Incorporating Sea Surface Temperature into Bioeconomic Fishery Models: An Examination of Western and Central Pacific Tuna Fisheries

: Ocean temperatures are increasing. Little work has been done to examine the effects that these changes will have on fishery production. The study at hand seeks to incorporate the influence of climate change into established bioeconomic fishery models. Stock biomass is approximated to be a function of sea surface temperature. Following a feasible generalized least squares regression using data from the Western and Central Pacific, the interaction between fishery effort and temperature is found to be statistically significant. From this model, various functional forms relating effort, catch, and temperature are specified. In particular, a function that returns an effort requirement given a target catch level and temperature forecast is generated.The importance of these tools for fishery management is explored through application to Western and Central Pacific tuna fisheries. Recommendations for extensions into future research are made and the foundation for a model of efficient effort allocation across time and the entirety of a management area, given changing temperatures, is specified.


Introduction
It is beyond doubt that ocean temperatures are rising globally. The past thirty years have seen mean January temperatures across the Pacific rise more than five percent 1 . These temperature increases cause dramatic habitat changes and will likely affect the biological processes and ecology of marine species. While there has already been a wealth of literature produced that examines the changes to abundance and distribution of various marine species, examinations of the interplay between these ecological changes and fishery processes and outcomes have only recently begun to receive their due attention.
Bioeconomic modeling methods are an ideal tool for this sort of analysis. The basis of the bioeconomic fishery model originated in the work of Schaefer (1954Schaefer ( , 1957. Schaefer's model looked at fishery catch as a function of the interplay between effort, a species' growth rate, and that species' environment's natural carrying capacity (Schaefer, 1954). This model was notably expanded upon by Lynn et al. (1981). In this paper the model was expanded to include the influence of an environmental factor on a particular fishery. In this case, the impact of the mangrove area on a Florida crab fishery was examined. This model is notable in that the actual functional relationship between stock biomass and the environmental variable (mangrove area) was not known. Rather, Lynn et al. posited several potential functional forms, based on certain hypothesized marginal properties, and then utilized the form which validated these assumptions and best fit their data. Other notable bioeconomic models that seek to incorporate environmental influences include Barbier & Strand (1998) and Ellis & Fisher (1986). For a survey of various bioeconomic fishery models which incorporate environmental variables see Knowler (2002) and Foley et al. (2012).
There has been significantly less work done in incorporating temperature effects into fishery models. Bell (1972) provided one of the first examinations of its kind, in which he sought to model the relationship between lobster catch and sea surface temperature as a linear function. This work provides the foundational basis for much of the research that has followed. Following in this vein are Sun et al. (2006), Garza-Gil et al. (2010), and Sanz et al. (2017); all studies in which the relationship between sea surface temperature and fishery catch were examined. However, none of these studies provide a replicable methodology that could be utilized for the purpose of the study at hand. Sun et al. (2006) did not seek to generate a general model relating temperature and catch, but rather sought to quantify the effect that the El Niño has on catch. Garza-Gil et al. (2010) focused their study on Sardines, which compared to various tuna species are not a highly migratory fish. As such, stock biomass data was readily available for their study. This allowed for the determination of a specific functional form relating temperature to the observed biomass. Sanz et al. (2017) examined a nonmigratory species and utilized a general Cobb-Douglas production function, rather than the Schaefer (1954) model used in the study at hand.
Developing a generalized methodology for understanding the relationship between ocean temperature and fishery catch is crucial given the reliance on fisheries in much of the developing world. The focus of the study at hand is on the various tuna species fisheries of the Western Central Pacific. Tunas are extremely responsive to changes in temperature (Brill, 1994). He established that tunas regulate body temperature by swimming, and as such may be forced out of desirable habitat due to overheating (Brill, 1994, p. 205). As such, warming ocean temperatures put the future of locality specific tuna fisheries in jeopardy. Bell et al. (2013) elucidated the problem that this may pose for many of the small island developing states of the Pacific Ocean. These tuna fisheries provide a significant portion of government revenue and GDP to these small island states through domestic fleets, domestic tuna processing, and licensing fees to foreign fleets (Bell et al., 2013). Further, most inhabitants of these small island developing nations derive the majority of their dietary animal protein from fish (Bell et al., 2009). In fact, for one nation, the Solomon Islands, 92% of inhabitants' animal dietary protein is derived from fish (Bell et al., 2009, p. 66). Given the paramount role that tuna plays both in the scope of economic development and for food security in these small island states, and the manner in which tuna species are influenced by temperature changes in their environment, serious examination of the relationships between catch, effort, and temperature is warranted. Lehodey et al. (2013) examined changes in spatial distribution of skipjack tuna catch in relation to oceanic temperature changes. However, their work did not seek to model the relation between catch and sea surface temperature, but rather to model spatial distribution changes. Monllor-Hurtado et al. (2017) modeled changes in tuna catch as a function of temperature and a latitude time interaction. Their model did not account for the impact of effort on catch or for the interplay between effort and temperature. They did, however, hypothesize that effort trends were related to catch trends as a result of fishery strategies changing with changes in stock abundance. Notably, Mediodia et al. (2020) recently released findings from a current working paper in which results obtained and methodology utilized in examining the relationship between sea surface temperature and tuna catch were similar to those of the paper at hand. Testing several functional relationships, a quadratic model was ultimately decided upon and utilized to quantify the effect of temperature on tuna, catch The study at hand examines the relationship between catch, effort, and temperature for the tuna species of the Western Central Pacific. This is accomplished through the vehicle of a bioeconomic model, consistent with the theoretical foundations of Schaefer (1954Schaefer ( , 1957. The incorporation of Sea Surface Temperature into this model allows for an adaptable form that is readily able to be utilized in the study of the fisheries of other migratory species. From this model optimal temperature and effort functions are derived. Statistical software packages were utilized to generate parameter values, which in turn were input into the optimal temperature and effort functions to provide optimal values for the fishery under examination. These optimal values are used to identify where overfishing is occurring, relative to temperature levels, and where decreases in catch-per-unit-effort can be expected based on temperature trends and predictions.

