An aggregate model of effort distribution for the eastern Pacific tuna fishery

Alejandro A. Anganuzzi¹
Inter-American Tropical Tuna Commission
8604 La Jolla Shores Drive
La Jolla, California 92093 USA

¹ Present Address: Indo-Pacific Tuna Programme, P.O. Box 2004, c/o NARA, Crow Island, Mattakkuliya, Colombo 15, Sri Lanka

ABSTRACT

A simulation model of the spatial distribution of fishing effort in the eastern Pacific tuna fishery is introduced. The model represents the changes in distribution of total effort as a function of perceived changes in relative abundance of the resource. Several stratification schemes allow us to refine the resolution of the model to account for major heterogeneity in fleet composition. Possible parameterisations of the basic model and associated estimation procedures for its parameters are also discussed. In the eastern Pacific fishery, the model performed reasonably well in predicting average distributions of purse seine effort. Potential expansions of the basic model include individual-based models, currently under development.

1. INTRODUCTION

Simulation models can provide valuable insights in assessing the importance of interactions between fisheries, since they allow us to summarise our current level of knowledge and formulate predictions that can be compared against observations. For example, hypothetical scenarios can be explored that depart radically from past situations, such as large changes in fishing strategies that might result in changes in fishing mortality patterns. In the case of the eastern tropical Pacific fishery (ETP), the Inter-American Tropical Tuna Commission (IATTC) developed a simulation model to assess potential interactions that might result from large changes in the relative importance of the different fishing modes (fishing in association with dolphins, fishing on floating objects or fishing free-swimming schools of tuna) (see Mullen et al., 1996) for a description of the main simulation model). In this context, a fishery interaction is defined as the potential changes in yield by one sector of the fishery due to the activities of another component of the fleet.

Under those circumstances, large departures in the spatial distribution of effort relative to the traditional patterns can be expected. In synthesis, the question is: how are fishermen going to spatially reallocate their effort if certain fishing tactics (traditionally associated with specific time-area strata) are abandoned?

In order to answer this question, we need to model the distribution of fishing effort as a function of the information available to the fishermen about the distribution of the resource. This implies that, as in any model of distribution of fishing effort, there are two processes that need our attention: 1) modelling the amount and distribution of the information among the fishermen, and 2) modelling the response of the fishermen to the available information. In this paper, the model implemented as part of the simulation model described by Mullen et al. (1996) is reviewed. In this model, effort is represented as an aggregate variable, aggregated over boats within a particular flag and vessel-size categories. The distribution of effort in the model changes spatially at certain time intervals, according to the existing perception of the local distribution of the resource.

Similar approaches for the spatial modelling of effort have been proposed in the past, with some variations. Caddy (1975) used a similar representation of the effort in a model of shellfish fisheries, Hilborn and Walters (1987) proposed the same basic approach when discussing fishery modelling with spatial structure. Gillis et al. (1993) discuss general principles that lead to aggregated models.

These models are suitable for situations where we do not have detailed information about the daily activities of the vessels. The data requirements are relatively low, and they represent a useful first step in modelling fleet dynamics. However, the models are too simple to reflect the consequences of individual variability in fishing strategies. Processes such as the sharing of information among fishermen or different attitudes towards risk might not be easily incorporated in these models.

Table 1. Summary list of the assumptions implicit in the model.

· Fishing effort is defined as number of days/boats, i.e., no individual vessels are identified. · Effort is stratified into categories related to vessel size, fishing mode and groups of nations. · Effort is spatially allocated by 2.5-degree cells and assumed to be distributed randomly within a 2.5-degree cell. · Total level of effort is constrained to the observed level for the month. · Effort is re-allocated once a day. · In a day, effort can move only within a local area defined by the original cell and the eight neighbouring 2.5-degree cells. · Costs of moving within the local area are negligible. · Information regarding the fishing conditions is updated every six days. · Information is perfect regarding the catch rates of vessels from the same effort category in the local area. · No information is available from cells beyond the local area. · Redistribution of effort in the local area is done proportionally to the catch rates experienced in the previous week

2. AN OUTLINE OF THE MODEL

The main assumptions of the model are listed in Table 1. In the model, effort is assumed to be an aggregated variable being reallocated dynamically according to the changes in the local conditions of the fishery. Fishing effort is defined as the numbers of days fishing multiplied by the number of boats, that is, a unit of effort is one boat fishing one day. The spatial resolution is set to a 2.5-degree cell, in accordance to the spatial resolution of the population processes incorporated into the main simulation (Mullen et al., 1996).

