INTER-AMERICAN TROPICAL TUNA COMMISSION COMISION INTERAMERICANA DEL ATUN TROPICAL


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1 INTER-AMERICAN TROPICAL TUNA COMMISSION COMISION INTERAMERICANA DEL ATUN TROPICAL Bulletin - Bolet~n Vol. 16. No. 3 TUNP0P, A COMPUTER SIMULATION MODEL OF THE YELLOWFIN TUNA POPULATION AND THE SURFACE TUNA FISHERY OF THE EASTERN PACIFIC OCEAN TUNPOP. UN MODELO COMPUTADOR DE SIMULACION DE LA POBLACION DEL ATUN DE ALETA AMARILLA Y DE LA PESCA ATUNERA EPIPELAGICA DEL OCEANO PACIFICO ORIENTAL by - por ROBERT C. FRANCIS La Jolla. California 1974

2 h U n n U _ U u u unnu u h n h u u _ n u _n n u h u U u CONTEN~rs - INDICE ENGLISH VERSION -- VERSION EN INGLES Page ~~S~~~~~ ~-~ INTRODU ~~I0 N nn THE SIMUL~TION MODELn h n n u u u u' u u 239 Cornputer program structure n EXAMPLE RESULTS. u n n n 244 SENSITIVI~Y ANALYSIS n_n_ h n_n_ u SUMMA~Y AND ~ON~LUSION n n n n A~KNOWLEDGMENTS u n_ U n n _n on nn n u 248 FIGURES - FIGUR~S h n - - u n_h. 249 T A~LES - TABLAS h u u u 257 VERSION EN ESPANC.L - SPANISH VERSION Pagina EXT'RACTO IN~R0 D U CClON h u U MODELO DE SIMULA~ION_h_U _u_.. Uh Introduccion. u u u_ h h 259 u 263. u u 263 Ecuaciones Estructura del programa de computo..; h EJEMPLOS DE LOS RESULTAD()S n u_h ANALISIS DE SENSIBILIDAD. u. U SUMARIO Y ~ONCLUSION_uu _h RE~ON O~IMIENTO u -- u h h u. u_u u_268 h 268._UUU hh u 273 u_273 LITERA~URE CITED BIBLIOC~RAFI~ ~IT~DA u u_uu u t\PPENDIX TA~LES - TABLAS DEL APENDICE u. uu_uu uu 276

3 TUNP0P, A COMPUTE,R SIMULATION MODEL OF THE YELLOWFIN TUNA POPULATION AND THE SURFACE TUNA FISHERY OF THE EASTERN PACIFIC OCEAN by Rubert C. Francis ABSTRACT Mathematical documentation of TUNP0P, an age-structured computer simulation model of the yellowfin tuna population and surface tuna fishery of the eastern Pacific Ocean, is described. Example runs of the model are presented and discussed, and the sensitivity of the model output to changes in various parameters is examined. INTRODUCTION The staff of the Inter-American Tropical Tuna Commission (IATTC) is charged with the responsibility of conducting scientific studies on the tropical tunas inhabiting the eastern Pacific Ocean in order to recommend measures which would result in the maintenance of these stocks at levels which support the maximum yields on a sustained basis. These studies have covered a number of species of tuna but have concentrated most heavily on yellowfin (Tbunnus albacares) and skipjack (Katsuwonus pelamis). From an early date it has been clear that fishing affects the abundance of yellowfin, but not that of skipjack. Models formulated by the Commission staff indicated that in the early 1960's the catches of yellowfin in the eastern Pacific were in excess of the estimated maximum amount that this stock could support on a sustained basis. In 1962 the Commission delineated an area within which recommendations for management of this stock would apply, the Commission Yellowfin Regulatory Area (CYRA, Figure 1). In 1966 a conservation program on yellowfin was established within the CYRA. This program, modified annually, is still in effect. Review of the motivation and evolution of the yellowfin management program is given by Joseph (1970). The purpose of this paper is to provide the basic mathematical documentation of a computer simulation model of the yellowfin tuna population and the surface tuna fishery of the eastern Pacific Ocean. In addition, example runs of the model are presented and discussed, and the sensitivity of the output from the model to changes in various parameters is examined. The objectives of the model, hereinafter referred to as TUNP0P, are threefold: (1) to investigate the yellowfin tuna population structure in the eastern Pacific Ocean; (2) to investigate the causes of changes in both the magnitude and size composition of the yellowfin catch in the CYRA 235

4 236 J:i-'IlANCIS during recent years; (3) to stud)' the potential yield of yellowfin in the CYRA under various management schemes. Mathematical models which have traditionally been used for stock assessment and management by fisheries scientists can be categorized into two groups; stock production models and age-structured models. Stock production models are used to attempt to relate directly the instantaneous rate of change of the stock biomass to the existing biomass level and the amount of fishing effort being applied to the stock at any instant in time. This type of model enables the investigator to estimate the proportion of the stock removed by a single unit of fishing effort, the intrinsic ability of the stock to increase and the maximum size that the stock could theoretically attain. Consequently this type of model can be used to predict the theoretical maximum yield that the stocks can support on a sustained basis, as well as the average yield under any given fishing intensity. Stock production models were first used in tuna research by Schaefer (1954, 1957) and were refined and expanded by Pella and Tomlinson (1969) and Fox (1970). The biological motivation for the logistic form of the stock production model is discussed by Pielou (1969). Models of this type are relatively easy to apply since all that is required as data is a time series of catch and effort statistics for the stock under investigation. Three key population properties upon which the validity of a production model analysis rest are: 1) The population must be a distinct self-sustaining unit. The population must tend to a stable size and age distribution at each level of fishing effort. 3) Each standard unit of effort must remove, on the average, a constant fraction of the population. In other words the probability that a given unit of effort encounters a given unit of the population is constant over time. Thus at any fixed level of population biomass, each unit of effort is assumed to be applied at a constant intensity and with a constant distribution over the exploited portion of the population age structure. In addition a constant fraction of the population is assumed to be available and vulnerable to the fishery at any time. It is of interest to examine the eastern Pacific yellowfin fishery over the past ten years as it relates to these properties in order to determine how well suited for production model analysis this group of fish is. With respect to property 1), the fishery has expanded in area from a predominantly coastal fishery during the early 1960's to a present day fishery which covers a great proportion of the CY~RA and areas to the west of the CYRA (Figure 2). Age..structured analyses (summarized later in this paper)

