CUIMR-R-80-082 c.3

110RE REALISTIC FISHERY MODELS:

CYCLES COLLAPSE AND OPTIMAL POLICY

Louis lf. BotsfordBodega !farine Laboratory

P.O. Box 247Bodega Bay, California 94923

Sinple classical approaches to nathenatical analysis of fisheries areinherently linited. Increased use of nore realistic population nodels,in the forn of age- or size-specific, densitydependent nodals based onphysiological energetics, is proposed haze. Recent, fishery-relatedresults using nodule of this type illuninate such issues as cyclic be-havior of populations, nultiple equilibriun levels, and optinal policy.Specific results include: size-selective fishing csn lead to unstablecycles, an increase in individual grovth rate can naintain depressedequilibri~ levels, optinal policy for size-specific, density&ependentnodels nay involve pulse-fishing, and nore realistic nodels of nulti-species problens in the Antarctic nay alter conclusions reached throughstapler nodels.

I!INTRODUCTION

The increasing inportance of food resources and the questionable past recordof fishery nanagaaent haply better nanagenent techniques are required for che fu-ture. awhile nuch effort has been expended tovard increasingly sophisticated an-alysis of existing nathanatical nodels of fisheries, very little has been appliedto inproving the aodels thenselves. Age- or size-specific nodule based on conceptsof physiological energetics nay provide better fishery nodels. The rationale be-hind use of these nodels in fisheries, zone of the nodels of this sort currently inuse, recent results obtained through their use and hov they night apply to nuiti-species nanagenent of Antarctic fisheries are discussed in this paper.

Existing fishery nodels cf. Richer 1977, Culland 1977! are of three basic

plate, realistic description of population dynanics. The logistic nodal, for ex-anple, reflects the self-liniting behavior of population grovth, but does not in-clude the nechanics of grovth, reproduction and nortality that account for thisbehavior. The dynanic pool nodal focuses on grovth and nortality of individuals assuned to be identical and not density-dependent! vhile reproduction is not

Current Address: Departnent of Wildlife and Fisheries Biology, University ofCalifornia Davis, CA 95616

Johnston/Bots ford; R/F-52! Proceedings of Conference onControi Theory Appi ied to Re-newa e esource Pro ems,Lecture Notes in Biomathematics,

FROM

Springer-Ver lag: New York 19 0

types that have existed in the sane fora for tventy years or nore: the logisticnodal Crahan 1935, Shaeffer l956!, the dynsnic pool nodal geverton and Holt 1957!,and the stock-recruitnent nodel kicker 1954!. Kach of these falls short of a con-

accounted for. The scock-rectuitaent model, on the orher hand, focuses on density-

dependenc reproductive processes and survival of the young at the expense of a de-

scription of grovth and mortality at older ages. 1n addition to these three basic

types of aodels, the latter cwo have been combined to provide a sore ccaplece viev

of fish populations cf. Beverton and Holt 1957, Walters 1969!. The behavior of

chase coabinacion models and opciaal fishery policy using chem are usually examined

through simulation vith only rare attempts at further analysis cf. Gecx 1979!.

Pormsletion of better models requires definition of what aakes a model better

or vorse. Models can be viewed as inductive arguments by analogy. The basis of

this argument is simple induccive enuaeracion. Since the aodel behaves the saae as

che object modeled in vays A, B, C ~ ic will behave che saae in vay D. Pot example,

A, B, C could correspond co time periods in the past and D to a time period in che

future. Arguments by analogy become stronger as the nuaber of instances of enumer-

arian of relevant siailarities increases Salaon 1973, p. 97!, or in ocher words as

che realism of the model increases.

With regard to population models this implies chat aodels become better as

the nuaber of ways in which they match teality increases. Hovever, the number of

siailarities betveen models and populations could be increased in several vayu.

