ape of the recruit curve (figs. 5A, 5B).

The series I'", and I'ware not, of course, com­plet.ely independent, since Y .. enters to some extentint.o the calculation of the parameters used to com­pute }';.,c. As a relative measure of goodness of fit,however, the coefficient is of some value. It ispossible to make at least two positive statementsregarding r as used here;

1. If the calculat,ed P value is greater than thesignificance level, the real value is certainly

greater. Additional degrees of freedom willbe lost according to the degree of dependenceof the variables. It is possible, then, toidentify markedly nonsignificant fits.

~. For calculated correlations within conven­tional significance levels, the value of r servesas a means of comparing two fits. If one set.of parameters generates a fit with a highervalue of r than another, then it should becloser to the truth than the other.

The parameters used in the simulations have vary­ing and subtle degrees of dependency on the catchdata. Thus, to assess accurately the value of P forthe coefficients would require a lengthy and com­plicated statistical analysis. This work appearshardly to be \val1'anted in view of the approximatenature of the simulations.

' ...... ~

,

Ie =4.00 Ie =4.25'-----.-..~

'--I--'

I I-'-'-'- TOTAL POPULATION, Pt.-- ACTUAL CATCH, Yw .'"

-----CALCULATED CATCH, Yw

/ ............./ --.

F=.30 ~O~~... '

'.\

\

'.\

180 F=.30 F=.20-----.- ~.........1-' ro-'

U> 160 ,Z

,-/0 ,

I-140 .-

u0:::I-w 120~

\u.. \~ 100~---1---+--+-+-+-+-.:.:-....-,+.'-;.'4·1-+---+..:o..,+·=-_-t-.~~-

"o ,g 80~--I---I--1-i----1-+--I--I~1-+--r-+-++--t~,:-,-I--1-4--

\ " ,<>! 60~---I----+-';---r--I--f---I--+--+--+-f-+-+-+---r'"""::,--i--I--1

~ l,t, II

« , I '

>-' 4°~v-I1T'~A'-.L, ---1-!-+-'--1' '-1 .......

o ' ---- i ~ ,I I , - ..•- ~'"- ~. -+__-+-__-;

"-~ 20 I---Tr ~~ ~- 1-"'::"-:':--::"""-.b-__- --+--_~

~9L3-2--1913-4--:c19J3::-:6~--:-19~3:8~~1:::9~4-::0~:::19~4;-;:2~-;;19f4;;--4;;----;-19~4~6~,1~9~408...LiCI9~5:n0-1c19~5~2;-~19~5~411Iq9~56~191958FISHING SEASON

FIGURE G.-Population and yield simulation for New England cod; 111 = 0.25, G = 1.47, and g = 0.34 for entire period.Other pammet.ers as shown in dm-wing.

ANALOG COMPUTER MODELS OF FISH POPULATIONS 41

GENERAL APPLICATION OF THETECHNIQUE

Application of the analog computer techniquedescribed in this report requires est.imat.ion of aseries of constants or parameters. It also requiresa number of approximations and some appraisal oft.heir uniqueness. These two topics will be dis­cussed in this section.

PRELIMINARY ESTIMATES OF PARAMETERS

An extensive literature is available on the estima­tion of parameters of fish populations under exploi­tation. The most thoroughgoing summaries anddescriptions known to me are given in Beverton andHolt (1957) and Ricker (1958). Treatment here islimited to the specific constants, variables, and rela­tions whieh must be estimated for simulations byanalog computer as described above. They will beconsidered in descending order of the degree ofcertainty with which they are likely to be known.

1. Annual yield. For most commercial fisheriesthis figure is likely to be known rather precisely.Since t.he original data come from weighouts at thetime the fish are first sold, t.heir accuracy has beenwatched closely by both fishermen and fish buyers.If possible, catches should be segregated accordingt.o biological stock unit.s.

2. Growt.h in weight. Lengths or weights atspecific ages are among the most commonly gatheredfishery data. If data are in lengths, they must beconvert.cd to weights through a length-weight curve.The length-weight relation is fairly stable as com­pared wit.h other paramet.ers and can be determinedfrom a relatively small number of samples coveringthe size range of t.he fish in question. The Gompertzgrowth curve can be fitted to the empirical datadirectly on t.he analog computer simply by adjustingpotentiometer set.tings for G and g unt.il a good fit. isobtained.

3. Inst.antaneous fishing mortalit.y rate, F. Thisrate may be difficult to est.imate with accuracy. Ifempirical estimates are lacking, however, it. may bepossible t.o produce an "educated guess" throughgeneral knowledge of the history of the fishery, itschanges in yield, t.he size of the current fishery, ....This value can be adjusted during simulation.

Once a value of F is available for one period, thevalues for ot.her periods can be est.imated if fishingintensity is known. Data on the number and sizeof units in the fleet are usually available. Thesefigures must be multiplied by an estimate of time in

operat.jon to obtain the value of fishing effort, f­Once a series of values of.f is on hand, together withF and f for a "base period," values of F for allperiods can be calculated from the relation F = qf,where q is a constant to be determined from thebase-period data.

