In this paper, we present a case study of the blue shrimp (Penaeus stylirostris) fishery in coastal marine waters of the central-eastern region of the Gulf of California ( Figure 1 ). Two fleets participate in this fishery: one small-scale fleet that operates with line seines (mesh size 63.50 mm) and “suripera” nets (mesh size 31.75 mm), and another industrial fleet that operates with trawl nets (mesh size 50.8 mm in wings, and 38.1 mm in cod end). The characteristics of the boats and fishing gear can be found in various documents (Villasenor-Talavera, 2012; DOF, 2018). As a source of information, there are daily catch records and effective fishing days which is a measure of fishing effort; both were reported in the official arrival records (CONAPESCA, https://conapesca.gob.mx/wb/cona/avisos_arribo_cosecha_produccion) during the period from 2007 to 2020. In all cases, information was arranged in monthly periods during the fishing seasons that ran from September to March. Monthly catch and corresponding fishing effort per fleet are included in the supporting materials.

The fishing season begins in September and ends in March (DOF, 2018), and the main objective of the closed season is to protect the breeding population. Since it is a population with annual longevity, it is assumed that the abundance of the population at the end of the fishing season corresponds to the spawning stock on which the recruitment success depends. In the absence of more information, the best approximation of the recruitment is the size of the stock, which can be approximated from the catch per unit effort at the beginning of the fishing season.

Blue shrimp require shallow and estuarine habitats as nursery grounds, and marine waters for maturity and reproduction. Interannual changes in abundance are affected by environmental patterns such as the Pacific Decadal Oscillation, sea surface temperature, rainfall, upwelling (Aragón-Noriega, 2007; Castro-Ortiz and Lluch-Belda, 2008; Almendarez-Hernández et al., 2015); as well as by regional circulation patterns that affect larval distribution and recruitment (Calderón-Aguilera et al., 2003), which confirms that the carrying capacity is variable between years.

The data includes records of the two fleets operating every month during the fishing seasons. More than 80% of the blue shrimp yields were taken at depths less than 20 m (54% between 10 and 13 m; and 97% up to 30 m) (Muñoz-Rubí et al., 2023). For this reason, and in the absence of other data, the operation scheme assumed for the analysis is that the two fleets use the same resource in time and space. Given this assumption and that the fleets operate with different fishing boats and gears, it is assumed that if the fleets had the same fishing power, the catch per unit effort per month should be U i , m = b · U j , m , where U is the catch per effective day of fishing, i and j represent the two fleets, and m is the monthly time unit during the fishing seasons. If the previous assumption were fulfilled, b , the slope of the relationship between U i , m and U j , m should be 1 , while if b ≠ 1 , the resulting value of b will correspond to the conversion factor among fleets. In this way, the U data is standardized between fleets.

The Leslie model (Leslie and Davis, 1939; Hilborn and Walters, 1992) is a very useful tool for estimating the catchability per year for a closed population, a criterion that can be assumed in our case study of a species with annual longevity, and an estimate of the size of the population at a time just before starting the stock assessment experiment. For fleet i , the model is described by the relation:

where U i , t is the catch per unit effort at time t , in our case a month; N 0 , t is the size of the population just before the start of the estimation at time t ; K t − 1 is the cumulative catch at time t − 1 ; and q i is the catchability coefficient, which is assumed to be constant.

In this way, for a given fleet i , the catchability estimator is, for each year, q i ,   y ( m ) , where y ( m ) represents the month m at each fishing season (year) y , with a similar solution for the larger fleet j . On the other hand, the ordinate to the origin in Equation 3, U 0 , would approximate N 0 since N 0 = U 0 / q ; similarly, the size of the population each month can be estimated as N m = U m / q , and the size of the population during the entire fishing season is

The state of exploitation of the population can be derived from the harvest rate, represented as H R = C / B . Theoretically, for a stable population, in a stable environment, MSY is obtained when H R = 0.5 (Gulland, 1983). For a population with annual longevity, this means that the remaining population can replace losses due to fishing on an annual basis. Based on this concept, an estimator of the state of exploitation of each fishing season can be obtained from the relationship C y / B y   = H R y and a similar calculation could be made for each month if it were necessary to know these data.