5 TUNP0P, A COMPUTER SIMULATION MODEL 237 indicate that the proportion of the stock available to the fishery has increased with this offshore expansion of the effort. Thus there is some question as to whether the stock now being exploited is the same basic unit as that which was being exploited in earlier years. With respect to property 2), the average size of fish in the catch has changed significantly for similar "near equilibrium" yields. In 1966 the catch of yellowfin within the CYRA was 182 million pounds and the average weight of fish in the catch was 22.4 pounds, whereas in 1967 the catch was 179 million pounds and the average weight of fish in the catch was 19.3 pounds. In 1970 the yield of yellowfin was 285 million pounds and the average weight of fish in the catch was 31.5 pounds whereas in 1971 the catch was 226 million pounds and the average weight of fish in the catch was 27.7 pounds. In 1967 and 1971, the t\vo years of lesser average weights, large quantities of skipjack tuna were caught in the CYRA. During a year of good skipjack catches the fleet is concentrated more along the coastline of Central and South America where the skipjack concentrate and where surface schools of small yellowfin are encountered. Under these conditions the average weight of yellowfin in the catch tends to be relatively low. Contrastingly in 1966 and 1970 the fishery was distributed more in the offshore region of the CYRA where larger yellowfin are encountered predominantly with schools of porpoises. Under these conditions the average weight of yellowfin in the catch tends to be relatively high. The following table gives the fraction of the total number of successful sets on yellowfin which were porpoise associated sets for each of the four years. Years Fraction Porpoise Sets It thus appears that the average size of yellowfin in the CYRA catch is strongly affected by both the geographical distribution of the effort and the fishing mode (porpoise or non-porpoise) in which that effort is predominantly employed. Finally, with respect to property 3), the catchability, or vulnerability of the available population to the fleet, appears to have changed with the refinement of purse-seine techniques and the movement of the fleet offshore. If the CYRA is divided into two areas (Figure 3), area Al being the historic area of the fishery and area A2 the more recently exploited offshore area within the CYRA, then the following table of the percent of the total CYRA yellowfin catch and standardized effort within A2 sheds some light on the

6 238 FIlANCIS above-mentioned problem. The estimates of effort have been adjusted or standardized for "apparent" changes in efficiency due to changes in successful set ratios. Year Percent CYRA catch in A2 Percent CYRA standard effort in A o o It is apparent that the fraction of the total CYRA catch for A2 increased more rapidly than the fraction of the total CYRA standardized effort applied to A2. Assuming a unit stock within the CYRA, these data indicate that as the fleet expanded into area A2 the overall efficiency of the fleet, in terms of catchability, increased. It thus appears that production modeling techniques should be applied to the recent years of the eastern Pacific yellowfin fishery with some degree of caution. Whereas stock production models represent the dynamics of stock biomass as a single unit, age-structured models use the relationships among growth, mortality and reproduction of the individual elements that make up the population to simulate stock biomass dynamics. The basic mathematical theory for these models was developed by Baranov (1918) and refined by Ricker (1945, 1958) and Beverton and Holt (1957) in attempts to relate the expected yield from a cohort of fish as it passes through a fishery to the interaction of growth, mortality and age at entry to the fishery. This type of model is commonly referred to as a "yield-per-recruit" model. Unfortunately, the traditional yield-per-recruit models of Ricker and Beverton and Holt fail to take into account variations in recruitment to a population, and therefore cannot be applied to represent the dynamics of total yield from a stock of fish being exploited over a period of time. TUNP0P is an attempt at expansion of age-structured modeling to investigate the long-term dynamics of abundance and age structure in a population subject to age-dependent exploitation and variable recruitment, the yellowfin tuna population of the eastern Pacific Ocean. In recent years similar models have been produced for other species by Walters (1969), Tillman and Paulik (1971), and Fox (1973).

7 TUNP0P, A COMPUTER SIMULATION MODEL 239 THE SIMULATION MODEL Introduction The primary function of a population simulator such as TUNP0P is to describe accurately the manner in which population biomass varies over time. There are five basic mechanisms which operate in an exploited fish population to produce biomass flux: recruitment, growth, natural mortality, fishing mortality and graduation (Fig. 4). In a given unit time period these mechanisms operate as follows in TUNP0P. Recruitment, the addition of fish to the youngest age class represented in the simulator, is assumed to occur in a "knife-edge" fashion at the beginning of the unit time period. The magnitude of the recruitment can either be computed as a function of stock biomass during some previous time period or can be specified as input to the model. It should be noted that recruitment to the simulator can occur at the beginning of any unit time period over which a simulation is being made. In the examples presented later in the paper two annual cohorts, recruited at half-year intervals, are simulated simultaneously. In this case a unit time period is a quarter of a year, and thus recruitment occurs at the beginning of every other unit time period. Gross growth is the gross biomass added to the population by the process of individual fish growth during the unit time period. Thus gross growth refers to the total individual growth by both those fish which survive to the end of the unit of time and those fish which are removed from the population during that unit time period. For the latter it includes growth only up to the time of removal, of course. Individual fish growth is represented by empirical growth functions which are input to the model. Separate growth functions can be utilized for separate cohorts as they pass through the fishery. Thus in the examples employed later, separate growth functions are used for the two annual recruitment groups or cohorts whose joint dynamics are being simulated. Natural mortality refers to losses from all causes other than fishing and graduation from the oldest age class. Natural mortality rates occur as input to the model. Fishing mortality refers to losses from fishing. The ability to harvest fish depends on the interaction between the availability of the population to the fishery and vulnerability, or effectiveness, of the effort in capturing the available population. If a given fish is unavailable to the fishery at the beginning of the time unit it remains unavailable for the entire unit. Thus availability is defined in the sense of Ricker (1958) and Yamanaka (1961). During any unit time period the fish which are present at the beginning of the time period can be divided into two groups: unavailable and available, the former being acted upon only by natural mortality and the latter being acted upon by both natural and fishing mortality during the time period. Vulnerability, which when multiplied by fishing intensity, or effort, produces the fishing mortality rate, refers to the likelihood that a unit of effort encounters an available unit of the population in a unit of time. It is commonly referred to as the catchability coefficient. Finally,