The aust obvious is to natch populacion level behavior over as long a cine period in

che past as possible in order to predict the future. A second vay chat is possibly

implicit in use of the logistic model is to choose a aodel chat aatches behavior of

othet populations and conclude, thetefore, chat it will natch the behavior of the

population of interest. Por exaaple, che logistic aodel fits laboratory growth of

s hoss hl ss, p ~s sppsh ~pso hil sp. ~ Moi p. pop l sloop

Hucchinson 1978!, therefore it vill reflect the dynamics of exploited tuna popula-

tioas. These cvo possibilities appeat co exhausc the possible arguments by enua-

eration in support of aodels Chat ate based on behavior at che population level.

However, chere are additional instances of enumeration that in aany cases can

be based on relevant existing knowledge. Behavior at che Lho~u cion level, the

level of interesc in fisheries, is determined by behavior ac che level of individ-

uals vithin the populacion; that is individual growch, mortality and reproduccive

rates. In many cases chase races are known or can be estimated fram existing dace.

Thus, aodels can match the real population in terms of individual behavior rather

chan only behavior ac the population level. Inclusion of information regarding in-

dividuals provides the potential fot better fishery models in the sense that it

provides stronger induccive arguments by analogy.

As models becoae aors realistic, they could becoae hopelessly detailed, re-

'qu«ing predigious quantities of data. Hov far should this process be carried7

Part of the ansver co this question is chat che mode] be ac least of sufficienc de-

tail to include the aechanisas responsible for all behavior of interest. A second

part of the ansuer is that uodels should be couples enough to ailou use of existing

data of relevance e.g., data concerning individual grovth, aortality and reproduc-

tive rates!. In uany practical applications staple uodeis are used because of the

alleged paucity of data. An alternative approach that appears nore reasonable

uould be to exasine the effects of having nore data and then to decide uhether to

gather tbez on an econmic basis see Discussion! ~ The point here is that current

fishery uodels are oversiuplified so that in uany cases they do not include all

available data, cannot exhibit the behavior observed in real populations, and tend

to lead ~ coaplacent attitude regarding hou ouch is actually knmm about a popula-

tion.

Xn the reuainder of this paper I discuss hou age-, size-specific uodels uay

better serve fishery analysts and uanagers. I vill shou that these uodels repre-

sent population behavior nore realistically than classical fishery uodels. In doing

so, I hope to draw sore uatheuatical attention to these aodels in the future.

AGE-! SIZE-SPECIFIC MODELS

Age-, size-specific uodels are those that "keep track" of the nuubar of indiv-

iduals of each age or size or both! in a population. These uodels appear in sev-

eral different fores in the ecological literature depending on the discrete or con-

tinuous nature of tine and age, and on uhich variables are of prinary interest. The

Leslie uatrix vas an early discrete tine, discrete age population nodal Leslie

I94$!. The videly-wed concepts of fertility tables and life tables is a continu-

ous ttue version of this nodal Lotka 192$!. A continuous-tine, continuous-age

version of this nodal, the Von Poerster equation or in uathesatical physics, the

continuity equation! ~ vas introduced as a population nodal by Von Poerster �9$9!.

Sinko and gtreifer �967! developed a sinilar nodal in tens of continuous tine and

both age and size. Changes in the nuuber of aniuals at each size and age vith tine

uere deteruined by grouth, uortality, and reproductive rates vhich in turn depend

on age, size. density, and environuent.

An tuuediate probleu in the use of uodels in fisheries is deciding vhich var-

iables associated uith individuals ~ .g., age, size! are necessary to describe pop-

ulation behavior. This decision can be based on tuo considerations: a! choose

the individwl variables that are of coaaercial significance, and b! choose those

individwl variables that uost critically affect population dynauics. llith regard

t'o the first, the individual characteristic of uost luportance coaaercially is in-

dividual size length or ueight!. Hath regard to the second, dynauic behavior of

populations is deteruined by individual grouth, reproductive and nortality races.

