Management strategies were evaluated using a spatial bioeconomic model (SBEM) (Anderson and Seijo 2010, Seijo and Caddy 2008). This model simulates the dynamics of an age-structured population with a heterogeneous distribution for a single species: H. corrugata. The distribution area of the stock was divided into 625 cells in a 25×25 array, each covering an area of 0.25 km² for a total of 156 km². The spatial distribution of stock abundance was obtained through the assessments in the fishing banks from 2000 to 2017 carried out by the National Fisheries Institute (INAPESCA). This database contains the number of organisms per 50 m² transect (sample unit), geographically georeferenced using a global positioning system. The database has 21576 vectors, with the following information: Year, Abundance (Number of organisms), Zone, X-coordinate and Y-coordinate (INAPESCA 2019). To represent and compare a continuous spatial density for the year in which the historical series begins (2000) and the year in which the moratorium starts (2017), the inverse distance weighted (IDW) interpolation method was applied through a geographic information system (QGIS.org 2022).

The commercial catch was included in the model through a database that incorporates the number of organisms captured, thus generating a specific spatially explicit harvest matrix from the year 2000 to the year 2017. The H. corrugata harvest database has 35513 vectors, with the following information. Year, Zone, Subzone, X-coordinate, Y-coordinate and number of organisms captured (Progreso 2020).

The growth parameters incorporating environmental variability and the 17 ages (species longevity) to be considered in the spatial model correspond to H. corrugata on the north Pacific coast of Mexico. Parameters are taken from Vargas-López et al. (2021). The model functions and parameters are presented in Table 1. The heterogeneous distribution in spatial recruitment was modelled using the recruitment function of Beverton and Holt (1957), multiplied by a negative binomial function (ɛ=15, µ=5), which allows spatially explicit patches to be generated with a probability of zero recruitment (Anderson and Seijo 2010). Additionally, this function has been used in simulation works of sedentary species that colonize different sites (González-Durán et al. 2018, Seijo et al. 2004, Seijo and Caddy 2008). Because of the highly selective nature of the fishery, no fishing mortality on sub-legal individuals occurs. An excel sheet developed by Anderson and Seijo (2010) was adapted to the specific spatial characteristics of the abalone fishery in the study region to conduct the spatial management simulations with the mathematical models described in Table 2.

This abalone fishery is managed by catch quota recommendations and in 2010 caught about 24 tonnes with an effort in fishing trips of 1125. As observed in other studies (e.g. Sanchirico and Wilen 1999; Cabrera and Defeo 2001; Hernández-Flores et al. 2018), the spatial allocation of fishing intensity (effort per unit of area) was based on the quasi-rents of the variable costs obtained in alternative fishing sites over time. The spatial allocation of effort over time was allocated over space in proportion to the site-specific profits obtained in the previous periods; when the income fell to zero in any area, the function stopped allocating fishing effort to it (Caddy and Seijo 1998). Thus, the number of daily fishing trips in each management strategy is determined by changes in abundance in the resource’s distribution area, the costs of fishing in alternative sites and the price of abalone. The Vt dynamic is calculated by numerically integrating (using Euler numerical integration with DT=1 in this case) the spatially adapted Vernon Smith (1969) function. A vessel makes one fishing trip per day, targeting only one species. Therefore, a single commercially exploited species determines the total costs and profits. The yellow abalone fishery is assumed to be price-taking, so its harvest does not affect the corresponding market prices. To simplify the analysis, constant prices were assumed over the simulation run. This dynamic bioeconomic model allows vessel exit in case of negative profits and restricts fishing not exceeding the maximum catch observed in 2010. Employment effects are negligible because community fishers have access rights to another high-value species (red spiny lobster, Panulirus interruptus), so they would not become unemployed when effort is at a level that maximizes the present value of resource rent, which involves less employment than operating at maximum sustainable yield.