8 FR,ANCIS graduation, the loss of fish from the oldest age group represented in the simulator, is assumed to occur in a "knife-edge" fashion at the end of the time period. Once a fish graduates from the simulated population it is no longer available to the fishery and is no longer considered in determining future recruitment levels. The population of yellowfin is partitioned into 18 quarterly age classes. The first age class is determined by the youngest age at which fish appear in the eastern Pacific yellowfin fishery in significant numbers. Recruitment into the simulated fishery occurs when fish enter this age class at the beginning of their second year of life. An average recruit is assumed to weigh 3 pounds and have a fork 'length of 40 em. The last age class contains the oldest fish which appear in the fishery in significant numbers and for which age can be determined by analysis of modal progressions. Thus graduation from the simulated fishery occurs when animals leave this age class in the middle of their sixth year of life. An average graduate is assumed to weigh 135 pounds and have a fork length of 145 em. All parameters are held constant over a unit time period, that interval of time that any individual animal spends in an age class. Thus in this case model parameters can only be varied on a quarterly basis. However, within a unit time period, computations are made on a continuous time basis. Thus to upgrade the time resolution of the model the population would have to be divided into more age classes, reducing the unit time period. The TUNP0P program is written so that the user can vary the length and/or number of time periods without changing the program. However the length of time that an animal spends in an age class must coincide with a unit time period in the simulator. Equations Since the basic structural unit of the simulated population is the age group, the algebraic manipulations which make the model function can be best understood if the computations performed to advance a group of fish through the unit time period (in this case a quarter of a year) that it spends in a given age group are revealed. Thus let {t,t + 1} unit time period commencing at time I (an integer) and ending at time t 1 during which a given cohort of fish is assumed to be of age i, Assuming constant rates of growth, natural mortality, catchability and availability as well as constant application of effort the unit time tj...c......,,'""'. the state variables are computed as follows: (t 1) number of fish entering age group i 1 at time t 1

9 TUNP0P, A COMPUTER SIMULATION MODEL 241 where Z, (t) instantaneous unit time total mortality rate on age group i during {t,t + 1} F i (I) instantaneous unit time fishing mortality rate on age group i during {t,t + 1} qi(/) catchability coefficient on age group i during {/,t 1} (INPUT), f(/) number of standard days effort during {t,t I} (INPUT), M i Pi(t) instantaneous unit time natural mortality rate on age group i (INPUT), fraction of N i (t) available to the fishery during {t,1 1} (INPUT). (2) u,(i) biomass entering age group i at time t where ui, (I) average weight of an individual at entry into age group i at time t (INPUT). (3) catch in numbers of age group i during {t,t 1} (4) catch in weight of age class i during {t,t + 1}

10 242 FRANCIS where Also note that G i (t) instantaneous growth rate of age class i during {t,t -1-1j In [llj i + 1 (t + 1) *]. ui, (I) _ R(t) N 1 (t) number of fish recruited into age class 1 at time t, In addition two more variables of interest are computed on each age group for each unit time period. (5) average biomass in age class i during {t,t -t- 1} Pi (t) e, (t) M i [ e ( G i (I)-Pi (t)-a1i) + (1 Pi (t) ) B i (t), [e(gdn-al) - 1 ] Gi(t) - M j (6) gross growth of age class i during {t,t + I} where total biomass added to age class i by fish which survive {t,t + I}

11 GG2 i (t) [t + 1 Jt TUNP0P, A COMPUTER SIMULATION MODEL 243 total biomass added to age class i by fish which die during {t,t 1} [ Pi (t) P, (t) + M,1 N, (x) ui, (t) (1) tx-t) - 11 dx (t)f,(t) M;]Nj(t) w;(t){pj(t) [G,~:;<t~z~:)(t) ~;;:I; Gj(t) 1 Zj(t) Zj~t)] (l-pi(t» [l+e~' ~J} if Gi(t) == Mi. o if z, (t) O. [pi(t)fi(t)) Nj(t)Wi(t) {pj(t) 1 1 ] "G i (I) r;(i) Zi (t) if M[ 0, r, (t) O. 1 ]} otherwise. ~J} Thus the population gross growth is simply the sum of the age-specific gross growths during that time interval. The reasons for computing gross growth are twofold. First it is of interest to look at the ratio of total biomass added to the population to total biomass removed from the population by both fishing and natural causes. Second, as TUNP0P may eventually be utilized as a population energetics simulator, a computation of gross growth must be made in order to subsequently compute the total amount of energy used by a population in the course of a unit time period.