A reviev of these vital rates in exploited aquatic populations has shovn that they

usually depend heavily on size ~ less closely on age and also on an inherent varia-

tiou among individuals Socsford 1978!. U!timately, the choice of variables de-

pends on the nature of vital races for the specific population being modeled.

Host of the results for this class of models have been obtafned for age-speci-

fic models. However, in aost aquatic individuals vital rates depend on size racher

than age. ln addition, growth rate, at least in !uvenf les, is density dependent.

These tvo coadicfons require a model chat at least includes a size variable co de-

scribe population behavior.

BEHAVIOR Aif D OPT MLL POLICY

Qhffe age-specific models have been in use for aany years, very liccle is

known about behavfor and optimal fishery policy for realistic non-linear age-,

sfze-specific models. This section is a qualicacive description of the salientfeacures of some recoat results for age-, size-specific aodels. Local stability

of aa age-specific aodel vfch density-dependenc recruitaenc is discussed first to

provide che basis for a global analysis of an age and size-specific model with

density-dependent recruitaenc and grovth race. These are folloved by some results

regardfag optimal ffshery policy for ~ general density-depeadeat, size-specific~ odelL. geologically relevant behavior of these aodels is stressed rather chan a

complete view of the mathematical characteristics.

lehavfor of linear versions of age-specific models e.g., che Leslie matrix!

is well known Lotka 1925, Leslfe 1945, Keyfitz 1978!. The dominant solutfon is an

exponeacial reproductive rate chat increases vich tine unstable about zero! or de-

creases vith tiae stable about zero! or under fortuftous circumstances remains

coascant!. Since the essencial mechanisms chat prevent populations froa going to

zero or fafinity are lacking fn this model, its behavior fs unrealfstfc. Ic is,

therefore, of limited usefulness in praetfeal problems.

gehavfor of non-linear age- or size-specific models has been analyzed for

only a fear cases of interest. Por age-specific aodels vith density-depeadenc re-

eruitmeat, early results vere obtained through siilation. Ricker �954! estab-lished che fact chat the slope of the stock-recrufcaeat curve at the replacement

poiac had ro be greater than ainus oae for stability of a ~sfn le age-class popula-

cl ~ lle thea eetea thee l halatc l ~let l. a wlaaa popel tc ~ c htllcp

vas possible vhea this slope vas less chan ainus oae. A second observation by

Rfeker �954! that vas lacer supported fn other simularions by Allea and Easasfbwakf

�974! and Keashutkia �964! vas that fishing made populatfons nore stable.

Socsford and Qfckham �978!, in an attempt to better understand, cycles in the

northern Caiffornfa Dungeness crab fishery, investigated these tvo issues further.Theft nodal is a continuous-tfae, coacinuous-age, age-specific model fn vhich ce-

crufcaeat vas the product of reproduccion aad a survival function that reflects

10

survival to recruitment. Each of the latter tvo depends on a veighced sum integral!

of older individuals in the population. This model is similar to those of Ricker�954! and Allen and Basasibvaki �914! except that reproductive rate and densicy-dependent survfval do noc necessarily depend on che saaa vefghted sua over olderage classes. There fs no ~ priori reason to expect that the mechanism responsiblefor densicy-dependent recruicaent e.g., cannibalism! vill depend on older indiv-iduals in che same vay chat fecundity does.

Equilibriua conditions vere determined and local stability vas analyxed forthe aodel lineariced about equilfbrium. Stability eondicions are fn a form thatilluminaces the forementioned aultfple age-class and fishing versus scabflfcy fssurm.

Stability for the compensatory aeehanfsas of fnteresr. depends on tvo nuabers: K,the "noraalixed" slope of the recruitment survival function at equilibrium, and K',a nuaber thee depends on the relative influence ac each age of older animals onrecruitment e.g., fecundity and propensity to cannibalise ac each age!. The aodelis stable for K i K' decreasing oscillation! and unstable for K < K' increasing

oscillation!.