To calibrate the SBEM, a comparison of the observed yield (Y_obs) and the calculated yield (Y_cal) for the first period (2000-2017) was carried out (Fig. 1). Statistical comparison of and was performed using the two-sample Kolmogorov-Smirnov (KS) test. The KS test statistic is D=0.333 and the corresponding p-value = 0.27. Since the p-value is greater than 0.05, we accept the null hypothesis. This indicates that the Y_obs and Y_cal datasets do not exhibit statistically significant differences. This allows us to infer that the suitable sensitivity of the SBEM foresaw a decrease in biomass and therefore calculated yields appropriate to this trend.

Once the SBEM was calibrated, management strategies for the yellow abalone fishery in the Mexican North Pacific were simulated and compared. The comparison between management strategies was based on the effect on biomass, predicted yield and present value of resource rent. The strategies evaluated are described in Table 3.

At the request by resource users in this co-managed fishery to lift the moratorium on the fishery, the aim was to identify the minimum effort (fishing days) at which the fishers can obtain an above-zero resource rent to cover the operating cost of four vessels and their opportunity cost of labour of moving to another high-value species such as the red spiny lobster Panulirus interruptus. As an assumption in the simulation period, the fishery management authority is considering this minimum effort when it reopens the fishery.

Simulations using these management strategies began with the assumption of lifting the moratorium on the fishery starting in 2023 and continuing for a simulation period of 17 years until 2040. This period is equivalent to one life cycle of H. corrugata.

The SBEM used a set of biological, economic and technological parameters that contribute to the calculated performance of biomass and present value of resource rent, so a sensitivity analysis was undertaken on the following parameters: parameter k of the von Bertalanffy growth , α and β stock-recruitment parameters, natural mortality M, price of species and catchability. The sensitivity analysis of the parameters was related to the performance variables such as final biomass B₂₀₄₀ and present value of resource rent. The results were expressed as correlation coefficients between the parameter and the output variable.

A Monte Carlo analysis was carried out using Crystal Ball Pro software (ver. 11.1.2.4.850) to estimate the risk of exceeding the limit reference point (LRP) for the yellow abalone fishery in the study area. With the appropriate probability density function (distribution of the parameter data to be analysed), this software allows the primary sources of parameter uncertainty to be represented by generating random variables of these parameters and estimating the risk of exceeding the LRP. Under controlled conditions (e.g. aquaculture), abalone growth is influenced by factors such as temperature and food availability (Britz et al. 1997; Morash and Alter 2016). Vargas-López et al. (2021) found that the relationship between growth and sea surface temperature (SST) was statistically significant for abalone species in wild conditions. Temperature regulates the expression of growth (Day and Fleming 1992, Pérez 2010, Vilchis et al. 2017); this effect is reflected by an acceleration of metabolism, which allows it to gain robustness faster, or a slowing of metabolism, which delays some vital functions (Essington et al. 2001, Renner-Martin et al. 2018). Also, SST had a direct effect on abalone growth. These changes in size have been described as the “third ecological response to global warming” (Daufresne et al. 2009). Uncertainty was incorporated within the SBEM in the parameter k of the von Bertalanffy growth . The SST variability predicted for 2023 to 2040 was undertaken by varying, with a uniform probability density function, the reported environmentally driven k’ values ranging from 0.32 to 0.38 for H. corrugata (Vargas-López et al. 2021).

Bioeconomic reference points are shown in Table 4. The biological LRP is determined by the biomass level that conditioned the closure of the fishery for the yellow abalone in 2017, while the biological target reference point (TRP) is determined by the biomass level at maximum sustainable yield. The economic LRP is determined by the resource rent that covers the operating costs per vessel (RR_pv) and the opportunity cost of work when another high-value species such as the red lobster Panulirus interruptus is targeted. This value will be known as the minimum resource rent (RR_min). The economic TRP is based on the proposal of the fishing sector, in which they suggest that the optimum resource rent RR_opt be 50% higher than RR_min.

To represent the variation in abalone density, two plots were made using IDW interpolation (Fig. 2A and B). At the beginning of the historical series (2000), there was extensive coverage of abalone in the fishing zone; in 2017, this coverage fell considerably, as shown in Figure 1A and B, and the average density per sample unit decreased from 0.144 to 0.062 ind/50m² during that period. These changes and the reduction in density were the reason for the moratorium agreement for the fishery.