12 244 FRANCIS Computer program structure Figure 5 is a flow diagram of the procedure which binds the model inputs and age-specific computations together to produce the desired model output. The input to the model includes the basic time parameters, fishing parameters and variables (age- and time-specific availability and vulnerability and time-specific fishing effort), natural population parameters (age-specific natural mortality, time-specific recruitment, and time and age-specific size (weight) at entry into each age group). Initial values for the state variables must be specified in order to activate the model. The output from the model includes age-specific data on numbers, biomass, growth and yield for each unit time interval over which a simulation is made. Also, the average weight of a fish in the catch and the ratio of gross growth to losses in weight due to natural causes and fishing are given for each unit time interval. Finally, the model produces annual statistics of age-specific catch in numbers and weight. EXAMPL]~ RESULTS Two example runs of the model were made to attempt to mimic a six-year ( ) history of quarterly age-specific yellowfin catch statistics in the CYRA. In run 1 quarterly variations in the level of fishing effort are taken into account in determining age-specific fishing mortality rates, whereas in run 2 these age-specific rates remain constant throughout the simulation. For both runs the model input parameters were estimated utilizing quarterly statistics on catch, effort and size of the fish in the catch (lengthfrequency data) for For each quarter of 1966 through 1971 the total catch of yellowfin from the CYRA was grouped into length intervals based upon data from length-frequency samples. For the purpose of this analysis two recruitment groups were assumed to enter the fishery each year. Modal groups from the length-frequency samples were r}e-c lrr... 'nrl to one of two categories, designated X and Y, on the basis of their recruitment dates into the fishery. Recruitment is defined to occur when a fish attains the length of 40 em (approx. 1 year old). Utilizing backcalculations by means of growth curves, fish which were estimated to have been recruited during the first six months of a year were assigned to the X group and fish which were estima.ted to have been recruited during the second six months of a year were assigned to the Y-group. Thus each quarter's catch is segmented into both X and Y age distributions. The concept of X and Y groups was introduced by Hennemuth (1961) although in this study the fish were assigned to the groups in a slightly different manner. The age-length relationships (slight differences between the X and Y groups) used to transform the length-frequency samples to agerouuencv samples (annual X and Y age groups) were estimated by Tom-

13 TUNP0P, A COMPUTER SIMULATION MODEL linson (personal communication). Estimates were then made of the quarterly age-(cohort)-specific catches for the years of concern by combining the age-frequency samples with the quarterly catch statistics. The Murphy technique (Tomlinson, 1970) was then used with these age-specific catch data to estimate the quarterly age-specific fishing mortality rates (F, (t) ) and year class abundances (N i (t)). Quarterly age specific catchability coefficients t«. (t)) were then computed by dividing the fishing mortality rates by the quarterly estimates of standardized fishing effort (/(t) ). The Murphy analyses were performed for the X and Y groups separately under the assumption that the instantaneous natural mortality rate (M) was constant at 0.8 on an annual basis for both groups. Simulations were made over a 24-quarter (6-year) time span ( ). Estimates of recruitment to the fishery, in millions of fish, derived from the previously mentioned Murphy analysis, are as follows: Year X-group Y-group Overall average These values are direct input to the simulations. Estimates of the quarterly fishing effort were derived from mean catch per unit of effort estimates for those 5 squares in the CYRA where predominantly yellowfin was caught. These values of CPUE were divided into the corresponding total quarterly CYRA yellowfin catch statistics to give estimates of total effective standardized yellowfin effort (Table 1). Thus when all of these estimates are employed in the simulation unmodified, the output consists of the exact catches obtained from the length frequency-catch statistics analysis, since the simulator is structurally similar to the basic Murphy model. In the first run of the model, mean values of age-specific catchability ('ii; i 1,...,18) estimated by the Murphy model were employed for the X and Y groups (Table 2). These values were then combined with the quarterly estimates of yellowfin effort to produce quarterly age-specific fishing mortality rates. In the second run of the model, mean values of age-specific fishing mortality rates (F i ; i 1,...,18) estimated by the Murphy model were computed and employed directly, again using separate X and Y group means (Table 2). In both runs all availability rates were equated to unity.

14 246 FR.A.NCIS Examples of the three basic categories of model output are given in Appendix I. These three are 1) a listing of all input variables and param 2) output from a unit time period simulation, 3) output from a year's simulation. All output is summarized by X and Y groupings as well as the combination of the two. The final table of the unit time period output the ratio of biomass removed from the population by fishing and natural death to the combination of gross growth plus recruitment to age class 1 minus emigration from age class 18. The final table of the annual output section gives the gross annual changes in population biomass due to five sources: Increases due to recruitment and growth, and decreases due to fishing, natural death and graduation. In addition the net annual change in population biomass is Graphs of the quarterly catch in number and weight and average in the catch are given in Figures 6, 7 and 8. Using the criterion for goodness of fit of sums of squared deviations, the following results were obtained: a) The sum of squared deviations of observed from simulated quarterly age-specific catch in numbers was 55 percent lower for Run 2 than Run 1. b) terly Run 1. The sum of squared deviations of observed from simulated quarcatch in weight was 54 percent lower for Run 2 than c) The sum of squared deviations of observed from simulated quarterly average weight in the catch was 74 percent higher for Run 2 than Run 1. From a linear regression of the form Z a bw between the simulated age-specific catch in numbers (Z) and the observed age-specific catch in numbers (W) to assay the model bias, it appears that whereas in Run 2 one obtains a consistent over-estimate of the observed by the simulated ii.e. "b" not significantly different from one and "a" significantly greater than zero), in Run 1 a bias exists which is not consistent (i.e. "b" significantly different from one) over the range of observed age-specific catches. It would thus appear in this case that while the magnitude of the catch in those age classes which contributed most heavily to the total catch in numbers and total catch in weight can be simulated adequately by applying constant age-specific fishing mortality rates (i.e. not taking into account the year to year fluctuations in quarterly effort), the variation in the relative contributions of the various age groups to the catch can be more adequately described by taking into account the interaction between constant age-specific vulnerability (catchability) and variable effort to generate a variable time series of age-specific fishing mortality rates. Figure 9 the quarterly sum of squared deviations of observed from simulated catch in numbers for the years 19H for the X and Y groups sep-