Since K depends on f and the equilfbrfua level, it doesn't change vith theadditfon or removal of older age classes unless equilibrium level changes!. The

value of K', hovever, responds sore directly to changes in age structure. Valuesof K' for a specified model can be obtained numerically K' fs the value of K forvhich the real part of che dominant eigenvalue is aero!. From nuaerical solucfons

under several different conditions and analycical solutions for tvo cases, the be-havior of K' ean be roughly described. The value of K' corresponds to Rfcker's�954! result for the single-age-class case a slope of -I!. As che number of age-classes increases, K' tends Co decrease. An approxiaate solution for K' shovedthat it increased vith che ratio of the aean age of influence of older animals on

recruftaent to the standard devfacion. Thus a narrov, peaked influence of older

~ niaals on recruitaent! over age is generally less stable than a broad, flat in-

fluence.

These results support the observation chat che slope of the stock-recruitaent

curve can be less than minus one as the model changes froa a single to a mulriple

agemlass model. Rovever, they do not support the earlier conjecture chat fishingstabflfses populations. Age- or sire-selective fishing removes anfaals of older~ ges, vhich narrovs the influence over age of older aniaals on recruitmenc and canmake the populations unstable. for exaaple, in several cases of a aodel of thenorthern California Dungeness crab population sfxe-selective fishing caused K' tobecome more positfve, thus making the populacion inherencly less stable.

In addition to che fact that fishing can aake populations unstable, a second

result of this vork is of practical importance. Since stability criteria can besimply expressed in terms of K and K', stability characteristics can be measured in

the field. 'Me are currently engaged in roaparison of different biological mechan-

isms e.g., cannibalism, an egg-predator wora whose population size is proportionalto the number of females, intra-specific competition for resources, etc.! to de-cermine vhich fs responsible for ehe cycles in the northern Calffornia crab fish-ery. To decermine vhether or noe each of these is the cause of cyclir. behavior,ve are essentially determining values of K and K'. Of course, che actual mechanismof cycles is more complex than a simple linearized analysis vould indicate, but

nonetheless the change in survival vith density aad relative influence of older

individuals on recruitment are key parameters eo be determined ~

Relaced analyses of similar models in a fishery context include those ofAllen and Sasasibvaki l974! and Levin and Goodyear �979!. Allen and Sasasibvakf,

using a discrete time and age version of the above model derived stability condi-

tions for some cases. Levin and Goodyear �979!, also using a discrete time aadage model vfth a Ricker stock-recruitment relationship, have examfned the effect

oa stability of mortality and reproduction at older ages. An important result fs

thee as mortality rate of older individuals is increased, it can first increase,

chen decrease stability. Thus even fishing that is not age- or size-selective canmake a population less stable. This result differs from a scatemene fn Socsford

and Vfckham �978! to the efface ehae non-selective fishing alvays stabilizes a

papulation. This statement fs in error.!

There have been far fever attempts at analysis of global stability of densfey-

dependent, age- or size-specific models. Botsford �980a!, fn aa attempt co explainche protracced decline of several exploited populations, aaalyzed a model similareo the above ~del of Sotsford and Qfckham �978! except chat ic vas size-specific

and included density&ependent grovth race. The populacions of specific interest

vere the Dungeness crab population of cencral California, the Eurasian perrh popu-laeioa of Lake Windermere in England and che Pacffic sardine off California. These

populations have three characteristics fn coazaon: a! follovfng exploitatfon, a

decline co lov levels from vhich chey have not recovered, b! an increase in in-

dividual grovch rate during or folloving the declfae, aad c! a compensatory effect

oa recruitmenc chat depends on older members of che population.

Fram the results dfscuased prevfausly boch equilibrium and stabiliey depend

an the relacive influence on recruitment! of older animals at each age. The ac-

tuel biological influeace on recruitment probably depends on size rather than age.

