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Application of a gnomonic model to estimate the life span and natural mortality in Panopea globosa.


Natural mortality estimates are commonly computed from empirical methods or catch curve analysis, and their values are assumed constant for age or size in a population; however, estimates of natural mortality usually vary spatially, temporally, or by size and age. Several factors affect natural mortality rates, such as predation, disease, senescence, cannibalism, starvation, or environmental factors. Seven gnomonic time divisions (GTD) were used to estimate the natural mortality of Panopea glohosa for specific portions of its life history: 1, egg to trochophore larvae (24 h); 2, early larvae (6.5 days); 3, late larvae (11 days); 4, early juvenile (35 days); 5, juvenile (3-9 mo); 6, late juvenile (1-2 y); and 7, preadult to adult (47 y). The statistical procedure based on gnomonic time divisions assumes units of time increase as a constant proportion of time elapsed from the end of the previous biological stage; in this manner, the method estimates a vector of natural mortality values by dividing the life cycle into specific time-based subunits. The results provided the following values of natural mortality at GTD: 1 = 537.42/y; 2 = 230.32/y; 3 = 134.35/y; 4 = 33.58/y; 5 = 2.55/y; 6 = 2.21/y; and 7 = 0.046/y. The consistency of the estimates derived were compared with previous reports of mortality rates and yielded similar values. The gnomonic time method proved to be particularly effective in estimating natural mortality based on the specific life history and life span of the geoduck.

KEY WORDS: natural mortality, gnomonic time, life span, Panopea globosa, geoduck, clam


The Mexican geoduck Panopea globosa (Dali, 1898) is distributed from Nayarit (22[degrees]22' N) in the upper Gulf of California in a natural range that extends south throughout the Gulf of California and then north along the west coast of Peninsula de Baja California to Bahia Magdalena (24[degrees]38' N)-a point that demarks the tropical-temperate transition zone along the Pacific coast (Gonzalez-Pelaez et al. 2013). Biological information on the population dynamics for P. globosa is limited. This species has a synchronic and short reproductive period initiated by decreasing seawater temperature 4 mo before the peak of productivity in the central and upper Gulf of California (Aragon-Noriega et al. 2007, Calderon-Aguilera et al. 2010). The age structure for geoducks sampled in the central Gulf of California varied from 2-27 y (mode, 10 y), whereas clams sampled from Bahia Magdalena contained older individuals, ranging from 3-47 y in age (average, 17.2 y) (Cruz-Vasquez et al. 2012). The average shell lengths reported were 155.9 mm, 147.7 mm, and 143.72 mm in the upper, central, and southern areas of the Gulf of California, respectively. In Bahia Magdalena, mean shell length was 158.3 mm (Aragon-Noriega et al. 2007, Rocha-Olivares et al. 2010, Gonzalez-Pelaez et al. 2013, Gonzalez-Pelaez et al. 2015). Morphological analyses for P. globosa showed that average shell length, width, and height of this species were significantly greater than those reported for Panopea generosa (Gould, 1850) from the Mexican coast (Rocha-Olivares et al. 2010).

For Panopea globosa, the rate of natural mortality is a key parameter in population dynamic models--a value commonly used as a constant in modeling quantitative fisheries stocks for animals of all ages or sizes large enough to be exploited (Caddy 1990). According to Andrews and Mangel (2012), using a constant for natural mortality is a concession resulting mainly from the lack of data on age or size-specific mortality, or simply from consistency and/or to compare estimates with previous population assessments. Errors in calculating natural mortality propagate throughout the population dynamics model (e.g., catch at age, catch-at-size models, or modified version of the depletion models). To determine natural mortality rates, several assumptions are made regarding loss to predation, disease, senescence, cannibalism, and starvation, among other factors. Mortality rates may vary both spatially and temporally, or by size and age, and the assumption of a constant rate of mortality across all age and size classes is highly questionable (Lorenzen 2000). Life history theory predicts that natural mortality decreases with age and size as, for example, refuges from predation are gained through greater size or increased burial depth. Mortality rates likely continue to decrease steadily as well for mature individuals that continue to grow, although at a reduced rate.