15 TUNP0P, A COMPUTER SIMULATION MODEL 247 arately. This figure gives some indication of within and between year variability in the accuracy of the two simulations. It is interesting to note that for both the X and Y groups in Run 1, the simulated catch tends to deviate most markedly from the observed catch at times when the average size in the catch is reduced (Figure 8). Of course these are the times when the greatest numbers of fish are caught and thus one would expect the deviations of catch in numbers to be relatively high. However it also might indicate that the model, as structured, is relatively insensitive to the annual and seasonal fluctuations in small fish abundance and/or availability. From the results of these runs it appears that to improve the accuracy of the model under constant age-specific catchability coefficients 1) better estimates of yellowfin effort must be obtained, and 2) time and age-specific availability factors must be employed in the model. SENSITIVITY ANALYSIS An analysis was performed to examine the sensitivity of the model output to systematic changes in input parameters and independent variables. Three output variables were utilized: annual catch in numbers, annual catch in weight, and annual average weight in the catch. Let Then x y(x) input variable or parameter with respect to which the sensitivity analysis is being performed output variable which is to measure the sensitivity of the model to changes in x, and change in x to be simulated. S(x,Y,6x) net sensitivity of J' to a change, in x y The ambient conditions from which the input variables and parameters were systematically varied were those values used in Run 2 of the previous section. Constant annual X- and Y-group recruitments (21.2 million and 20.6 million fish annually to the X- and Y-groups respectively) were used so that all yield values could be computed under equilibrium conditions. Thus letting.l'1 (x) x /:» o.io- annual equilibrium catch in numbers instantaneous natural mortality rate

16 248 FRJ\NCIS then S(x, Yl, Lx) fraction change in annual equilibrium catch in numbers in response to a 10% increase in natural mortality. The sensitivity of annual equilibrium catch in numbers, catch in weight, and average weight in the catch was tested with respect to 1) annual age-specific catchability coefficients 2) fishing effort (fishing mortality) recruitment 4) natural mortality 5) availability The results of the analysis are presented in Table 3. In the general vicinity of the values of the model input variables and parameters utilized in Run 2, the output from the fishery. in particular yield in weight, appears to be much more sensitive to changes in recruitment and natural mortality than it is to changes of a similar magnitude in availability, age-specific catchability, or fishing intensity (effort). It is interesting to note that a lo-percent decrease in availability produces a 17 percent greater decrease in equilibrium catch in weight than does a 10-percent decrease in fishing effort. Also a 10-percent change in natural mortality produces a larger than 10-percent change in the equilibrium yield in weight. Finally, it appears that in the vicinity of the values of the model input values and parameters used in Run 2, the equilibrium output from the fishery is more sensitive to changes in catchability on age 3 fish than on any other annual age group. SUMMARY AN:D C()NCLUSION The purpose of this paper was to provide the basic reference document for TUNP0P, a computer simulation model of the yellowfin tuna population and surface fishery in the eastern Pacific Ocean. In addition example runs of the model and the results of a limited sensitivity analysis were presented. It is hoped that this model will aid in the revelation of the basic structure of the yellowfin tuna population in the eastern Pacific and its response to exploitation by the surface fishery in recent years. ACKNOWLEDGMENTS The author is most grateful to Dr. William Bayliff, Inter-American Tropical Tuna Commission, Mr. Joseph Greenough, National Marine Fisheries Service, and Dr. Robin Allen, Fisheries Research Division, New Zealand Ministry of Agriculture and Fisheries, for their helpful comments on the manuscript.

17 TUNP0P, A COMPUTER SIMULATION MODEL 249 ~ EASTERN PACIFIC YElLOWFIN TUNA REGULATORY AREA UNITED STATES \ \. \ '-., t ~ J i j20 :t o N COLOMBIA o,'\ I 200f I t CHliTO 1 PERU I I 30 S--!------( I I Area defined according to the recommendation of the Inter American Tropical Tuna, Commission, QUito, Ecuador May 16 18, 1962 I I I I I J 4WL.. --l....l L-..!:~,"_---' 40 0 FIGURE 1. FIGURA 1. Commission Yellowfin Regulatory Area-~CYRA. Area Reglamentaria de la Comisi6n de Atun Aleta Amarilla-ARCAA.

18 l'v C..H 120" 110" 100" 90" BO" " 140" BOo "11 i 0( :t::::l::i:::+l40 40"ffTTfTR=TT I! I I I, Iii I I I I I I, I :r 1/40 30" CATCH I I I I I 1 I : : I I I I I I I I I I I I I I I I 1 I I 1 I IvI 1 k CATCH 30" ~ EFFORT, NO CATCH ~ EFFORT, NO CATC H "* Tf-+ 20" I I I I 1 I I I I 1 I I I I I IvI 1 I I I I I I I!vla)l I I I+-H- Ii. IF.. Iii i I I I I N'" 10 l-:rj ~ ~ 0" I I I 1 I 1 I I I I I I I II " 1 ' Iii I I I I " I III ill! I " I I I I 1 I! 1 I I I ji'" 0 Z ++H-++ i±--1\ t-+-+--w-+-h--+-+ t! I+t-it -I +- l-i---+-l--j--+--h---h-l--t--1ttt=p o f f f h H r.n 10" 10" AIII!IIIIIIIIIIIIIIIIIIIIIIIIIII',111111III'illllll'III!I'IIO'v~ 10 20" H-+ I I I. I I I I I ' I I I I I I i0oi120 20"11 I I I I I I I I I I I I I I I I I I I I I I 1 I I I 1 I I 1 I I I 1 I 1 1 I I 1 1 I I 1 I I I I I 1 I II! I I I I i I I I I I I, 'I 1"'1"'1"'1 ' Ii. n ! I i I I I I I I I I 1 1 I I 1 Ii!' I II! I ' I I I I I I I I I!! I I I! I j J I ' I I I I I I I I ' I I I I I I! I I! I 1 ' I I ' I I I I! I I! I I i I I I! ' I! ' I I I I I! " 150" 140" " BOO FIGURE 2. Areas of exploitation in 1960 and FIGURA 2. Areas de explotacion en 1960 y 1972.