Theory and Model
The Shaefer (1954) model forms the basis for much of the work that has been done on bioeconomic fishery modeling. His model take the form of: Where: is the catch in period t, is a catchability coefficient, is the maximum potential biomass of the stock in period t, is the fishery effort in period t, is the stock's natural growth rate, This model is often modified to include a lagged dependent variable, catch, term. This is because stock adjustment may not be instantaneous. This model takes the form: Where: 1 is marshland acreage in the previous period Incorporating (3) into (2) provided: Where: For the study at hand, the relationship between maximum potential stock biomass and sea surface temperature, = ( ), was hypothesized to have the following marginal properties: > 0for < * , < 0for > * , and 2 2 < 0 * is some optimum temperature level The functional form for the maximum potential biomass and temperature relationship was posited to be 2 : Inputting (5) into (2) yields the following: Where: is , is an error term, indicates the particular geographic position of the observation Multiplying this out provides: This can be simplified to 3 : Where: In theory both temperature and effort can vary without bounds 4 . As such, the optimum combination of effort and temperature can be easily determined. First obtained is the catch maximizing optimal effort level for any given temperature. This conditional effort function is: 2 This model and other functional forms were tested using catch per unit effort as a proxy for biomass. The model utilized was best able to explain variation in CPUE.
3 It bears stating that the model will be input into statistical software with positive signs on all coefficients. As such, when interpreting regression results the signs of several of the coefficient values will need to be inverted. The optimal effort level and temperature combination obtained is: * = (10) and (11) The marginal rate of technical substitution of effort for temperature can be found easily enough: The necessary algebra is a bit more complicated in determining the functional form conditional effort function if both temperature and catch level are held constant. Some manipulation, and the quadratic formula yield: Since ocean temperature levels are beyond the control of fishery managers, the conditional effort functions will be the primary vehicles utilized to evaluate the sustainability of the tuna fisheries of the Western Central Pacific, and to make recommendations as to what changes should be made to best protect these fisheries in the face of rising ocean temperatures.