Natural mortality rates have long been estimated using different approaches. The most common technique is based on meta-analysis, which uses life history parameters across a variety of species and environments to develop predictive relationships based on regression analyses in which natural mortality is maintained as the dependent variable. These models are usually simple and are based on biological and physiological parameters, and environmental variables (e.g., Gunderson 1980, Pauly 1980, Hoenig 1983). Another method commonly used is the catch curve analysis. In this case, estimates of mortality are based on the collection of age composition data and make the assumption that mortality from fishing is negligible (Ricker 1975, Vetter 1988). For field studies, it is often possible to collect mark-recapture data. These approaches are often preferable because a known number of individuals can be released and followed over time, with population loss resulting from either mortality or emigration (e.g., Hilborn 1990, Deriso etal. 1991, Dorazio& Rago 1991). The delay-difference models or catch-at-age models are frequently fitted to data, and natural mortality can be estimated as a parameter in the age-structured models (Megrey 1989, Quinn & Deriso 1999). Another method is based on the collection of dead organisms, when this is feasible (Shepherd & Breen 1992). The use of ecosystem-based models may also be of use in estimating natural mortality rates as well (Walters et al. 1997, Arreguin-Sanchez 2000).

Caddy (1990) proposed an alternative methodology for estimating natural mortality for different developmental stages of the life cycle of an organism (from egg to adult). This approach is referred to as gnomonic time division. Gnomonic time divisions are based on a gnomon--a geometric term defined as a value that, added to another value, leaves the resultant value similar to the original (Caddy 1996). Thus, in terms of elapsed time, any two gnomonic intervals in a subdivided life history may be considered equivalent if each interval represents a constant proportion of elapsed time from birth to the initiation of that specific time interval. This method was first applied to populations of shrimp and loliginid squid, both short-lived species with annual life spans (Caddy 1996, Royer et al. 2002, Ramirez-Rodriguez & Arreguin-Sanchez 2003). Martinez-Aguilar et al. (2005, 2010) noted that gnomonic intervals correspond closely to time frames associated with successive life history stages and proposed an improvement to Caddy's methodology that permitted analyses for species with life spans greater than 1 y. In comparison, traditional estimates of natural mortality are usually based on constant time units (e.g., 1 wk, 1 mo, 1 y). In this study, the gnomonic model was applied to different life history stages of the geoduck based on ontogenic development from the egg stage to adult, with each stage characterized by a different time interval. As a result, the duration of the developmental stages must be expressed in similar time units, referred in this study as gnomonic time (Fig. I).

For Panopea globosa the gnomonic approach was used as an alternative method for estimating natural mortality and it offers an alternative to traditional methods used for Panopea species that are focused on adult population parameters (e.g., mark-recapture, catch curve analysis, metapopulation methods) (Calderon-Aguilera et al. 2010, Morsan et al. 2010) (Table 1). Because natural mortality for different developmental stages is usually unknown for many species, including geoducks, the gnomonic model provides a statistical procedure that allows for estimates to be made for different developmental stages ranging from early larvae to reproductive adults. This has resulted in a new statistical tool and an increased capacity to estimate relevant information on natural mortality rates in early life stages of economically important molluscs, especially in cases when data are lacking or incomplete.