19 'runp0p, A COMPUTER SIMULATION MODEL [J Al Historical Fishing Area ~ A2 Offshore Fish ing Area ~ zz z "// 1200 I 15 jj 10 ~ t::r:::-..c::r:::±:1:r: 5;::x:o~r±1:::I:20=r =:C:::r::±1=r: =::I ::::::X=:r=±1:="l10:=l :::=r::::r:::::±'=:j 0 ==5 :x=::r:::di=::j0::=0:r:::::o:=:::±9:::::::j5c::0::c:::r:::id!::=90c: :r:=:r:=c::*=r:::r::::t::::t:::h=:j:::::::::t:.::t::::b=t:::::t:=:i::~fr Ii FIGURE 3. FIGURA 3. I-listorical and offshore fishing areas. Areas de pesca hist6ricas y fuera de la costa.

20 IRecru ilmenl - R (I I ~I Pop. Numbers - Nj (t) Pop. Biomass - Bi (I) Z Graduation I "e >' Pop. Numbers - N + (t+l) i 1 Pop. Biomass - B i + 1 (t+1) - E (t +1) l::v c..n l-.:j,-- I... ~ IAvailabi lily - Pi (ti I Not. Mort. - Mi ~ ---;,. FISHERY ---I /". --~ I Ii // I I I I I I biomass flow> Catch in Numbers-YN j (t) I IColch in Weight - YWj (tii I _cont~~ I I I ~ ::D. ~ Z n H w L ~ FIGURE 4. FIGURA 4. Basic TUNP0P mechanisms. Mecanismos basicos de TUNP0P.

21 TUNP0P, A COMPUTER SIMULATION MODEL 253 _ {no (no age classes), n,t (no. unit ~Ime.in~~rvOIS for s.,m~lotion ) q I ~ t), f ( t ), P(t), MI' R( t ), WI (t), I -,,, no, t - I,..., nt N, (1); 1= 2,...,1'10 GI(t), i=l,..."na;t=i,...,nt set runn I ng time to time mterva I (t, t +1) Nj(t), Bj(t}, "f3j(t), GGi(t), YNI(t}, YW,(t): i =1,..., n a Totals, average weight in catch, ratio of gross growth to morta lity losse~ IF final Un-It time interva I of year YES Annua I outpu t Annual age - specific catch numbers and weight in NO Compute Nj(t+l), j='2,..., no; graduation E(Hl) FIGURE 5. FIGURA 5. Flow diagram of TUNP0P computer program. Organigrama del programa de c6mputo TUNP0P.

22 FRA.NCIS X_GROUJ.. NUMBERS xl0 t 6 1\ 2 I \ I \ 1~ "X i I ox 0--- OBSE:.RVED 4O- RUN I x... RUN 2 3 V-GROUP ~UMBERS xl Q \ ~ Q Q FIGURE 6. FIGURA 6. Quarterly catch in numbers. Captura tri.mestral en cantidades.

23 TUNP0P, A COMPUTER SIMULATION MODEL , , \ 8 \ \ \ 6 X-GROUP lb. xlo " i. '''ls: >\. \ \.'.\. 4 \\ /.:\ /...\ /'... \y / 2.:-~... 'X x X 0-- OBSERVED RUN 1 x.. RUN 2 6 V-GROUP lb. xlo' 4~.)(\ ~~, 2 ". " o j(.x _.-'---'--->O:---'--.l..----'------''----'----.c_-'-,-..L-'---,-I--'-.'_-,-'-----I.'--L1_-''--.L--.-'----l Ot Q FIGURE 7. FIGURA 7. Quarterly catch in weight. Captura trimestral en peso.

24 256 FRANCIS XGROUP~ Ib ~ x... '\;'...~ \\ ~ ~....~. ~.\: '\..;? ~ /""e ~ OBSERVED 40 e-- RUN 1 x.... RUN 2 30 V-GROUP lb. 10 I o --'_..~..L... _ ~._.~ L.~.. ~ _l.~,._j.~",~~..~~l,... o~~,_a..'-".. ~_.. _,_- J Q2 Q3 Q (H Ql ' FIGURE 8. FIGURA 8. Quarterly average weight in the catch. Peso promedio en la captura trimestral. X-GROUP FIGURE 9. FIGURA 9. I.--- RUN I I i\ x. RUN 2 I 1.0 I \ / 0.5 ;' \ \ \, ~j/.../{...x... /\ '_"..."... --'-----'_-"'---'--A~=.::... X.'_---i/_ ~.=...:.:;\.:IL:_.s 1::-:~;:; '--L~ ~4L:<':_ :& J L_...l_+i_L_~.~=- t \\,... T ':" :\ X...X 1\ /; : \ 1.0 I \ if ~ \ 1\ It l\ v-group I \ l\ I \ l\ le_- 0.5 /' " x \ X" \ ~..x \ ~ C ~ ~~i~..., \~.~_~~~ ~,~L~~~"._ 7~\f~~~~, \..., QI (;4 01 Q Ot \ Sum of squared deviations of observed from simulated quarterly age-specific catch in numbers. Suma de las desviaciones cuadradas de la eaptura trimestral en cantidades, observada a edad especifica con relaci6n a la simulada.