Modelled effort is stratified into classes related to vessel size and fishing strategy: 1) boats of less than 900 tons of carrying capacity; 2) boats above 900 tons; and 3) boats that usually do not fish in association with dolphins. Within these categories, effort was further subdivided into three fleets corresponding approximately to the different coastal regions of the ETP, and a fourth fleet for non-coastal nations.

The total spatial range of the fishery in the model, assumed to comprise 327 2.5-degree cells was divide into five areas: four that represent the economic exclusive zones of groups of coastal nations and a fifth area, that represents international waters. This stratification was used as a way to represent some of the logistic restrictions that small vessels face (they cannot go too far from port), and also as a way to explore potential consequences of changing fishing patterns (like restricting access to certain areas).

In the model, effort has a certain initial distribution in space depending on fishing mode, fleet and vessel-size class. Starting from these initial conditions, effort is allowed to change spatially and, in principle, between fishing modes, following a set of rules that establish a relation between the local situation of the fishery and the rates of flow of effort. Monthly total effort by fleet and vessel size class remains constant and equal to the observed levels for the month, that is, temporal fluctuations of total effort levels are not modelled.

The time resolution of the effort allocation model is set to one day. In one day, modelled effort from one cell can move only to the eight surrounding cells or stay in the original cell. This is equivalent to assuming that vessels cannot move further than approximately 225 miles on any day, which on average corresponds to the observed movement rates of the vessels. In Figure 1 the distribution of distances between noon positions of consecutive days for all vessels in 1994 is shown. The proportion of distances that are lower than 225 miles is over 90% for vessels less than 900 tons, and 86%, for vessels larger than or equal to 900 tons, which indicates that the applied restriction in movement is reasonable.

The rule for reallocating the effort is based on the relative fishing success in the area in the previous weekly period ("weeks" are defined as a fifth of a month in the simulation model). This represents the information state of the fishermen, that is, the model assumes that fishermen are using this information to decide on where to fish.

At the end of a day, the amount of effort spent in any single cell is redistributed in the local area (defined as the original cell and its eight neighbours) proportionally to the catch rates experienced (within the same fleet and size classes) in the previous week. Implicit in this rule is the assumption that vessels have precise information about the conditions in the local area and no information about areas further away. Also, vessels are assumed to update their information state only once a week, and that information retains its value until new information is available (i.e., vessels respond in the same way to the same information throughout the week).

Figure 1. Distribution of distances between noon positions of purse-seine vessels, data from 1994. (A)

Figure 1. Distribution of distances between noon positions of purse-seine vessels, data from 1994. (B)

The costs of moving in the local area are assumed negligible, that is, there is no additional cost in moving to an adjacent area rather than staying in the original cell. The actual implementation of this model requires two steps: 1) computing the expected revenue or value per unit effort (VPUE) as a function of location and fishing modes, and 2) re-distributing the effort proportionally to some function of the expected VPUE. We will review each of these in the next sections.

2.1. First Step: Computation of the information State

In the model, it is assumed that the information that fishermen have about the desirability of each location is based on the expected VPUEs, which are essentially the catch rates (measured in landed value) from the previous week. These represent the landed value of the expected catch composition, and depend on the local abundance and catchability of the yellowfin semi-annual age groups and the abundance of skipjack, and on the value (price) of the different fish sizes. Therefore, separate VPUEs are calculated for yellowfin and skipjack.