For example, fecandfty and cannibalism would most likely depend more on size than

on age. lf the aceual mechanism vere size-dependent and the grovch race changed,

chen equilibrium and stability could change appreciably.

Sotsford �98Da! aaalyzed several aspeces of global stabflicy. He determinedchat an increase fn growth rate could lead to lover equflibrfum rerruitment. ThisIov« recruitment level vas due ta individuals reaching, "cannibalistic size" in tvo

12

years at higher gtovth rate rather than four years ac lover grouch rate. If theyare exposed to constant aorcalfcy pressure during each of chase years, aors of thea

survive to "cannibalistic size" per recruit! at the higher grovth race.

After establishing the possfbilfty oi different equilibriua recruitaent rates

corresponding to different individual grovth races, a aodel of che central Calif-

ornia htngeness crab population vas siaulated to see ff populacions could actuallysvfcch froa one equilibriua level to another. The siaulacions included gradually

increasing fishing aottality for Cuo kfnds of exploitacion: in one only aales arefished and fn the ocher boch asles and feaales are fished. Por the aalesmnly case,

the fished systea could be pushed into a different doaain of attraction, correspond-

ing Co lov equilibrfua tacruitaenc and high individual grovth rate, by several years

of poor recruftaenc. That che unfished population vas less susceptible co this in-put supports the contention of Hurphy �968! regarding relative faperviousness offtetopatous and seaelparous populations co environaental fluctuations. Pot checase in vhich boCh aales and feaales are fished, this sane type of behavior vas ob-

served. Bovever, an additional type vas also present: the gradual increase infishing pressure eventually led co a point ac vhfch recruitaant decreased and in-

dividual grovch rate increased fn a regenerative fashion uncil a nev equilfbriua

vas reached at lov recruitaent and high grovth rate.

Hhether ot not this kind of aechanfsa fs actually che cause of depressed lev-

els reaains to be detetained. Hovever, the aain point here is that ic is a possible

~ echanfsa that has not yet been considered as a potential cause of these declines.

Purthezaore, flluainacion of this aechanisa is not possible vichout age-, size-

specific aodels. An faportant aspect of this aechanisa is thee it refleccs earlier

cadence on resilience of ecosysceas Rolling 1973!. Here is ~ possible exaaple in

vhich aan has reduced populations to a lov level froa vhich recovery vill be dfffi-

ult, ic not fapossible.

Opcfasl harvest policy of age- or size-speciffc aodels has been addressed

largely vich che use of che linear Leslie aacrix aodel. The results obtained areof liaited value because of the linearity of the nodal. Recause the nodal is lin-

ear, only ~ population vith doafnant eigenvalue greater than one i.e., an increas-

ing population! could yield a sustained harvest. Since the populacfon is increas-

ing exponentially, optiaal policy involves vafting until just before the and of the

planning period, then harvestfng che vhole population. The opcfaal harvest problea

vith this nodal has been aade aors reasonable by appending constraints such as

aaintainfng a constant level! Reddingcon and Taylor 1973, Rorres and Pair 1975,Rorres 1976!. The liaiCations of chio approach and this nodal ara discussed in

geddington and Taylor �973!, Handelssohn �976!, Reed �979!, and Rotsford �980b!.Hors realistic results can be obcained by including the actual biological "con-

straints" on populatfon grouch rather than attaching artificial constraints to an

unrealfstfc aodel.

Botsford �980b! formulated the optiual harvest probleu in terna of a size-

~ pacific model with ~ density-dependent growth and recruitment rates. These rates

could depend either on a weighted sun of other individuals in the population sim-

ilar to the crab model discussed above! or on a food variable that in turn depended

on consumpticsa by the population. Hecessary conditions for values of size limits

and fishing effort that maximized discounted future profits were determined.

These results reduced to those obtained by others for the special cases of the lin-

ear, age-specific model ind the single age class nodal.