Biomass sensitivity results for the three management strategies under consideration are shown in Table 5. The range of values are the following: natural mortality (-54.1%-39.5%) and parameter k of growth function (21.2%-35.2 %), followed by beta (-12.7%-0.9%) and alpha (9.7%-22.3%) parameters of recruitment function. Biomass sensitivity to catchability ranged from -0.4% to 6.0%.

The present value of resource rent sensitivity to the above-mentioned parameters for the three management strategies under consideration is presented in Table 6. The range of sensitivity values are the following: natural mortality (-61.1%-38.8%), catchability (10.9%-14.6%), parameter k of growth function (6.5%-13.9%), price of species (7.5%-20.4%), and alpha (8.6%-13.7%) and beta (-6.9%-0.4%) parameters of recruitment function. In addition to the Monte Carlo calculated magnitude, the indication of positive or negative sensitivity to changes in parameters resulted in the direction expected for final biomass and present value of resource rent.

When performing one thousand simulations of the calculated values for the observed period, incorporating uncertainty in the parameter k’ of the von Bertalanffy growth function, variability in the calculated values was observed; this was mainly determined by the wide range of the assumed value (k’) in the uncertainty analysis (Fig. 1A). It can be inferred that during the period 2004 to 2010, the values of Y_obs and Y_cal were very similar. After this period, there was a difference to consider. As of 2011, the SBEM was already calculating lower catch values than those observed. It is noteworthy that for 2013, the Y_obs coincided with the outlier value of the Y_cal trajectory calculated for that year. From 2011 to 2016, the Y_obs was above the values calculated through simulations. The SBEM thus suggests that catches from 2013 to 2017 decreased from approximately 15 to 8 t, a situation leading to closure of the fishery after 2017.

The biomass levels projected by the SBEM considering the MSE are presented in Figure 1B. This graph clearly shows the differences between biomass trajectories. Different levels of fishing effort determined by the management strategies evaluated generate levels of final biomass that are important for consideration by the stakeholders.

In the simulation period, when the moratorium was lifted and harvesting of the resource began, a decrease in biomass levels was observed in the three management strategy trajectories. Subsequently, the dynamic catch quota effect was observed four years after reopening the fishery. Derived from this, the E_min management strategy tended to equilibrium biomass in less time than the other management strategies. The E_min and E_maxNPV management strategies allowed higher biomass levels than the E_msy management strategy at the end of the simulation period. As expected, E_min (lowest effort) was the management strategy in which the highest biomass level was observed at the end of the simulation.

The yield trajectories projected by the SBEM considering the three management strategies show decreasing trends, although in different magnitudes, towards the end of the simulation period. (Fig. 1C)

As expected, different levels of effort generated different levels of forecasted yield (Fig. 1C). E_msy predicted yields above E_maxNPV because E_msy operates in open access concerning the number of daily fishing trips per season (status quo). E_maxNPV has a catch quota restriction based on the assumption of not exceeding the maximum recommended quota in the observed period of the fishery. Nevertheless, it must maximize the economic rent generated by the resource. For this reason, the trajectory of this management strategy was lower throughout the simulation than E_msy but higher than E_min. The magnitude of the difference in yield trajectories should be recognized by authorities and fishers when they consider reopening the fishery due to the requirements in the number of vessels needed to catch the forecasted values: E_min predicted an initial yield of 9 t, and a final yield of 6 t, while, E_msy predicted an initial yield of 31 t and a final yield of 12 t. This management strategy suggests higher catch quotas than those recorded in the observed period.

To compare the resource rent for the alternative MSE, resource rent per vessel is likely to be considered as a decision variable by resource users and authorities. At the beginning of the simulation, mainly because of the low mobility, high catchability and consequently high value and low costs of the resource in the fishing operation, the resource rent generated by the fishery was 50% higher in E_maxNPV than in E_min and 75% higher than in E_msy (Fig. 1D). However, it gradually declined as the dynamic catch quotas developed. At the end of the simulation, resource rent per vessel in E_maxNPV and E_min were higher than resource rent per vessel in E_msy.

Other alternative discount rates could be used for different contexts to calculate the present value of resource rent. Table 7 shows the effect on present value of resource rent for the three options for three alternative rates of discount.