4. Instantaneous natural mortality rate, lIf. Esti­mates of 1II may be available from tagging or bio­logical data, or an assumed value may be selectedfor the initial trial. If an assumed value must beused, guidance can often be obtained from thegrowth characteristics of the fish (Beverton andHolt, 19.59; Beverton, 1963). For fish in the com­mercially available stock, values of 1If are often low,in the vicinity of 0.1 to 0.3. Frequently, reasonableapproximations can be obtained by considering 1Ifto be constant during the study period, for all agesof fish .

.5. Stock-recruitment relation. Of the items re­quired for simulation, this one is usually the mostdifficult to obtain. If data are available on age com­position, it may be possible to estimate the abun­dance of the youngest (or youngest important) yearclass in the commercially available stock. A seriesof such estimates can then be related to the estimatedsize of the spawning stock fr years earlier, as was doneby Clark and Marl' (1955). The resulting scatterdiagram may reveal a pattern that can form thebasis for one or more recruit.ment curves. It issignificant that only the portion of the curve cover­ing the stock sizes encountered can affect the out­come of the simulation.

SUCCESSIVE APPROXIMATIONS ANDUNIQUENESS

Once the preliminary estimates of the parametersare assembled, simulation trials can begin. Becauseall the work is visible on the computer chart as itproceeds, it is often possible to see in what directionthe parameters must be changed to produce a betterfit of calculated to actual catches. Experimentationis readily accomplished, since all parameters may bechanged simply by resetting potentiometers (eithercoefficient potentiometers or the several small onesin the Diode ·Function Generator). For trials in theapplications above, each trial for 26 or 27 fishingseasons c.onsumed about one-half day. This workincluded preparation of the chart and all other neces­sary operations leading to a plot of annual popula­tions and yields. It is thus possible to accomplishseveral trials within a reasonable period of time.

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+.5

).0

0

1I

-.5 I1.4 I0, II, F HELD CONSTANT,

1.3 \ M VARIED\ 1

\I1.2 \

\ I1.1 \

I\

1.0 \1 I, M HELD CONSTANT,F VARIED

.9 "-I: ..........

......... .8 '0 ....~

1(>- ".7 " '\'\

.6 '\'\

'b.5

0Z .50 .55 .60 .65

I {~ .25 .30 .35 .40.25 .25 .25 .25

II{~ .30 .30 .30 .30.20 .25 .30 .35

:FIGtTRE 7.-Effect of varying F or M in simulation trialswith cod. The value r is the coefficient of correlationbetween calculated catches ii' ...) and actual catches (1'..).Since r is affected only by Z, and not the ratio of its com­ponent.s F and M, it has only one value for each pair ofcombinations. The fract.ion f ../}'.. represents the ratio ofthe mean calculated cat('h Cf..) to the mean actual catch<.1- ...). Vertical line of dashes indicates r.ombination ofvalues used in third co.:1 simulation trial.

1. Initlal cost of equipment.-The analog com­puter and plotter equipped as I used them costabout $8,000. This figure is substantially more thana good desk calculator ($1,000 to $2,000) but notover one-tenth the cost of comparable digital equip-

1.0

COMPARISON WITH OTHER TECHNIQUES

By appropriate transformations of the fOl'mlllas,any of the calculations reported above can be per­fOl'med on either conventional desk calculators orelectronic digital computers. It is pertinent, there­fore, to consider the relative advantages and dis­advantages of analog computation as compared withother techniques.

In any trial-and-error approach involving severalvariables, the question of uniqueness must be faced.It is fair to ask, if one set of parameters has produceda reasonable result, may not another set produce onethat is equally reasonable?

Although it is not possible to answer the abovequestion for the general case, some light may appearfrom a further examination of the trials on codreported above. In assessing goodness of fit, I tookaccount of the ratio of mean heights of the calculatedand actual catch curves, as well as the correlationbetween them. Thus I had two criteria of "good-

ness of fit": Y,jY", and r. To provide some ideaof the discriminative value of these criteria, I madeadditional simulations in which only F or 111 wasvaried and all other parameters were held constant,Curves of calculated values (fig. 7), bracketingthose used in the third trial above, are revealing.For this particular set of combinations, only oneother than the third approaches its "goodness offit." The combination of F=0.35 with M =0.25produced about as good a fit as the com~ina~ionof

the third trial; r was slightly higher, and Y,./Y Ii' wasslightly lower. The difference in Fof 0.05, however,is well within what might be considered reasonableerror in a rough approximation.

Obviously, curves of the type in figure 7 cannot"prove" the uniqueness of fit obtained with anyparticular combination of parameters. Since F, ill,and the shape and height of the recruitment curvecan be continuously varied, the number of possiblecombinations is infinite. Every effort should bemade, therefore, to use reasonable values of bio­logical parameters. Thus we know that F, M, andthe recruitment curve are sOllnd because biologistsgenerally recognize that fish stocks are affected byfishing, natural mortality, and recruitment. Thequestion of "reasonable values" is more difficult,but the range of possibilities may be narrowed byuse of empirical data as indicated under"PreliminaryEstimates of Parameters."