The resulting necessary conditions were similar to those obtained by others,

but extended earlier results in the direction of realism and completeness. Clark

et al. �973! described optimal policy for a Beverton&olt nodal of single age

class as a balance between the rate of increase in value of stock due to "economic"

reasons i.e., discount rare times net biovalue! and the rate of increase in value

of a population due to biological causes growth and mortality rates for the age

class!. 6 corresponding expression can be interpreted similarly for optimal policy

using the logistic nodal, but the biological variables involved r and K! are not

as easily associated with real populations as individual growth and mortality rates.

The former model is not complete in that it lacks a self-sustaining, reproductive

pz'ocean, while the latter is vague and less realistic. The model used in Botsford

�980b! extends the realism of the terner to include the density-dependent, self-

sustaining nature of the Latter. The corresponding necessary condition is again a

balance between the discount rate times net biovalue and rate of increase of stock

due to biological factors. However, in this case the latter is the sum over sizes

to be fished of rates of increase in biovalue due to individual growth, reproduc-

tive and mortality rates, as well as the negative values due to food consumption

and density-dependent effects of older animals. Decisions made on the basis of

this expression include a much more detailed consideration of individua1 growth and

~ etabolism. For axmpie, the effect of changes in metabolic efficiency with age

Paloheimo and Dickie 1965! and the effect of changes in relative market value ver-

sus future reproduction on harvest policy can be evaluated.

The results also showed that optimal policy solutions differ radically from

those obtained with sinplez' models. For example, optimal policy for the logistic

is for certain parameter values! a policy of constant effort and population level

Clark 1976!. However, uhan the realism of explicit multiple age classes with

density

14

policy tried. Halters �969! numerically obcained the maximum yield of models vith

fndividual growth and scock-recruitment relacionships for two cases: one with and

one vichouc fishing selectivity. In the former case, a constanc policy vas optimal

vhile in the latter a pulse fishing policy was optimal. Pope �973!, using a sim-

ilar model with no gear selectivicy, also numerically determined optimal yield to

be a policy involving pulse fishing. Cetz �979!, also using a model that includes

s ~ Cock recruitaenC relationship and individual grovth, recently obtained a numer-

ical solucion for aaxfaua yield vfch a description of harvesting similar to chat

used bere; constant fishing aortalfty for all sixes greater than a mfniaua size to

be determfnel. In hfs solution fishing mortality and ninfnum size vere assumed

constanc vith cia».

ANTARCTIC KRILL, BALEEN WHALE FISHERIES

14sltfspecfes fisheries presenc problems chat have noc yet been addressed in

texas of age or size speciffc models. Nay et al. �979!, in their analysis of the

Antarctic fisheries based priaarily on popularion-level, logistfc-type models

pointed out che necassfty of bringing aors «oaplex models to bear on these problems.

In this section, several probable dffferences between the results of 1 ay et al.

�979! and results of an age- or size-specific modeling approach based on che be-

havior of inc ividuals are discussed.

Consideration of age-speciffc characteriscfcs of individuals vould probably

change the estimate of hov much kriII is available for harvest. The current esti-

aace, 150 million tons surplus krill per year, is based on estimates of food con-

~ uaption races based on physiological energetics of vhales Lockyer 1972! and es-

tiaaces of che difference betveen che tocal vhale bioaass b'afore exploftacion and

present biomass 0 ackintosh 1973, Laws 1977!. Basing the estimate of surplus krill

on a biomass deffcft and percentage consumption rate per unit biomass aay incur sig-

nificant error. Since the age and size structure of che populacion have changed

vfth exploitation, che total annual aetabolic requirement per unft biomass may also

have changed. This fs due co Cvo causes. One result of exploitation is a shift fn

sfxe distribution tovard smaller sizes. For most aquatic anfaals the race of food

consumption per unit body veighc is greater for saaller, younger individuals and

decreases as the individual grovs older. Thus a change fn size structure vould

change consuaption rate per unit veight. A second cause of change in consumption

rate per unit bioaass is increased consumption rate itself. Shen fish are provided

a greater raCfon of food, they often eat nore up to a point! and grov faster.