Data
Data utilized came from a variety of sources. Catch data was taken from the Western & Central Pacific Fisheries Commission's (WCPFC) "Public Domain Aggregate Catch/Effort" dataset.
This data was provided in a monthly time series of 5° by 5° gridded coordinate squares. Longline effort was used in this study's analysis due to the random sampling effect inherent in longline fishing methods. There is little targeting of effort beyond selection of a generalized geographic area in which to set the longline, which can be in excess of 20 miles in length 5 . The variety and abundance of species that are caught on the longline, in effect, will represent a more or less randomly generated sample of the area's biodiversity.
Temperature data was drawn from the NOAA Optimum Interpolation (OI) SST V2 dataset. Three unique species of tuna were examined separately: yellowfin, albacore, and big eye. It bears stating that effort is not species specific. The same unit of effort is capable of catching any of these tuna species, or any number of other marine species, with likelihoods varying solely based on distribution and stock abundance. Also as a caveat to the data, the minimum effort level reported was 100,000 hooks in a coordinate square in a particular month. Small scale fishing effort was not recorded or not not reported by WCPFC. However, the average longline has some 3,000 hooks 6 . As such, the minimum level of effort recorded is sufficient to capture the bulk of commercial fishing activity in each roughly 87,000 square mile 5° by 5° coordinate square.

Empirical Methodology and Results
The model derived in equation (8)      The parameter values derived from these estimates can be input into equations (9) through (14) to generate useful predictions regarding the management of the region's fisheries. Equation's (10) and (11) will provide the optimum effort and temperature combinations for each species. These combinations are a useful benchmark against which to measure a fishery's current state. These results are reported in table 7. It is of note that Yellowfin catch is maximized at such low temperature levels, near the minimum point at which catch is observed. This is more a factor of the mathematics of the model, in which the extremely high level of effort plays a larger part. This illustrates that these hypothetical optimum effort and temperature combinations are not an ideal standard for fishery management. Table 7. Optimum effort and temperature and effort combinations Equation (9) provides a much more useful measure for setting fishery goals with the objective of maximizing catch. Approximating necessary effort levels to maximize fishery yields in the face of changing ocean temperatures allows for fishery managers to plan accordingly against the forecasts of oceanographers and climate scientists. The results, reported below, show that the optimal effort level respective to temperature for each species is unique. Catch of Bigeye at low temperatures was nearly negligible in the data, as was the catch of Albacore at high temperatures.
Optimal effort levels trend differently for each species group. Maximizing Albacore catch would require higher effort levels at lower temperatures, while both Bigeye and Yellowfin catch are maximized with higher effort levels at higher temperatures. These effort levels are observably quite high, due to their representing the highest level of extraction possible without consideration for the long term economic or biological sustainability of the yield. As such, they solely represent catch maximizing effort levels at various temperatures in an open access environment. The primary value of these estimates is in illustrating the optimal trends in effort respective of temperature; that each species of tuna responds differently to different temperature levels. Perhaps most useful of the tools derived is equation (14) which allows fishery managers to determine requisite effort levels based upon catch targets and temperature forecasts. Catch targets can be set with aims based in maximum economic yield fishery management (MEY) or maximum sustainable yield fishery management strategies (MSY). Whatever target levels are set, this equation will allow fishery managers to determine the requisite effort level needed to meet that target in the face of climate predictions. Those seeking to minimize total effort while reaching a particular fishery goal for the entirety of an exclusive economic zone (EEZ) can set up a simple optimization problem to solve for the most efficient allocation of effort across their EEZ, using the form: Where: TE= total effort, the summation of effort in each coordinate square of the EEZ TC= a preset desired total catch target level across the EEZ = the catch in each coordinate square j=an individual coordinate square in the EEZ n= the total number of coordinate squares in the EEZ While the math for deriving this solution will undoubtedly grow quite complex, the relatively simple set up provides a useful framework for managing fisheries in the face of increasing ocean temperatures. This model can be readily adapted for the allocation of effort across time with little modification: Returning to the simple case of equation (14), where we are simply looking at the necessary effort for a single coordinate square, an example will help elucidate the usefulness. Suppose those who manage the fishery immediately surrounding Fiji's capital Suva (found in the 15°S by 175° E coordinate square) desire to increase their monthly Yellowfin catch from 28,000 to 30,000 kg in the face of an expected quarter degree rise in ocean temperatures, to 29.6°.In this situation, the fishery should seek to deploy around 260,000 hooks to reach this goal. It also bears stating that this model can readily be adapted for management in the multi-species fishery case. To reach a particular target level for all species, all that needs to be done is to calculate the effort required for each species in the same time period and coordinate square. The maximum of these values will be the best estimate for the effort needed to reach all of the species target goals, as longline effort is largely indiscriminate.Such a strategy will likely result in excess catches of the lesser required effort species. As such, conservation minded fishery managers would do well to employ the opposite strategy, utilizing the minimum effort value obtained and keeping all catch levels below targets.

