Natural mortality, M, is a critical parameter in studies of population dynamics and in the modelling of the fish stocks for the development and implementation of management programmes. Biased estimates of M affect the assessment analysis of resources, which can have severe consequences (Caddy 1991, Ramírez-Rodríguez and Arreguín-Sánchez 2003).

There is biological evidence showing a typical M-at-age trajectory in penaeid shrimps; there is an early, steep drop over a short time period, which includes the early life stages (eggs and larvae), rapidly converging on an M-value asymptote for ages approaching maturity (Caddy 1991, 1996, Ramírez-Rodríguez and Arreguín-Sánchez 2003). The existence of an asymptote gives rise to the conventional assumption of a constant M for stock assessment purposes. This happens for two main reasons: many fisheries target adult individuals and it is difficult to estimate variable mortality based on field data from harvested stocks. Although this assumption would reasonably apply to individuals of the same age and area (Caddy 1996), other conditions may lead to errors. Such critical cases are the sequential fisheries, particularly those of penaeid shrimps in tropical waters, in which the harvesting may occur throughout most of the life history of the species.

The Mexican Pacific shrimp fishery is the most important fishery in Mexico. It is primarily based on three shrimp species: white (Litopenaeus vannamei Boone, 1931), blue (L. stylirostris Stimpson, 1874), and brown (Farfantepenaeus californiensis Holmes, 1900). The last two species comprise 80% of the landings (INP 2012). Three fleets typically exploit these shrimps: the estuarine or inshore fleet, which targets mainly juveniles, comprises fishermen who built barriers across the estuarine channels called tapos to prevent juvenile shrimps from escaping to offshore ground. They catch them using atarrayas or throwing nets from land or in small boats or canoes (typically 5 m long) with no engine, or outboard engines typically of 15-45 hp. The coastal fleet, which targets late-stage juveniles and pre-adults, uses a boat or panga (typically 7 m long) equipped with one trawling net. Finally, the offshore or industrial fleet targets adults and uses vessels (typically 18 to 23 m long) equipped with two trawling nets with a 30- to 40-day autonomy at sea. The sequential nature of this fishery imposes the need for proper M-at-age estimations to assess stocks and inform management.

Estimates of M for Mexican Pacific penaeid shrimps are mainly based on the exploited component of the offshore stock, which is mainly comprised of adults, so there are few estimates for the juveniles, and almost no estimates for very young stages (Table 1). The extreme variations in the M interval values reported in the literature for the three species range from M_postlarvae = 46.8 year^–1 to M_adults = 0.78 year^–1 in white shrimp, M_preadults = 6.34 year^–1 to M_adults = 0.48 year^–1 in blue shrimp, and M_preadults = 8.5 year^–1 to M_adults = 0.24 in brown shrimp. With the exception of the estimate by García-Borbón (2009) (which was not included in the above ranges), who used the gnomonic model, these estimates were obtained through empirical methods assuming a constant M with age and time (Table 1).

The assumption of a constant M-value can be addressed by applying the gnomonic interval model proposed by Caddy (1991, 1996) and later expanded and improved by Martínez-Aguilar et al. (2005). The gnomonic model has been applied to short-lived species such as squids, Loligo forbesi and L. vulgaris of the English Channel (Royer et al. 2002) and Dosidicus gigas of the Gulf of California (Martínez-Aguilar et al. 2010), and to penaeid shrimps, Farfantepenaeus duorarum and F. californiensis of the Gulf of Mexico and the Gulf of California, respectively (Ramírez-Rodríguez and Arreguín-Sánchez 2003, García-Borbón 2009). Additionally, the model has been applied to some fish species, including the pacific sardine Sardinops caeruleus (Martínez-Aguilar et al. 2005) and the red grouper Epinephelus morio, of the Gulf of Mexico (Giménez-Hurtado et al. 2009).

Given the importance of the Mexican Pacific shrimp fishery and the sequential exploitation of the stocks, the objective of this study was to estimate the variation in natural mortality by life history stage in the three most important commercial penaeid shrimp species: white (L. vannamei), blue (L. stylirostris), and brown (F. californiensis).

Obtaining realistic estimates of the natural mortality of species with short lifespans, such as penaeid shrimp, represents a challenge for the fishery biologist. The white (L. vannamei), blue (L. stylirostris), and brown (F. californiensis) shrimps considered in this study are no exception. Though they have been intensively studied because of their commercial importance, there are still many aspects of their biology that are unknown, especially the early life stages, including their instantaneous rates of natural mortality. Independent estimates of M have been calculated in previous studies, but the majority of the literature is concentrated on the pre-adult and adult stages. The empirical equations used to estimate natural mortality were developed for adult stages and assume a constant M, and even when some authors have calculated M for juveniles, these estimates have usually been consistent with the biology of the species.