Estimates of natural mortality for Panopea globosa were computed using gnomonic time divisions (GTD) according to the definition provided by Caddy (1996). The GTD method assumes that the time units for the ensuing life history stage increase as a constant proportion of time elapsed from the end of the previous developmental stage. The method estimates a vector of natural mortality (M) values (numerical array expressed as column) and is computed by dividing the life cycle into specific time subunits. The input data for estimating natural mortality for each gnomonic time interval ([[DELTA].sub.i]) (referred to as natural mortality at age) requires information on the number of developmental stages encompassing the organism's life cycle. It is described as (1) a series of time intervals of increasing duration starting at time t = 0; (2) the time elapsed during the first ith stage ([t.sub.1]), corresponding to the first gnomic time interval ([[DELTA].sub.i]), then [[DELTA].sub.i] = [t.sub.1]; and (3) the mean fecundity of the species ([delta]). Thus, the duration of the second gnomonic interval was defined as [[DELTA].sub.i] = ([alpha][T.sub.i-1]) + [t.sub.i-1], where [alpha] is a proportionality constant. Successive gnomonic intervals were computed as [[DELTA].sub.i] = ([alpha][t.sub.i-1]) + [t.sub.i-1], where i [greater than or equal to] 3 up to the nth gnomonic interval, the duration of which was estimated as [[DELTA].sub.n] = ([alpha][t.sub.n-1]) + [t.sub.n-1]. If the species is annual in longevity, then [t.sub.n] = [n.summation over (I=1)] [[DELTA].sub.i] represents 1 y, or 365 days (if days are defined as the unit of time).


Caddy (1990) described appropriately chosen gnomonic time intervals over an organism's life history as a series of time intervals in which a constant proportion of the population dies within each selected interval. It would be convenient to choose intervals such that the product of [M.sub.i] and [[DELTA].sub.i] is constant for all intervals, so G = [M.sub.i][[DELTA].sub.i], where G is the constant proportion of death for each interval (Caddy 1996, Martinez-Aguilar et al. 2010). The natural mortality would then be estimated as [M.sub.i] = G/[[theta].sub.i] - [[theta].sub.i-1], where [[theta].sub.i] = [[DELTA].sub.i]/[t.sub.n] and represents the time duration of a gnomonic interval as a proportion of a year. The number of individuals of the first gnomic time interval ([N.sub.1]) was computed as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Successive values of A) were computed as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where i [greater than or equal to] 2 up to the nth gnomonic interval. The parameters a and G were estimated when two routines of the residuals sum of squares were minimized as follows: (1) the [alpha] value was accepted when the sum of duration of the specified number of gnomonic time intervals was equal to [t.sub.n] assuming the life history was completed; and (2) the estimate of G was accepted when the number of individuals ([N.sub.1]) surviving for the last gnomonic time interval (the oldest individuals) was equal to two, according to constraints as outlined by Caddy (1996).

For Panopea globosa, the number of gnomonic intervals was determined by dividing its life history into a predetermined number of biological units (life stages). The partitions were based on knowledge of the specific life history of the clam, using a modification of the criterion established by Goodwin and Pease (1989) for Panopea generosa: 1, egg to trochophore larvae (24 h); 2, early larvae (6.5 days); 3, late larvae (11 days); 4, early juvenile (35 days); 5, juvenile (3-9 mo); 6, late juvenile (1-2 y); and 7, preadult to adult (47 y). Because survival of P. globosa is more than 1 y, the modified method of GTD proposed by Martinez-Aguilaret al. (2005, 2010) was used. In this case, they substituted [t.sub.n] = [n.summation over (I=1)] [[DELTA].sub.i] = 1 with [t.sub.l] = [n.summation over (I=1)] [[DELTA].sub.i] = total longevity (in days), and the [[theta].sub.i] value was computed as [[theta].sub.i] = [[DELTA].sub.i]/[t.sub.1]/365 to serve as an annual basis for the [M.sub.i] vector estimates. In this study, mean annual fecundity ([delta]) was estimated from the fecundity of Panopea females during a year divided by total longevity (in years). This simple computation therefore represented the average number of eggs produced for an annual cohort and provides a necessary first step, because the method depends on the depletion of one annual cohort (Caddy 1996). Specific studies on the fecundity of P. globosa are not available, so the estimated fecundity for Panopea generosa (Conrad, 1849), as reported by Goodwin and Pease (1989), was used as a surrogate for P. globosa--a clam very similar in biology and life history parameters (Cooper 1998). For P. generosa, female clams release between one million and two million eggs per spawning event, although Goodwin and Pease (1989) report that the largest release observed from a single female during one spawning episode was 20 million eggs. According to Goodwin and Pease (1989), the life history of the Pacific geoduck passes through seven life history stages that are common to most pelecypods. Growth rate, thus size at age for any life history stage, is variable and depends on environmental conditions.