From this information, a Kobe diagram can be obtained using as a reference point an estimator of the limit rate of population renewal, P R R L i m and the state of exploitation given by the estimated harvest rate. The P R R L i m , y is obtained from the relationship between the recruitment rate, ρ y , at the beginning of the fishing season with respect to the survival ratio of the cohort, s y , represented by the ratio between the abundance in the last month of the fishing season with respect to the accumulated biomass at the end of the fishing season, B y . The base assumption in this relationship is when s y = ρ y ; that is, the stock level is analogous to that defined as the replacement level, or the level of recruitment necessary to maintain a population with constant density (Shepherd, 1982; Daan, 1988; Mace and Sissenwine, 1993; Garcia, 1996). With the information available from our case study, s y and ρ y can be estimated as follows:

and

where B m a r , y represents the biomass in the last month of the fishing season (March) corresponding to breeding adults, while B o c t , y + 1 represents the biomass in the month just before the start of the fishing season corresponding to recruits.

Then, P R R L i m , y = ρ L i m , y s L i m , y = 1 , and the Kobe diagram can be constructed with H R y in reference to s y or to ρ y levels when s y = ρ y , conditions represented by s L i m , y and ρ L i m , y

One of the characteristics of short-lived populations is their dependence on the environment. In the case of penaeid shrimp, this feature has been widely reported in the literature (Garcia, 1988; Garcia, 1996; Arreguin-Sanchez et al., 2015; Diop et al., 2015; Lopes et al., 2018), including climate variability and its effects on blue shrimp populations the Gulf of California (Galindo-Bect et al., 2000; Castro-Ortiz and Lluch-Belda, 2008; Perez-Arvizu et al., 2009; Santamaria-del-Angel et al., 2011; Páez-Osuna et al., 2016; Cota-Durán et al., 2021). In the case study, given the evidence reported in the literature, a brief analysis of the contribution of environmental variability in the context of the Kobe diagram, particularly related to P R R L i m , is provided. In the present case study, two environmental variables associated with specific biological processes were considered: primary production represented by the concentration of chlorophyll-a, C h l _ a , which is associated with survival throughout the fishing season, as an index of food availability, and sea surface temperature March ( S S T m a r ), which is associated with reproductive success and, consequently, with recruitment success.

Since only monthly catch and effort data are available each year and blue shrimp is a species with annual longevity, it is necessary to make some assumptions about stock assessment. A sequentiality in the operation of the fleets is generally assumed, meaning the small-scale fleet retains smaller organisms than the industrial fleet. In this context, it is also assumed that the operation of the small-scale fleet starts before the industrial fleet, depending on the migration process of the species and its growth. In the case of the blue shrimp, as seen in the available data supply (capture in weight but without size structure and effective fishing days as fishing effort), the fishery does not exhibit this operational scheme (as can be seen in S1). Fishing in coastal waters up to 5 fathoms (9.15 m) deep is prohibited; and from that depth, both fleets begin to operate. In the case study, the data do not show a time lag among fleets and sequentiality is not considered (see Supplementary materials S1 ).

Reproduction occurs between June and August (Aragón-Noriega and García-Juárez, 2002; Calderon-Aguilera et al., 2003; Lopez-Martinez et al., 2005; Aragón-Noriega, 2007); therefore, recruitment to the fishery is reflected at the beginning of the fishing season since there is a single annual cohort. According to the available data (catch and effort data), in September, when the fishing season begins, the small-scale fleet retains 43% of the annual catches, while the industrial fleet retains 39%. On the other hand, the Leslie model (Leslie and Davis, 1939; Hilborn and Walters, 1992) was initially developed for individual counts. In our case study, the model is solved with biomass data so that Equations 3 and 4 are defined as follows:

and

where q ′ ≠ q given the units in each case, one referring to the number of individuals and the other to biomass units, and k represents the cumulative capture in weight. Parameters q t ' and B 0 , t in equation (3’) can be estimated by regression or some analogous procedure.

This method assumes that the assessment experiment is carried out on a closed population, where recruitment occurs only at the beginning of the season and no later, and there is no migration to the area during the time of the experiment. In the case study, this assumption is acceptable since it is an annually closed population. Additionally, an assumption that is not generally made explicit is natural mortality, in the case of the model, it is assumed to be constant so that the trend for the “depletion rate” represents changes in fishing mortality rate.