In the case of yellowfin, for which we have an age-structured representation in the general simulation model, we have, for a given period or week;

(1)

where

i = 2.5 degree cell,
c = a cohort (semi-annual age group),
f = a fleet category
g = a vessel-size category,
Y_ifg = yellowfin catch rate in commercial value,
N_ic = is the abundance in numbers for cohort c in cell i,
q_icfg = is a catchability coefficient,
w_c = average weight of a individual from semi-annual age group c, and
p_y(w_c) = ex-vessel price of a unit of weight of a yellowfin from cohort c.

Zifg = Uifgps(w)

(2)

where

Z_ifg = skipjack catch rate in commercial value,
U_ifg = skipjack catch rate, and
p_s(w) = ex-vessel price of a unit of weight of skipjack,

Vifg = Yifg + Zifg

(3)

That is, the value assigned to any 2.5-degree cell is the sum of the expected values for skipjack and yellowfin. The value for yellowfin is a sum of the abundance of the different cohorts present in the cell, weighted by the catchabilities (specific to the cohort, set type, fleet and vessel size), average weight and price per unit weight. Thus defined, the values V' represent the average economic gross revenue that could be obtained from a particular 2.5-degree area by a unit of effort of a given fleet and vessel-size class. These values, computed at the beginning of each reallocation period for each cell, are the driving force behind the effort reallocation algorithm.

2.2 Second Step: Redistribution of the Effort

Once we have a map of the values that define the desirability of each area, we need to devise a procedure for reallocating the effort. Again, there are several possible ways to proceed. In one approach, we could redistribute the effort over the entire fishery. However, this would be equivalent to assume that vessels can move over large areas without any cost or time restriction in the time step chosen in the model (a week). Therefore, we must incorporate some restrictions to the movement allowed in one time step. As we mentioned above, we took the alternative approach of distributing the effort from a given 2.5-degree cell, i, in the local area W_i, determined by the cell itself and its eight adjacent neighbours.

Let us denote by e_ifg(t) the amount of effort in time t and cell i (the source cell). This effort will be redistributed in each of the j target cells, which are those in the neighbourhood

Wi (jÎWi), according to

(4)

That is, effort from any source cell is pro-rated among the surrounding cells according to the relative local VPUE. The effort in any cell will be the sum of contributions from the other cells in the neighbourhood W_i. Therefore, the final effort in cell i, for flag group f and size class g fishing in mode s during period t+1, is obtained from

(5)

This reallocation procedure is repeated for all cells in the area of the modelled fishery that are admissible. An example of non-admissible cells are those that are outside the area defined by the 327 cells. They are not included in the above calculations and, as a consequence the boundaries of the region operate as reflective boundaries.

The procedure is repeated six times and the distribution of effort at the end of the six time steps is the one used to calculate the catches in the main simulation model. The reason for choosing six steps is not arbitrary but related to the assumption that vessels do not move more than 2.5 degrees in any direction in one day. Since in the main simulation model the catches are only calculated every fifth of a month (roughly six days), the effort is allowed to move for those six days. Therefore, although the effort can only propagate one cell at each time step, it can cover a much larger distance (up to six 2.5-degree cells) before the information state (and all the other processes in the main simulation) are updated.

The basic reallocation procedure relates to the model used by Caddy (1975). The main difference is that the historic distribution of effort is not taken into account, and that the re-distribution is done locally rather than on the basis of global information. The resulting distribution tends to an equilibrium close to that of the ideal free distribution model (Gillis et al., 1993), in which we have effort distributed approximately proportional to the relative abundance. Therefore, a mechanistic interpretation analogous to that allowed by these two models applies to our allocation procedure. Namely, that the main assumption is that the probability that a unit of effort moves to a given square in the local region is proportional to the relative catch rates experienced in the previous time period.