Confidence Intervals

To estimate confidence intervals (Cl) of the parameters (oc and G) and natural mortality for each GTD, a simulation was carried out without resampling the residuals. Because [delta] was a variable in the model, there were no residuals for [delta]. Two thousand vectors of natural mortality for gnomonic time of Panopea globosa were estimated based on the following assumptions: (1) the main source of uncertainty and variability was the mean annual fecundity and (2) a noninformative distribution (U) of [delta]. As a result, the baseline was restricted to [delta] [approximately equal to] U ([[delta].sub.min], [[delta].sub.max]) (Martinez-Aguilar et al. 2010). Hence, the mean annual fecundity was a uniform distribution on [delta], where the lower ([[delta].sub.min] = 10 million eggs) and upper ([[delta].sub.max] = 30 million eggs) boundaries were the smallest and largest values observed for [delta] (unpubl. data). Last, the mean ([[bar.[mu]].sub.i]) of the simulations was estimated, and it represented the mean value for natural mortality estimated for gnomonic time. The SD ([[sigma].sub.i]) represented the SE for gnomonic time. The bias (B) and percent bias (%B) were estimated as follows: B = [bar.x] - [[theta].sub.i], and %B = x-[[theta].sub.i]/[[theta].sub.i] x 100%, where [[theta].sub.i] is the best ith parameter estimate from each gnomonic time (Efron & LePage 1992, Jacobson et al. 1994, Haddon 2001). The CI was estimated using Student's t distribution (P < 0.05) (Neter et al. 1996).


In accordance with knowledge of the life span of Panopeci globosa, the seven gnomonic divisions selected a priori represented a suitable representation of realistic time subunits for this species. Each gnomonic interval at time t + 1 was observed to be less than the gnomonic interval at time t, and satisfies a basic assumption of the gnomonic model. Consequently, the vector of natural mortality also decreased with each increment in gnomonic time, and represented adequately the seven ontogenic stages in the life history of this clam (e.g., trochophore larvae, early larvae, late larvae, early juvenile, juvenile, late juvenile, and preadult to adult).

The duration of gnomonic intervals was 17,166 days (47 y for Panopea globosa). The model generated estimates of natural mortality at gnomonic time as follows: gnomonic time 1, 537.42/y; gnomonic time 2, 230.32/y; gnomonic time 3, 134.35/y; gnomonic time 4, 33.58/y; gnomonic time 5, 2.55/y; gnomonic time 6, 2.21/y; and gnomonic time 7, 0.046/y (Fig. 2). The highest values of natural mortality were estimated for stages encompassing development from egg to trochophore larvae, with successively lower mortality rates estimated for later increments, corresponding to later larvae-through-adult stages. The average, SE, bias, percent bias, and CI associated with the gnomonic time intervals are presented in Table 2.