ANALOG COMPUTER MODELS OF FISH POPULATIONS 43

ment. It is, of course, possible to perform work ondigital machines under contract or rental arrange­ments at no initial cost. This arrangement is alsopossible for analog machines.

2. Time 'required for comp'llta.lion.-For the totaloperation as outlined above, limited tests indicatedanalog-graphic computation to be about four timesas fast as desk calculation. With digital computers,the calculation time is a matter of minutes. If t.imerequired for preparat.ion of data for computer cal­culation, programming the computer, and exchangeof data with the computer center are considered,however, total time may well approach that for theanalog method.

3. Visibility of work during computation.-In theanalog-graphic method, stork size, recruitment rate,and yield are all visible in graphic form as thecomput.ation proeeeds. This advantage is impor­tant to the biologist, since it quirkly reveals absurdresult.s, or permits him to end a computation thatis leading away from realit.y. Work is invisibleduring digital romputation, and the final "readout"is usually in the form of a table that may have tobe plotted for study.

4. Scale adjustment.-Quantit.ies generated withinan analog computer must be kept within the voltagelimits of the machine. This limitation leads to aconsiderable amount of "fussing" to achieve properscaling of the variables. This problem is only minorin digital calculation and therefore represents acomparative disadvantage for the analog computer.Fortunately, once scaling has been adopted for agiven formula, it can usually'be used with only oneor two rhanges when shifting to a new set of empiriraldata for the same formula.

5. Accuracy of rcsults.-Because of the nature ofcomponents in an analog computer, the final resultsare usually accurate only to two or three significantdigits. At the present stage of development offishery science, the empirical data available are notsuch as to justify carrying more digits. In fact,two-digit accuracy in fishery predictions would beconsidered more than satisfactory by most fisheryadministrators. Thus, the accuracy limitations ofthe analog machine as compared with digital com­putation do not at present represent a seriousdisadvantage.

6. Summary comparison of methods.-From theabove brief listing, the analog technique is seen tohave advantages in comparatively ;::nv initial costof equipment, moderately rapid computation rate,

and visibility of results. It has limitations in accu­racy and in scaling requirements and is slower thana digital computer. Decision as to which techniqueto use must depend on the situation of the individ­ual investigator. Fartors bearing on the decisioninclude the salaries of persons doing various parts ofthe work, the accessibility of the research stationto a digital computer, and the types of empiricaldata available.

UTILITY OF THE TECHNIQUE

In this report I have described what may be auseful working tool for the fishery biologist. Theexample of application given demonstrated the typesof basic data needed and the way in which theycould be adjusted to improve "goodness of fi,t" ofcalculated to actual catches. As with any tech­nique, its utility can be assessed only by thosemaking use of it.

Where extensive biological data are available,simulat.ion is valuable in determining the effect.s ofinteraction among the vltrying mort.ality, growth,and recruitment rates. The accuracy with whichact.ual cat.ches can be reproduced should serve as acheck on the validity of the sampling, analy.sis,and interpretation involved in the derivation ·ofpopulation parameters.

SUMMARY

1. The objective of this study was to develop ananalog-computer simulation technique for modelingexploited fish populations.

3. The mathematical formula for survival of ayear class expressed the effect of fishing and naturalmortality rates and incorporated a Gompertz curveof growth.

3. Survival curves for successive year classes weregenerated on an analog computer through use ofthe differential form of the survival formula. Acombined analog-graphic technique summed theweights of survivors in each season to give the weightof the fishable stock.

4. Yield was calculated by applying the rate ofexploitation to the fishable stock.

5. Properly lagged recruitment was determinedfrom the stock weight through a stock-recruitmentcurve.

6. Mechanics of the technique were demonstratedby application to the Atlantic cod.

7. This technique may be applied to any fisheryfor which good measures or estimates of catch,

U.S. FISH AND WILDLIFE SERVICE

growth rat.e, fishing and natural mortality, andst.ock-recruitment. relat.ion can be obtained.

8. The problem of uniqueness was st.udied fromsimulat.ions in which F and 1Il were varied over arange of values. The failure of results t.o proveuniqueness brings out t.he importance of usingreasonable values of paramet.ers.

9. As compared wit.h ot.her techniques, t.he analog­graphic approach described here offers low init.ial costof equipment., moderat.e computat.ion speed, readyaccessibilit.y of equipment., and good visibilit.y ofresults during comput.ation. It. has limit.at.ions inaccuracy (two 01' t.hree digit.s) and in requirementsfor scaling variables.

ACKNOWLEDGMENTS

Suggest.ions from my several reviewers were freelyfollowed in revising t.he text.. Reviews were fur­nished by Vaughn C. Ant.hony, Raymond J. H.Bevert.on, Anthony C. Burd, Paul H. Eschmeyer,Reynold A. Fredin, David J. Garrod, ,John A.Gulland, Richard C. Hennemut.h, Ralph Hile,George Hirschhol'll, Pet.er A. Larkin, Garth 1.Murphy, and William E. Ricker. Paul E. Huber,of Electronic Associat.es, Inc., kindly reviewed thesections on analog comput.ation. John L. McHughprovided guidance during final revision of themanuscript..

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