Mhales also grov fencer in response to greater food availability. In addition,

they shov earlier maturation and exhibft higher pregnancy rates. Boch Fin and Sei

whales have been observed co mature at approximately half the age at vhich they

matured prior to exploitatfon I.ockyer 1972, Lockyer 1974! and higher pregnancy

15

rates have been observed fn these Cvo specfes and the blue whale Laws 1977!,

fs likely that these vhales are consuming nore food per unit veighc chan they vere

earlier.

A second vay fn vhich age- or size-specfffc models vould dfffer from the

simpler approach is in the actual behavior of the nodels themselves. These could

be significant, qualicacive differences. For example, in equation �! of Hay et al.

�979! the race of decrease in krill due co predation is proportional co the prod-

uct of sizes of the tvo populations. 'ln actual fact, hovever, the consumption rate

of an iadividual vhale is aot based on the cotal number of krfll in the Antarctic,

but rather oa che density of krfll available to hfn or her!. The functional de-

pendence of consumptfon rate on prey density vould probably increase vich prey

density to a maximum rate as in lvlev 1961!. The point here is chat at krill

levels that are aoc extremely lov, the last tern in equation �! of Hay et al.

�979! vould be proportional only co whale population size. The effect of this

change oa results could be a considerable change in static and dynamic aspects.

A third probable difference fn results is in dynamic behavior of the system.

Ac least some vhale populations apparently respond to changes in food level by

changing reproductive parameters. The response of the population a change fn

population size! toe chaage in environment food level! occurs after a lag equal

to che time required for aa increase in reproduccioa to be felt as an increase fn

populatfoa of vhales of significant size f.e., adults!. With increasing exploita-

tion of whales, this lag is decreasing, hence response time or "characcerfstic re-

turn time" fs decreasing see Hay ec al. 1978, p. 241 for a discussfoa of this

phenomenon!. Thfs efface vould act approximately fn opposition co the nechaaisms

that leagthen respoase time discussed fn May et al. �979!.

A fourth difference rhac fs nore qualftacive chan che others fs simply an

facreased emphasis oa measurement of results ac the level of the individual rather

than at the population level and inclusion of these fn manageneac decisions. Re-

ceat changes fn individual vhales and individuals in predator populations e.g.,

che lover age of sexual maturity of crab-eater seals Laws 1977! can be evaluated

quantitatively in terms oi their effects on prey populacions consumption rates! as

well as their ova population dynamics. The International Whaling C~ission cur-

rently includes age structure in chair population models Allen and Kirkvood 1977!.

perhaps these could be expanded to include mulci-species consideracions. These

could be based, at least partially, oa currenc knovledge of physiological energeticsof whales Lockyer 1972, Brodie 1976!.

In s~ry of this section, ft is impossible to definitely stace ac this cine

chat simple models vill lead us astray in the Antarctic. Hovever, there are addf-tioaal data available that could be used, and an approach that favolved nore real-istic models would emphasize gachering nore data ac the individual level. The

fnportant point here fs that, vhile lagistic-type nadels nay provide canvenient

metaphors, we should not allov chen co lull us into such a conplacenc fraus of nind

that ve believe Chat they realistically predict the future response of this systen,

Par exanple, ve have renoved a predator and canpeticor fran the Antarctic whales!,

and prey abundance as vali as conpeticor abundance ~ .g., seals! have increased.

This state of affairs nay tenpt us co predict chat ff we vere to stop fishing vhales

che syscen would return to ics ptevfous state. Hhecher or not it does is gofng to

depend on predator-prey and conpeticor inceraccions at che individual level. Ac-

conpanyfng any nev changes in our effects on the systea vith detailed nanitoring

of changes in age- ~ size-structure and the characteristfcs of individuals aeons a

better policy than changing policy, chen vaiting to observe populatfon level ef-

fects.