The gnomonic model obtains M_i estimates that better represent the biological reality than other models because it differentiates realistic life stages that correspond to different behaviours and habitats used by species throughout their life histories. Furthermore, Martínez-Aguilar et al. (2005) also proposed an algorithm that allows for calibration of the life stages.

The M_i values estimated for the three penaeid species were different, and this may be the result of their different life histories. The blue shrimp showed the lowest M_i values in four of the seven intervals (Table 5), and the least variation among the interspecific stages was observed in the egg and pre-adult stages (coefficient of variation [CV] = 2.5% and 4.86%, respectively; for others, CV ranged from 11% to 48.8%). The low variation in the egg stage is probably associated with its short stage duration (hours), and the pre-adult stage shrimp inhabit the continental shelf, so that stage is probably associated with relative low densities and, consequently, relative low vulnerability to predation as individuals are widely dispersed prior to their subsequent reproductive aggregation. From the nauplius to the adult stages, the CV of each M_i between species diminishes and continues to decrease with the duration of the stage until the pre-adult stage (Fig. 2). We hypothesize that this tendency suggests a similar pattern of behaviour between species in relation to their habitats, in which they share similar life conditions as they age. The pattern probably only changes for the adult and egg stages because reproductive processes and conditions are different between species, an aspect that is known in the literature (Magallón and Jaquemin 1976).

White shrimp showed the lowest M_i values of the larval mean range values ($\overline{M}$_1-4 =382.73 year^–1) as this species is probably less vulnerable to predation as a result of its rapid larval development in inshore habitats in contrast with the other two species. On the other hand, the brown shrimp lives in the offshore habitat throughout its lifecycle and showed a medium mean larval mean range value ($\overline{M}$ _1-4 =396.37 year^–1), which is probably associated with its life history as most of its metamorphosis occurs in the water column, where it is exposed to higher predation. Blue shrimp showed the highest M_i values for the mean larval stage ($\overline{M}$_1-4=461.54 year^–1); this is due to the second interval, or nauplius stage, which had an estimated value of M_i=684 year^–1, almost double the values of the other species (white shrimp, M₄=278 year^–1 and brown shrimp, M₄=358 year^–1). Blue shrimp showed a shorter duration and smaller size at this stage than the other two species (Table 4).

The blue shrimp is the dominant species in captures from the coastal zones, including the mouths of bays and estuaries (Rodríguez de la Cruz 1976, INP 2013). These zones are mostly exposed to environmental factors that contribute to higher larval mortality, including wave action and coastal currents that move larvae away from more favourable zones (Dahlberg 1979, Rumrill 1990). Furthermore, these zones serve as transit areas for the predators of penaeids. Local studies have reported predator species such as swimming crabs of the genus Callinectes and fishes, such as the snooks Centropomus robalito and C. nigrescens; the flounder Cyclopsetta panamensi; the catfish Ariopsis seemani; the snappers Lutjanus argentiventris and L. novemfasciatus; and the milkfish Chanos chanos (Sepúlveda-Medina 1981, Amezcua and Portillo 2010).

Previous shrimp stock assessment studies have primarily focused on the intervals most vulnerable to the offshore trawling fishery: pre-adults and adults. The gnomonic estimated values for both stages showed low variability among the shrimp species, possibly due to the convergence of all the species in the same zone during their migration to deeper marine waters (García and Le Reste 1986).

Globally, the M values reported for the adult penaeid stock varied between M=1.2 and 5.4 year^–1 (García and Le Reste 1986), encompassing the estimated values from the gnomonic model. The M values reported in the literature for adult Mexican Pacific shrimps ranged from 0.09 to 8.92 year^–1. Caddy (1996) observed that low adult values, such as M_i=0.2-0.5 year^–1, are possible but imply extremely high mortality in the egg and larval phases. Similarly, in accordance with Gracia (1997), the author found that adult M values higher than 5 are excessive and incompatible with an annual species. Thus, the magnitudes of the Mi values estimated with the gnomonic model in this study appear to be reasonable and compatible with the life histories of the three shrimp species.

The gnomonic model as proposed by Caddy (1996) and improved by Martínez-Aguilar et al. (2005) appears to provide biologically reasonable estimates of natural mortality during all life history stages. Fitting the estimates with independent data allowed for the identification of key life stages. Additionally, though some of the independent estimates of M for some life stages appear to be biologically inconsistent, they do not greatly affect the final estimation (even when included in the fitting process) because the pattern of the duration of time between life stages is governed by the properties of the gnomonic model, where parameter maintains the proportionality between the duration of the different life stages. The duration of the first life stage is most critical as an input for the gnomonic model because it affects parameter α.