25 TUNP0P, A COMP1JTER SIMULATION MODEL 257 TABl.lF~ 1. Estimates of standard size-class 3* fishing days for TUNP0P Run 1. rrabj--ia 1. F":stimaciones de los elias ele pesca de la clase 3* de arqueo en la ira. pasada de TUNP0P. QUARTER - TRIMESTRE YEAR -- ANO ,992 6,494 6,214 4, ,008 5,616 2, ,891 9,414 3, ,867 6, ,786 8,109 1,676 1, ,971 10,351 5,806 12,373 See Shimada & Schaefer (1956) for size-class definition. Vease Shimada y Schaefer (1956) para la definicion de las clases de arqueo. TABI.JE 2. T..~BLA 2. Mean age-specific quarterly catchability coefficients and instantaneous fishing mortality rates ( ). Coeficientes de la media de la edad especifica trimestral de la eapturabilidad, e indices de la mortalidad instantanea par pesca ( ). Quarterly age qi x 10-7 F. 1, :Edad trhnestral X \T X y _._._--,-~, ~--' _.._ _.._ ~_._-----~ < () _._~ _

26 258 FRANCIS TABLE 3. TABLA 3. Sensitivity of annual equilibrium catch variables to systematic changes in model input parameters and variables. Sensibilidad de las variables de la captura anual de equilibrio, con relaci6n a los cambios sistemlaticos de los parametros y variables en la entrada del modelo.,~ x D.X I S(.Y" Yi,' 6x) 'l~,.; 2 qi; i == 1,..., 4 (year 1) -.lox ~ t- 10x q i; i == 5,..., 8 (year 2) _._ -.10x x qi; i == 9,..., 12 (year 3) -.10x x qi; i == 13,...,16 (year 4) --.lox x f -.10x x -_._--_._ R -.10x x M i ; i == 1,..., x x Pi,; j == 1,..., x Y1 == annual equilibrium catch in nunlbers equilibrio anual de captura en cjlfras Y2 == annual equilibrium catch in weight equilibria anual de captura en peso Y3 == annual equilibrium average weight in the catch peso promedio del equilibrio anual de captura.

27 TUNP0P, UN l\iodelo COMPUTADOR DE SIMULACION DE LA POBLACION DEL ATUN DE ALETA AMARILLA Y DE LA PESCA ATUNERA EPIPELAGICA DEL OCEANO PACIFICO ORIENTAL por Bobert C. Francis EXTRACTO Se describe la documentación matemática de TUNP0P, un modelo computador de simulación basado en la edad de la población del atún aleta amarilla y de la pesca atunera epipelágíca del Océano Pacífico oriental. Se presentan y se discuten ejemplos de las pasadas del modelo, y se examina la sensibilidad de los resultados de salida con relación a los cambios de varios parámetros. INTRODUCCION Los investigadores de la Comisión Interamericana del Atún Tropical (CIAT) tienen a su cargo la responsabilidad de realizar estudios científicos sobre los atunes tropicales que habitan el Océano Pacífico oriental, con el fin de recomendar las medidas que servirán para la conservación de estas existencias a niveles que puedan soportar una producción máxima en forma continua. Estos estudios han abarcado un número de especies de atunes pero se han concentrado especialmente en el atún aleta amarilla (rabil) (Tbunnus alhacares) y el barrilete (listado) (Katsuwonus pelamis). Desde hace mucho tiempo ha sido evidente que la pesca afecta la abundancia del aleta amarilla, pero no la del barrilete. Los modelos formulados por la Comisión indicaron que a principios del decenio de 1960 las capturas de aleta amarilla en el Pacífico oriental excedían la cantidad máxima estimada que esta población podía soportar en una forma continua. En 1962, la Comisión designó un área a la cuál se debían aplicar las recomendaciones para administrar esta población, el Area Reglamentaria de la Comisión de Atún Aleta Amarilla (ARCAA, Fig. 1). En 1966, se estableció en el ARCAA un programa de conservación sobre esta especie. Este programa, modificado anualmente, sigue vigente. Un examen del motivo y evolución del programa administrativo del aleta amarilla fue presentado por Joseph (1970). El propósito de este estudio es suministrar la documentación básica, matemática, de un modelo computador de simulación de la población del aleta amarilla y de la pesca atunera epipelágica del Océano Pacífico oriental. Además, se presentan y discuten ejemplos de pasadas del modelo y se examina la sensibilidad de los resultados de salida con relación a los cambios de varios parámetros. 259

28 260 FRANCIS El objetivo del modelo, llamado de ahora en adelante TUNP0P, es triple: (1) investigar la estructura de la población del atún aleta amarilla en el Océano Pacífico oriental; (2) investigar las causas de los cambios tanto en magnitud como en la composición de talla de la captura de esta acy\tl!01ltl en el ARCAA durante los últimos años; (3) estudiar bajo varios esquemas administrativos el rendimiento potencial del aleta amarilla en el ARCAA. Los modelos matemáticos que los investigadores pesqueros han usado tradicionalmente para evaluar y administrar las poblaciones de peces, pueden catalogarse en dos grupos; modelos de producción de la población y modelos de la estructura de la edad. Los modelos de producción de la población se emplean para tratar de relacionar directamente el índice instantáneo de cambio de la biomasa de la población al nivel de la biomasa existente y la cantidad de esfuerzo de pesca que se aplica a la población en cualquier momento de tiempo. Este tipo de modelo capacita al investigador a estimar la proporción que ha sido removida de la población por una unidad individual de esfuerzo de pesca, la habilidad intrínseca de la población a aumentar y el volumen máximo que la población puede teóricamente alcanzar. Por lo consiguiente, este tipo de modelo puede emplearse para pronosticar el rendimiento teórico, máximo, que la población puede sostener en forma continua, como también el promedio de producción bajo cualquier intensidad pesquera determinada. Los modelos de producción de la población fueron primero usados en la investigación atunera por Schaefer (1954, 1957) y fueron refinados y ampliados por Pella y Tomlinson (1969) y Fax (1970). El motivo biológico de la forma logística del modelo de producción de la población es discutido por Pielou (1969). Los modelos de este tipo son relativamente fáciles de aplicar ya que lo único que se necesita en cuanto a datos es una serie cronológica de las estadísticas de captura y esfuerzo de la población que se está investigando. Tres propiedades claves de la población, sobre la que se basa la validez del análisis de un modelo de producción son: 1) La población debe ser una unidad precisa autónoma. 2) La población debe inclinarse a tener un volumen y una distribución de edad estable a cada nivel del esfuerzo de pesca. 3) Cada unidad normal de esfuerzo debe capturar, en promedio, una fracción constante de la población. En otras palabras, la probabilidad de que una unidad determinada de esfuerzo encuentre una unidad determinada de población, es constante durante el tiempo. Así, a cualquier nivel fijo de la biomasa de la población, se supone que cada unidad de esfuerzo se aplica a una intensidad constante y con una distribución constante sobre la porción explotada de la estructura de edad de la población. Se supone