In Figure 3, a comparison of net present value (NPV) per vessel for the MSE is shown. The NPV was considerably higher in E_maxNPV, three times higher than that in E_msy. The high yield at the beginning of the simulation, the high value of the species, and the low total costs are the variables that determine this considerable difference. Another critical factor conditioning this high NPV is the spatial allocation of effort determined by the SBEM.

It is crucial to compare NPV values per vessel, thus generating a fundamental decision criterion for fishers and authorities. Dividing E_msy NPV by 25 vessels, we obtained USD 436570 per vessel. In the case of E_min NPV divided by four vessels gave USD 866662 per vessel. In the long term, the E_min management strategy generated a higher NPV per vessel than E_msy, requiring a lower number of vessels and optimal results in final biomass and yield. In the case of E_maxNPV, the NPV value divided by the six vessels (six is the total number of vessels needed to perform 471 fishing trips in the fishery season) gave a total of USD 1073520 per vessel, which is the maximization of the NPV in the MSE.

Table 8 shows the management strategies, effort units, final biomass, final yield and final NPV per vessel per strategy at the end of the simulation period. This table offers clear visualization of the SBEM outputs, which implicitly include the fishery’s social, economic and biological considerations-crucial characteristics for decision-making.

A Kolmogorov-Smirnov (KS) test was conducted to compare the observed yield and the calculated yield in the first period (Fig. 1A). The KS test statistic is D=0.333 and the corresponding p-value = 0.27. Since the p-value is greater than 0.05, we accept the null hypothesis. This indicates that the Y_obs and Y_cal datasets do not exhibit statistically significant differences. This allows us to infer that the suitable sensitivity of the SBEM foresaw a decrease in biomass and therefore calculated yields appropriate to this trend.

After generating 1000 simulations, we calculated the probability of B₂₀₄₀ falling below the LRP of B_min and the TRP of B_msy. In the case of resource rent, we calculated the probability of having an RR_pv greater than the LRP of RR_min and the TRP of RR_opt at the end of the simulation period (RR₂₀₄₀). The biomass at the level in which the fishery went to closure in 2017 (B_min: 190 t) and the minimum resource rent to cover the operating cost of one vessel and its opportunity cost of labour of moving to another high-value species (RR_min: USD 15000) were used as LRPs; the biomass at maximum sustainable yield (B_msy: 244 t) and a fisher-proposed optimum resource rent per vessel (50% higher than RR_min) (RR_opt: USD 23000) were used as TRPs (Figs 4 and 5).

In the case of biomass, the three management strategies showed probabilities of falling below the TRP (dark area under the probability chart), but E_msy was the one that had the greatest risk of falling below it. Operating at this level of effort increased the risk (86%) that the biomass at the end of the simulation would fall below the desirable level point. In addition, with this strategy, there was also a 30% risk of falling below the LRP, the same biomass level that caused the closure of the fishery in 2017. The other two strategies involved zero risk of falling below this undesirable biomass level.

The resource rent is a crucial variable in fishermen’s and authorities’ decision-making. For this reason, two reference points were evaluated (TRP and LRP). Both E_min and E_maxNPV had a 100% probability (grey area under the probability chart) of being above the LRP. E_maxNPV was the strategy with the highest probability (47%) of exceeding the TRP of USD 23000 per vessel. E_msy was the strategy with a 0% probability of being above both reference points. In economic and sustainable terms, it is not a suitable strategy to be employed, because it shows a high risk of falling below biomass at maximum sustainable yield and it generates zero probabilities concerning the economic benefit that fishers could obtain from the resource in terms of reference points. A summary of the probabilities of exceeding LRP and TRP are shown in Table 8.

Once these reference points have been evaluated, as Anderson and Seijo (2010) comment, the fishers and authorities (decision-makers) and their inherent attitude towards risk will determine which strategy should be implemented in the reopening of the fishery and whether to reduce risks of returning to an undesirable biomass level that would suggest another fishery closure. It is necessary to identify the strategy or strategies that provide resource users with the maximization of the economic rent generated by this species while at the same time avoiding overexploitation.