The model assumes a constant average annual catchability rate, although conceptually, several sources of catchability variation have been mentioned in the literature (Arreguín-Sánchez, 1996; Arreguín-Sánchez and Pitcher, 1999). Consider in equation 3’, for a year y , that U y = q y E y , where U y represents the relative abundance of the stock in biomass, and E y the fishing effort. Then, q y is a coefficient representing the interaction between the stock and the fishing effort, such that anything that affects U y and E y also affects q y .

There is not a single level of replacement given the variability of the stocks (Shepherd, 1982), especially in exploited stocks showing interannual variation; in these cases, there is a different level of replacement according to each annual recruit/adult combination. This concept is especially valid for species with annual longevity and is especially critical for P R R L i m , y estimation. In this sense, the conventional assumption of a stable carrying capacity is not appropriate for these estimates, and the annual value of P R R L i m , y may be estimated each fishing season.

Figure 2 shows the monthly trends of the observed data for catch ( C m , y ), effort ( E m , y ) and catch per unit effort U m , y for each fishing season and fleet, while data are provided as Supplementary material S1, and the global patterns for all fishing seasons are shown in Supplementary materials S2 , Supplementary figure S2.1 .

As previously mentioned, two fleets participate in the fishery, one small-scale, s s f , and the other industrial, i n d ; and it is assumed they compete for the same resource. In both cases, the effort is recorded as an effective day of fishing; however, given the difference in equipment, boats and fishing gear, a conversion was made to standardize the fishing effort across fleets. An example of such standardization is given in S2 , Supplementary figure S2.2 .

Assuming the fleets operate simultaneously in time and space and that the U is a measure of relative abundance, then the fishing effort of a fleet could be estimated from the following equality.

Since this equality is not fulfilled (because of the differences in fishing power and gear used by the fleets), the effort of the small-scale fleet can be estimated in the units of the industrial fleet as follows:

In this way, the U s s f _ s t , m , y the standardized ( st ) U in terms of the units of effort of the industrial fleet.

Once the information was standardized, the catchability was estimated for each fleet and year. For this, the Leslie and DeLury model (Leslie and Davis, 1939; Hilborn and Walters, 1992) was used showing some examples in Supplementary figure S2.3 within S2 . Figure 3 shows the values of catchability by fleet and year. The order of magnitude for the first three fishing seasons considered (2005/6, 2006/7 and 2007/8) was clearly higher than that for the other seasons. However, when considering the 2010/11 fishing season, an apparent decreasing pattern is observed, which is more accentuated for the industrial fleet.

The capture is defined as C = q E B (Gulland, 1983), such that B = C q E = U q . This relationship is used to estimate the biomass, monthly, by fleet, or for the total fishery (adding the estimated biomass individually for each fleet), and for all fishing seasons. In this way, estimates of the decline of the biomass of the shrimp population from the beginning to the end of the fishing season and for the accumulated biomass during the entire fishing season are obtained ( Figure 4 ). The estimates of q y and B y are shown in Supplementary material S3.

Given the values of C m , f , y and B m , f , y , an estimate of the corresponding harvest rate for each month was obtained, H R m , f , y , and from its average, an estimate for the year, H R y . On the other hand, from the annual catch C y and biomass B y , a second annual harvest rate estimate was obtained, H R y ' . Figure 5 shows the trends of the values of the two harvest rates for fishing seasons.

The estimation form for H R y and H R y ' resulted in slightly different values for each fishing season ( Table 1 ). The same Figure 5 shows the relationship between both estimates, resulting, on average, in H R y > H R y ' , which, according to the relationship shown in Figure 5 , H R y ≈ 1.029 · H R y ' .

Two quantities of interest for management can be estimated from the monthly biomass data for each fishing season; one of them refers to the recruitment to the fishery, which occurs in September, at the beginning of the fishing season, and represents the reproduction success and the other is the survival of the cohort, which represents the remaining biomass at the end of the fishing season. The proportion of the stock surviving at the last month of the fishing season is an indicator of the remaining biomass of adults whose reproductive success will lead to recruitment to the following fishing season.