DISCUSSION

The above exanples of real and pro]e«ced differences between results obtained

fran sinple, populacian-level nodels and nore canplex age- or sfze-specific nodels

Chat include behavior at the fndfvidual level praved the basis for argunents for

fncreased use of the latter. Oae porc of the accunulatfon of nev knovledge in sci-

ence involves cescfng alcernacive explanacfons for observed phenonena. !f all pos-

sible a ~riorl explanations are noc included, che inference achene is linited.

!lore realistic nodels are needed to provide all possible explanations. Por exanple,

age- and size-specific nodels led co a nev possible explanation for depressed equi-

libtia that had noc yet been considered in any specific instance of this phenonenon.

Pornulation of opcfaal palfcy can be yfeved sfnilarly. If all possible occurrences

e.g., depressed equilibria, fishing causing inscabflicy! are not included in the

set of candidate policies, non-opcfnaI policies nay result e.g., instability, con-

scanc racher than pulse fishing!.

There are several specific cricfcisns of nodels as conplex as those being

proposed here. One of these is that they require too such data. Shaeftet and

geverton �963! actually used the anount af data available as a criterfan for choos-

ing becveen the logistic and che nore conplex Bevatron-!lait nodal. This issue can

be evaluated by discussing the actual gain af adopting each nodal. Plots of real

dace fn che foas of yield versus effort for a logiscic nodal fncrease fton the

otfgfn co a claud of points chat nay or nay not decrease as yield increases. This

~ ay appear co be a "good fft" for tvo reasons: a! at lov levels yield increases

vich effort, and b! yield vill eventually decrease as effort increases further.

Neither of these reflects such about population dynanfcs.

Oa the other hand, in use af nore complex nadels one can foIlov the strategy

set fatth by Sinko and Streifer �969! for age-, size-specific nodels and HacFadyen

�973! for ecological nodels in general. Canpfex nodels can be construcced based

l7

on available data and reasonable assumptions vhere data are lackfng, and can eben

be used eo determine behavior and optimal policy. Sensitivity of ehe results eo

poorly knovn parameeet values «an chen be determined and more data collected if

needed. This approach seems fat better rhan adopting a less than realistic model

because of liaited dace. In either obtaining ehe necessary data or ae lease re-

~ lising chat not enough dace exfsc eo solve the problem, one avoids che complacency

to uhich logistic-type models can lead,

A secoad crfticfsa of complex aodels is that ebey are eoo coaplex eo under-

stand. Rothschfld and Suds �977!, fn discussing models aors complex than the

logistic. recently argued chat "the coaplexicy of chase sore coaplex aodels, vhile

~ till less coaplex chan narute, soon oucserips coaprehension and then at this point,

the complex model fs, to the huaan mind, just as unfathomable as nacure itself." I

disagree and would contend tachet chat attempts to understand more realiseic models,

though they aay be complex, vill lead to increased understanding of the complex

processes fn populacfons.

A third crfeicfsa fs thee aacheaacical analysis of complex models is eoo dff-

ficult, chat is, not as many techniques are available for analysis of non-linear

partial differential equations as are available for simpler mathematical systeas.

awhile this aay be currently true, I don'e think tha» enough attention has been paid

by aathmsticfans eo use of realistic, complex aodels in solving real problems.

This is, in essence, ay purpose in presenting che ideas in this paper. I hope to

elicit greater attention co these aodels in the future.

ACKNOWLEDGMENY

Thfs uork is a tesulc of research sponsored by HOAA Office of Sea Grant,

Department of Comerce, under Grant PHOAA-HOI-184 R/P52. The U.S. Government fs

authorised to produce snd distribute reprfncs for goveneencal putposes notvfth-

standing any copyrfghe notation Chat asy appeat harem.

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