3. RESULTS

Originally, the general algorithm described above included a partition of the effort among fishing modes, that is, modelled effort was not only redirected spatially, but the switching between fishing modes was modelled as well. However, in some of the earlier trials of the models it became obvious that modelling this side of the fishing strategy was not going to be possible, at least at this stage. Modelled effort showed a strong tendency to switch to fishing on free-swimming schools and on floating objects. This is not surprising since log sets have high catch rates and both fishing modes are often associated with large catch rates for skipjack, which raises the value attached to those fishing modes. Part of the problem originates in the way the skipjack population is represented in the model. As there is a current lack of knowledge about the population dynamics of this species in the eastern tropical Pacific, a very simple model of the skipjack dynamics was used. This model does not incorporate local depletion of the population (in contrast with the yellowfin model) and, therefore, there is no limit to the catch that could be obtained by increasing effort. Large amounts of modelled effort tended to drift towards those two strategies, because skipjack catchabilities are higher in school and log fishing than in dolphin-associated fishing,

Therefore, the proportion of effort going into each fishing mode was constrained to be equal to the observed proportions of effort in the fishing modes allowed under a given simulation scenario. For example, a non dolphin-fishing scenario would contain effort split among log and school fishing modes in accordance with their relative proportions. In general, the model described by equation (4) shows a moderately good fit between the observed and modelled distribution of effort, as it can be seen from the maps in Figures 2-4. Those maps show the distribution of modelled effort on the left panels and the distribution of observed effort on the right panels. In Figure 5, the correlation between observed and modelled effort over the simulated period (1980-86) is shown. The correlation was computed on the basis of the total effort by 2.5-degree cells for a given month. The pattern of these correlations indicate that the model can mimic the distribution of effort reasonably well in the first three years, 1980 through 1982. The performance of the model is worse in 1983 and the beginning of 1984. This is most likely related to the major changes in the distribution of the resource that took place as a consequence of a very strong El Niño event. Effort was more dispersed during this period as a consequence of the unusual distribution of the fish that made it more difficult for the fishermen to locate the good fishing areas. This resulted in many areas with low observed catchabilities that slow down the dispersal of effort in the model. Beginning in 1984, the agreement between the predicted and the observed distribution improves gradually until it reaches in 1985 and 1986 levels similar to those for the 1980-81 period. The distribution of the discrepancies by month (Figure 6) shows that the main disagreements are concentrated in the months of October and November. This problem probably originates in the apparent inability of the model to predict well the seasonal occurrence of a traditional good fishing area close to the coast of South America. The low correlation shown for the month of March originates from a single cell with very high effort (see Figure 1), which is the consequence of an extraordinary event in 1984 where most of the catch originated in that cell.

Figure 2. Distribution of predicted (right panels) and observed (left panels) fishing effort by month, over the period 1980-86, January through April.

Figure 3. Distribution of predicted (right panels) and observed (left panels) fishing effort by month, over the period 1980-86, May through August.

Figure 4. Distribution of predicted (right panels) and observed (left panels) fishing effort by month, over the period 1980-86, September through December.

Figure 5. Monthly correlations between observed and predicted effort for the period 1980-86. The dotted line is a smoothed line to highlight the trends.

The lack of fit in the southern area points to one of the weaknesses of the procedure as implemented: the problem is that the catchabilities are based on monthly data on catch and abundance, rather than averaged over a longer period. This restricts the movement of the modelled effort: only those 2.5-degree cells where there was some catch for a given month are candidates to receive effort since these are the only ones that have positive catchabilities (and, therefore, non-zero values). If areas (such as the mentioned southern area) are surrounded by areas of zero (or near-zero) catchability, effort is never (or seldom) able to diffuse into them.

Figure 6. Correlations between observed and predicted effort for the period 1980-86, pooled by month.

4. CONCLUSIONS

The approach to model fleet dynamics presented here allows several variations. In the ETP simulation model, the distribution of effort was assumed to be proportional to the VPUEs, that is, according to equation (4). However, this is not the only way of parameterising the model. In principle we can take some functional of the VPUEs that would provide a better fit to the observed distribution of effort. There are several ways in which to set up the problem to allow for some parametric control.