Equations 5 and 6 represent estimators of the survival ratio and recruitment rate, respectively, obtained from the estimated monthly biomasses. Figure 6 shows the recruitment rate and survival ratio, both as a function of the remaining biomass at the end of each fishing season B m a r , y (S3). For the estimations we assume biomass B can be approached by catch per unit of effort data U , for the needed times.

The relationship between the recruitment rate, ρ y , and the remaining spawning biomass, B m a r , y , toward the end of the fishing season suggests a stock-recruitment relationship where density-independent processes predominate Beverton and Holt (1957). The same type of relationship appears with respect to survival ratio s y , showing some stability around s y = 0.062 (crossing point in Figure 7 ). According to Figure 6 , and regarding the α value referred to survival ratio in Figure 5 , the remaining biomasses greater than 1,500 tons would not have a significant impact on survival, while remaining biomass, less than 1,000 tons, could affect the survival rate. Figure 7 shows the relationship between ρ y and s y , where the bisector of these relationships reflects, analogously, the replacement level (Mace and Sissenwine, 1993; Garcia, 1996) in such a way that the crossing point between the trend of the observed relationship with the bisector can be interpreted as a limit reference point, in this case defined as the limit rate of renewal of the population, P R R L i m , which represents the replacement level. It is assumed that the s y corresponding to P R R L i m is the size of the remaining spawning biomass necessary to produce the recruitment level that will eventually replace the stock and the spawning biomass necessary for the persistence of the population. The estimate for P R R L i m in Figure 7 , in terms of the survival rate, was s y = 0.073 , while the maximum given by the upper confidence interval (95%) is P R R L i m , m a x , s y = 0.106. Note that point β in the figure representing P R R L i m , m a x could be identified, if necessary, as a limit based on a precautionary approach.

As previously mentioned, the estimations of s y and ρ y , as well as the estimated annual harvest rates, H R y , allow the construction of a Kobe diagram, as shown in Figure 8 , where the reference levels correspond to P R R L i m ; in this case, relative to the survival ratio, where s y = 0.073 . and for s y , m a x = 0.106 . Additionally, Figure 9 shows the Kobe diagrams corresponding to P R R L i m relative to the recruitment rate, with ρ y = 0.073 . and ρ y = 0.106 .

In the Kobe diagrams, the harvest rate, H R , is used as an indicator of the state of exploitation of the population, representing the ratio among catch and the available biomass. In this context, the values of H R > 0.5 assume overfishing since the remaining population will not be able to replace the losses due to fishing. Additionally, a limit harvest rate value derived for shrimp from ecosystem attributes (organization and function) is taken as an precautionary criterion, in this case H R = 0.43 (Arreguín-Sánchez, 2022a).

Statistically significant polynomial relationships were found between survival ratio, s y , and C h l _ a ; and between recruitment rate, ρ y , and S S T m a r (see figures in S2 and data of the anomalies of C h l _ a and S S T m a r in S3). These relationships were used to estimate theoretical values for s y and ρ y and estimate a new P R R L i m influenced by the environment ( Figure 10 ).

From the information of the anomalies of C h l _ a and S S T M a r , their rate of change was obtained, which was used to adjust the relative values of s y and ρ y (see figure illustrating the difference with and without the environmental contribution in S2 , Supplementary figure S2.4 ).

With the above information and to estimate the partial contributions of the environment in the estimation of the limit reference point, the relationship between ρ y and s y was reconstructed in the absence of an environmental effect to estimate an alternative P R R L i m ' . The result is shown in Figure 11 , where for P R R L i m ' , the value of s y ' = 0.068 was estimated. Accordingly, the difference between P R R L i m and P R R L i m ' (with and without environmental effects) is approximately 7%, where P R R L i m > P R R L i m ' .

The position relative to the state of exploitation of the blue shrimp resource throughout the evolution of the fishery in the Kobe diagram changes slightly when considering the effects of the environment. Figure 12 shows the Kobe diagram as a function of s y and ρ y , showing the relative position of the P R R L i m in respect the replacement level and the corresponding harvesting rates. In addition, Figure 12 shows the P R R L i m by considering the population and ecosystem perspectives. Such P R R L i m contrasts can be useful for management, particularly by considering precautionary approach into the decision-making process.