Discussion
The above section has visited the applied value of the equations derived in the model specification and theory section of this paper. Ocean temperature is shown to have a significant effect on fishery yield, at least in the specific context examined: tuna fisheries of the Western and Central Pacific Ocean. The equations derived in this paper are of clear value to fishery managers, and point to the role that consideration of temperature must play in fishery management moving forward as the planet is faced with rising ocean temperatures.
All of the applications above have been focused towards the future, centered on the value that these tools have in planning for future fishery yield. However, they also have a diagnostic value in identifying strategies that are not best suited for our era of climate change. As with the derived models' value in planning for the future, much of the diagnostic value could be improved upon with the logical progression of the models beyond the scope of this paper, stemming from the point of equations (15) and (16). These models provide a basis for evaluating past allocative efficiency of effort resources and for identification of EEZs in which resources could have been better utilized to achieve the same or better results. While this avenue of discussion is beyond the scope of the study at hand, and perhaps provides a good starting point for future research, it is evident, at least logically, that the models herein described have the potential to improve upon the capabilities of current fishery management methodologies. The addition of sea surface temperature into existing bioeconomic modeling methodologies is essential to better mirror the realities of the marine ecosystem.
Equation (13), which allows us to examine necessary changes in effort level to maintain a constant catch level as temperature changes, demonstrates the direction in which stock biomass is trending with temperature change. It is able to demonstrate the simple fact that in many locales increasing temperatures are resulting in decreasing catch per unit effort.
With the model as derived and limited herein we are able to identify locales that are being less than efficient in their deployment of fishery effort, relative to their sea surface temperature.
However, all of these equations for deriving particular optimal effort levels given temperature solely provide for immediate term catch maximizing levels of effort. This, for obvious reasons, is not the best tool for sustainable fishery management. In order to sustainably manage these fisheries in the face of climate change, target catch levels need to be set that are aimed at maintaining long term stock viability in a dynamic environment. Only with these targets appropriately and accurately set, are the tools derived in equations (14), (15), and (16) able to contribute to fishery sustainability.
Setting appropriate target catch levels that take into consideration stock viability, climate change, and shifting demand due to changing human population dynamics will allow the models developed herein to be their most useful. Climate change must factor into both ends of the equation, temperature needs to be considered both in determining the ideal supply and in determining how best to achieve that supply level. The models derived have demonstrated, for all three tuna species examined, a high level of statistical significance for the interaction between temperature and fishery effort. It seems intuitive that the environment affects the production function of this natural resource. However, quantification of this relationship is necessary to effectively address the problems that climate change is causing for stock depletion, food security, and small island developing state economic viability.

Conclusion
This study has established the significant relationship that ocean temperatures play in the production function for three subspecies of highly migratory pelagic fish. The models derived provide a theoretical framework for extending the analysis to a broader range of species. From this model, various equations have been derived that are able to provide fishery managers with useful metrics and tools for planning and evaluation of fishery management strategies. It is crucial that as the effects of climate change become more pronounced fishery managers do not neglect the role that ocean temperatures play in the fishery production function.
The study at hand faced some data limitations, and would have benefited from higher resolution data. This would have allowed us to utilize smaller geographic coordinate squares and to more precisely estimate effects. Future study with higher resolution data would do well to also look at the spatial trends as well.
Several clear routes for future research have been made apparent by this study. Foremost, equations (15) and (16) have the potential to provide an extremely useful fishery management framework if properly explored. This is especially true if they are able to be paired with catch targets that accurately account for the changes to species' ecosystems and human population growth. Further, other environmental factors would do well to be incorporated into the general model of equation (5) and all of the other functions derived from it. Factors such as changing levels of ocean acidity, plankton distribution, and oceanic pollution would add immensely to the value of this model.
Despite the data limitations and the narrow scope of the model's derivations, it has contributed significantly in providing a base formula to aid fishery managers in determining what level of effort need be employed to reach management goals. Climate change is a factor that needs to be considered in all aspects of natural resource management for the sake of food security and economic development. This examination of the Western and Central Pacific's tuna fisheries has established the role that temperature plays in the marine resource production function.
Funding: This research received no external funding. However, upon acceptance for publication, an open access publishing grant from West Virginia University's Open Access Author's Fund will be applied for.
Conflicts of Interest:The author declares no conflict of interest Appendix A Table 9. Results from testing equation (5)