For example, we can take a power function of the values, such as

(6)

where the parameters a and b can be estimated from the data. For values of b > 1, effort will tend to concentrate more on the locations that have the better prospects. The situation where b < 1 is equivalent to assuming a certain degree of saturation, that is, that the marginal rate of profit is a decreasing function of value. If the resolution of the data allows it, the proportion of effort going into each target square can be estimated directly through a generalised linear model with a logit link (in our case, we would need the daily information about effort movement and changes in local abundance.)

Alternatively, we could decide that fishermen are only going to move to locations that exceed a certain threshold value, therefore

(7)

This functional must be used with caution because, if V₀ is chosen to be too high, it will reduce the area that effort can move to (effort can only move to places where the V_ifg > 0). This might create areas that become isolated from the rest of the fishery range: effort cannot enter nor leave these areas. If functionals such as those described in equation (6) or (7) are being used, we need to obtain values for the parameters involved. In some cases, this can be done by either by a linear regression technique (in the case of equation (6), after a suitable transformation), or, more generally, through a least squares minimisation procedure. For example, in the case of the threshold function described by equation (7), we can estimate V₀ by

(8)

Again, there are multiple ways of implementing the estimation. For example, the sum of squares in the above equation is computed on the basis of the effort being projected for the whole modelled period, given a certain set of initial conditions. This is a procedure analogous to an observation error estimator. Alternatively, we can reset the initial conditions at certain intervals and project the effort distribution over shorter periods, similar to a process error estimation procedure. We can also further stratify the data in order to obtain season- or area-specific parameters.

The model presented here has been a useful first approximation to the problem of modelling effort in the ETP. However, the data available for this fishery allows us to go a step further and develop models that would exhibit richer behaviour. Currently, we are in the final stages of developing of a model that represents the decision-making of individual fishermen in the fleet. Such a model allows us to explore a broader range of problems.

Both classes of models are based on the same principle: the decisions related to the fishing operations are based on the amount of information available to the fishermen. In practice, we know neither the information state nor the decision rules of the fishermen. This is not unlike the problem in population dynamics models where we do not observe the stock state (a latent variable) but we try to infer it from the realisations of a stochastic process (the catch). Both the population size and the catch are stochastic processes for which we do not know the variability, and therefore some compromising assumptions are necessary. We are forced to make the same kind of compromises in modelling fleet dynamics. The model presented here leans towards conditioning of the information state, assuming that the main source of variability lies in the fishermen's response. A different compromise might yield different results.

In the case of the aggregate model, the interpretation of the discrepancies between predicted and observed effort distribution must be done with caution, because they are complicated by the confounding with some of the other processes modelled in the main simulation that are related to the distribution and abundance of the fish, and that determine the information state of the modelled fleet.

This points to a more general problem with models of effort, essentially that, unless we can approximate well the information that is available to the fishermen, we will not be able to correctly discriminate between the two sources of error. We can be making erroneous assumptions about the fishermen's information state, about their responses to that information, or both.

5. REFERENCES CITED

Caddy, J.F. 1975. Spatial model for an exploited shellfish population, and its application to the Georges Bank scallop fishery. J. Fish. Res. Bd. Can. 32(8):1305-1328.

Gillis, D.M., R.M. Peterman and A.V. Tyler. 1993. Movement dynamics in a fishery: application of the ideal free distribution to spatial allocation of effort. Can. J. Fish. Aquat. Sci. 50(2): 323-333.

Hilborn, R., and C.J. Walters. 1987. A general model for simulation of stock and fleet dynamics in spatially heterogeneous fisheries. Can. J. Fish. Aquat. Sci. 44(7): 1366-1369.

Mullen, A.J., A.A. Anganuzzi, R.G. Punsly and G.J. Walker. 1996. A spatial model to investigate the effects of catches of tunas in the eastern Pacific Ocean which might have ensued from curtailment of certain fishing methods. In: Shomura, R.S., J. Majkowski and R.F. Harman (eds.). Scientific Papers from the Second FAO Expert Consultation on Interactions of Pacific Tuna Fisheries, 23-31 January 1995, Shimizu, Japan. [This volume]