In the introduction section, mention was made of the methodological conflict when attempting to use the biomass dynamic models for short-lived species (see equation 2). To address this problem, other approaches have been proposed, such as depletion models, which are based on the Leslie and Davis (1939) model following the solution proposed by Chapman (1974) given by the relationship,

which, as proposed by Pope (1972), assumes that capture occurs in the middle of the time periods being t a given time interval, i refers to the number of time intervals, M is the natural mortality rate, and the catch, C , is expressed in number of individuals.

Depletion methods have been applied to various fished stocks (Rosenberg et al., 1990; Roa-Ureta and Arkhipkin, 2007; Maynou, 2015; Roa-Ureta, 2015; Arkhipkin et al., 2021); where the solution of equation (10) requires a little more information compared to the procedure proposed in this work. For example, to express C in numbers, it is necessary to know the average weight of the individuals, or the size structure of catch, and the weight-length relationship. Likewise, an independent estimate of M is required. With this base, equation 10 is solved for N 0 and q . In the procedure proposed in this paper, the q ′ and B 0 were estimated based on weight (biomass) and were obtained by regression, although any other analogous procedure may be applied if deemed convenient. Additionally, in this work, some population performance indices were proposed, derived from U t data, such as survival and recruitment rates. These indicators were used to estimate biological reference points and construct Kobe diagrams to identify the state of exploitation of the exploited stocks, and to suggest management trends.

Limited data are a challenge for many fisheries for which information on the state of exploitation of fish resources is needed, and typically, one must use the existing information, methods, and analysis tools available. In the case of the blue shrimp fishery in the central-eastern Gulf of California, the information available consists of daily catch records and effective days of fishing for the two fleets, one small-scale and the other industrial, that operate on the continental shelf. Although sequentiality in this fishery is a common assumption, when grouping data monthly by fleet, we found that there was no evidence of sequentiality, since the fishing operations start at the same time, and fit the paradigm of two fleets competing for the same resource. One of the characteristics of sequentiality in shrimp fisheries is that the fleets exploit different life stages, such that the yields of one fleet depend strongly on the fishing intensity of the other through survival; thus, the limits of fishing for one fleet must be defined in such a way that enables the operation of the next fleet. In the available information, there was no evidence of this situation; in contrast, assuming only one recruitment pulse, the fleets seemed to respond simultaneously to it in time and space so that competition between fleets appeared to be a reasonable model.

If we assume that the fleets operate simultaneously in time and space, the approximation of the standardization of the fishing effort is also reasonable in terms of units of effort (in this case, effective fishing days); however, the fishing power, fishing equipment and gear differ between the fleets, a situation that is reflected in differences in catchability. In the case of the blue shrimp fishery, the probability of retaining a unit of fish resource for a unit of fishing effort is different according to the relative fishing power of each fleet. This is evident when applying the Leslie method (Leslie and Davis, 1939; Hilborn and Walters, 1992); if the fleets had the same fishing power, they would exert the same fishing mortality per effective day of fishing, and the removal of individuals during the capture operations would be similar, followed by catchability. This did not always occur (see Supplementary material S3 ); the values for the industrial fleet ( Figure 3 ) were always greater, almost 8 times greater for the period from 2010 to 2020.

On the other hand, although variability in catchability has been reported throughout the fishing season and with sizes (Aranceta-Garza et al., 2020), since there is no detailed information available for this study, the constancy assumption is considered acceptable. Note in this context that, beyond the differences in the relative fishing power of each fleet already mentioned above, the cohort effect, which implies variability based on size and season, impacts the two fleets similarly when operating simultaneously in space and time. This last concept is a key assumption for which there is no information available that can be used for estimation.

The estimation of the stock size was initially carried out by fleet, considering the differences in magnitude of the catchability coefficient, and the global estimator of biomass of the stock was obtained by simple addition. This estimate corresponds to the biomass available for fishing and not necessarily to the total biomass of the stock, although, presumably, most of it is accessed by the fleet. Given this condition, for management purposes, the estimated biomass should be considered equal to the stock biomass.