29 TUNP0P, UN MODELO COMPUTADOR DE SIMULACION 261 además que una fracción constante de la población es accesible y vulnerable a la pesca en cualquier momento. Es interesante examinar la pesca de aleta amarilla del Pacífico oriental durante los 10 últimos años, con respecto a estas propiedades, con el fin de determinar qué tan apropiada es esta especie para el análisis del modelo de producción. Con respecto a la 1) propiedad, la pesca se ha extendido, en área, de una pesca predominantemente costera durante el principio del decenio de 1960 a la pesca actual que abarca una gran proporción del ARCAA y de las áreas al oeste del ARCAA (Fig. 2). El análisis de la estructura de la edad (sumarizado más adelante en este estudio) indica que la proporción de la población accesible a la pesca ha aumentado con la expansión del esfuerzo fuera de la costa. Por lo consiguiente existe alguna duda sobre si la población que se está explotando ahora es la misma unidad básica que la explotada en años anteriores. Con respecto a la 2) propiedad, la talla promedio de los peces en la captura ha cambiado significativamente con relación a rendimientos similares "próximos al equilibrio". En 1966, la captura de aleta amarilla en el ARCAA fue de 182 millones de libras y el peso promedio por pez en la captura fue de 22.4 libras, mientras que en 1967 la captura fue de 179 millones de libras y el peso promedio por pez en la captura fue 19.3 libras. En 1970, la producción de aleta amarilla fue 285 millones de libras y el peso promedio por pez en la captura fue de 31.5 libras, mientras que en 1971 la captura fue de 226 millones de libras y el peso promedio por pez en la captura fue 27.7 libras. En 1967 y 1971, los dos años en los que hubo un promedio inferior de peso, se capturaron grandes cantidades de atún barrilete en el ARCAA. Durante un año de buenas capturas de barrilete, la flota se concentra más a lo largo de la línea de la costa de la América Central y del Sur donde el barrilete se concentra y donde se encuentran pequeños cardúmenes superficiales de aleta amarilla. Bajo estas condiciones, el peso promedio del aleta amarilla en la captura tiende a ser relativamente bajo. En contraste, en 1966 y 1970 la pesca se distribuyó más en la región exterior del ARCAA donde se encuentra aleta amarilla de más talla, predominantemente con cardúmenes de delfines. Bajo estas condiciones el peso promedio de aleta amarilla en la captura tiende a ser relativamente alto. La siguiente tabla indica la fracción del número total de caladas positivas de aleta amarilla asociado con delfines en cada uno de los cuatro años. Años Fracción de caladas con delfines

30 262 FRA:NCIS Consecuentemente, parece que la talla promedio del aleta amarilla en la captura del ARCAA se encuentra fuertemente afectada tanto por la distribución geográfica del esfuerzo como por la moda de pesca (co:n delfines o sin delfines) en que se emplea predominantemente el esfuerzo. Finalmente, respecto a la 3) propiedad, la capturabilidad o vulnerabilidad de la población disponible a la flota parece haber cambiado con el refinamiento de las técnicas cerqueras y el movimiento fuera de la costa de la flota. Si el ARCAA se divide en dos áreas (Fig. 3), siendo el área Al el área histórica de pesca y el área A2 el área fuera de la costa que se ha explotado más recientemente en el ARCAA, entonces la siguiente tabla del porcentaje de la captura total de aleta amarilla en el ARCAA y del esfuerzo normalizado en el A2 refleja alguna luz sobre el problema antes mencionado. Las estimaciones del esfuerzo han sido ajustadas o normalizadas por cambios "aparentes" en la eficacia debida a cambios en las proporciones de las caladas positivas. Porcentaje del esfuerzo Porcentaje de captura en normalizado en el A2 Año el A2 en el ARCAA en el ARCAA 1962 O O Es evidente que la fracción de toda la captura del ARCAA en el A2 aumentó más rápidamente que la fracción del esfuerzo normalizado en el ARCAA aplicado al A2. Suponiendo que existe una unidad poblacional en el ARCAA, estos datos indican que a medida que la flota se extendió al área A2 la eficacia total de la flota, en términos de capturabilidad, aumentó. Por lo consiguiente parece que las técnicas modeladores (modelos de producción) deben aplicarse con precaución a la pesca de aleta amarilla realizada en años recientes en el Pacífico oriental. A diferencia de los modelos de producción de la población, los cuáles representan la dinámica de la biomasa como unidad individual, los modelos de la estructura de la edad usan la relación entre el crecimiento, mortalidad y reproducción de los elementos individuales que forman la población para simular la dinámica de la biomasa poblacional. La teoría matemática básica de estos modelos fue desarrollada por Baranov (1918) y refinada por Ricker (1945, 1958) y Beverton y Holt (1957) al tratar de relacionar

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