Two parameters were of great interest, especially when dealing with a stock with annual longevity. Survival to the last month of the fishing season, represented by s y , indicates the remaining biomass, which, in turn, given the life history, represents the breeding stock whose reproductive success will result in the cohort the following year, which will, in turn, give rise to the stocks for fishing. The other parameter is the recruitment rate ρ y , which represents the magnitude of the cohort available for fishing. The interaction between these two concepts establishes a relationship between the remaining biomass of one fishing season and recruitment for the next season (recruitment rate). Thus, the equality s y = ρ y represents the replacement level; that is, the magnitude of recruitment is equivalent to the magnitude of the spawning stock necessary for the population to persist. Consequently, this balance represents the limit reference point for the exploitation of the stock, estimated as P R R L i m = 0.073   (crossing point α in Figure 7 ), meaning the remaining stock at the end of the fishing season must be 7% of that at the beginning of the fishing season. In this sense, exploitation can be developed until reaching that limit level. From Figure 7 , and in a precautionary sense, we suggest the limit P R R L i m , m a x = 0.106 (crossing point β ). In terms of survival, this fishing limit was defined as 10% of the remaining biomass at the end of the fishing season. This concept is similar to the strategy defined as proportional escapement (Beddington and Cooke, 1983; Beddington et al., 1990; Rosenberg et al., 1990). Figure 13 illustrates the decline in relative abundance per fleet and total biomass throughout the fishing season. In all cases, the fishery was within the precautionary levels of exploitation.

The proportions of abundances at the end of the fishing season with respect to the beginning of the fishing season resulted in an escapement of close to 15% of the population, a figure slightly higher than that from the estimated biomass, which was 12%. In contrast, P R R L i m suggested values of 7% and 10%, based on a precautionary approach. This means that, in terms of escapement, the fishery is operating at safe levels of exploitation.

The effect of the environment was considered in two ways: through the primary production index expressed annually as C h l _ a concentration and interpreted as an indicator of food availability that could affect survival; and through the surface temperature of March, S S T M a r c h , acting as an indication of the quality of the environment at the time when the reproductive process begins. The anomalies of the two indices were used to identify the contribution of the environment to the variability of the survival and recruitment processes. According to the results, the environment contributed approximately 7% to the estimate P R R L i m obtained from the observed data; that is, the one P R R L i m without the environmental contribution was defined by lower values of s y and ρ y . As environmental variability, in terms of management, brings uncertainty, the recommendation is to use these estimates of P R R L i m , within a precautionary management scheme. For the blue shrimp, the s y and ρ y estimated without environmental effects could also be used to isolate the population response to fishing.

In the Kobe diagrams, when the survival ratio, s y , was used as a criterion, we found that, for the last 15 years, only three exceeded the limit harvest rate of H R = 0.50 , and five exceeded the harvest rate corresponding to the limit reference point based on the ecosystem criteria ( H R = 0.43 ) ( Figure 12 ). When recruitment rate, ρ y , was used as a criterion, no season fell within the critical zone, above H R = 0.50 , and five were located above H R = 0.43 ( Figure 12 ). The interpretation of the Kobe diagram indicates that the state of the blue shrimp fishery in the central-eastern region of the Gulf of California, given in Figure 12 , is typical of a sustainable fishery that must be managed by limiting its operation. In the case study, the limit levels for the survival ratio, s y , and the recruitment rate, ρ y , fell within the limits of exploitation that permit a sustainable fishery, even under a precautionary approach.

Currently, management measures consider the definition of the closed and start of the fishing season, size of first capture, prohibition of fishing within nursery areas, and controls on fishing gear and methods (DOF, 2018). Although the blue shrimp fishery in the central-eastern region of the Gulf of California is exploited at a level of sustainability close to its maximum production capacity, there is no harvest limit as a reference for management. In terms of the criteria defined in this work, this limit can be established to guarantee sustainability. According to the results, and the recent history of the fishery, it can be established as the limit harvest rate HR=0.43; which implies, in annual terms, that the proportion of capture regarding the biomass of the population must be a maximum of 43%. In practice, this quantity could be established by estimating the initial size of the population at the beginning of the fishing season (September) and monitoring the decline in abundance until reaching a limit of 10% of the initial abundance, which corresponds to the P R R L i m , estimated in this work considering the survival ratio, recruitment rate and the environmental effects.