Natural mortality rates for Panopea spp. have been estimated based on mark-recapture data and meta-analysis (Table 1). Bradbury et al. (2000) and Blewett and Bradbury (2005) analyzed mark-recapture data for Panopea generosa individuals older than 3 y from Hood Canal and South Puget Sound, Washington, and reported values of M = 0.016/y (CI, 0.007-0.025/y) and M = 0.006/y (CI, 0.001-0.011 /y), respectively, for each location. This estimated range was referred to as the lower natural mortality for this species. Another estimate from the same region was determined for P. generosa with M = 0.023 (Bradbury & Tagart 2000), whereas Sloan and Robinson (1984) reported a natural mortality of 0.035/y for the Pacific geoduck from southern British Columbia, Canada, waters. Calderon-Aguilera et al. (2010) used catch curve analysis and meta-analysis for estimating natural mortality of P. generosa for individuals ranging in age from 3-96 y. Depending on the method used, they found that, along the Pacific coast of Baja California, Mexico, natural mortality ranged from 0.027-0.046/y. Morsan et al. (2010) estimated the natural mortality of Panopea abbreviata (Valenciennes, 1839) in the Gulf of San Matias, Argentina. In that study, three methods were used to estimate natural mortality based on catch curve analysis, mean age, and maximum longevity data, with mortality ranging between 0.062/y and 0.233/y, with estimates based on mean age (M = 0.212-0.233/y), which is greater in general than natural mortality estimated by other methods. Breen et al. (1991) estimated natural mortality for Panopea zelandica (Quoy and Gaimard, 1835) in New Zealand populations using catch curve analysis and estimated M = 0.20.


Natural mortality measures the rate of population loss, and this may vary spatially and temporally, or by size, age, or other factors, including environmental variability. With regard to the gnomonic method, the statistical procedure depends first on the duration of early embryogenesis, which can vary over a range of physical environments. As an example, increasing seawater temperature reduces both the time of egg incubation and the duration of the yolk sac stage (Pepin 1991). For pelagic fish eggs, Floude (1987) reported that temperature has a significant effect on the daily mortality rate. If the development from egg to larvae is slow, then the organism may be subject to predation mortality for longer periods, resulting in increased natural mortality (Pitcher & Hart 1982). Kennedy et al. (1974) analyzed the effect of temperature on survival of cleavage stages, trochophore larvae, and straight-hinge veliger larvae of Mulinia lateralis (Say, 1822), and reported a direct relationship between mortality and increased temperature. As the clams aged, temperature tolerance increased; cleavage stages were observed to be most sensitive to warmer temperatures, and straight-hinge larvae were the least sensitive. In the current study, the environment and its effect on egg stage were not assessed specifically; however, the differences between estimates of natural mortality as reported by Breen et al. (1991) and Morsan et al. (2010), for example, showed a difference of more than an order of magnitude greater when compared with estimates of natural mortality reported here, or as reported by Bradbury et al. (2000) and Calderon-Aguilera et al. (2010). The effect of latitude and climatic variability on natural mortality in Panopea spp. from selected locations throughout the breadth of their natural ranges should be a focus for future studies involving demographic and stock assessments for geoducks.

For geoducks, to date only empirical methods and catch curve analyses have been used to estimate natural mortality. According to Thorson and Prager (2011), catch curve analysis is applied frequently to data-poor fisheries, where a variety of contradictory information (each of low quality for stock assessment) is available. Published reports on natural mortality in geoducks, using a variety of approaches and methods, have not generally assessed changes in values of natural mortality. The results presented here suggest that an approach based on gnomonic time represents biological developmental stages realistically throughout the life cycle of long-lived clams, including Panopea globosa. The vector associated with natural mortality was observed to decrease with gnomonic time and therefore provides a method for making reliable estimates of natural mortality. For comparative purposes, estimates of natural mortality made for preadults and adults (gnomonic time 7) ranged between 0.0447/y and 0.0492/y (Table 2), and are similar to values reported by Sloan and Robinson (1984). Bradbury et al. (2000), Calderon-Aguilera et al. (2010), and Morsan et al. (2010) (Table 1). The gnomonic time method has the advantage of using information on mean annual fecundity and the duration of the egg stage for estimating a vector of natural mortality for the life span of the species. In addition, the statistical procedure enables the determination of natural mortality for different stages of development for geoducks and represents a useful method that integrates mortality rates over a lifetime. As a result, the use of gnomonic-based methods may become an important tool in cases when stock assessments are necessary yet characterized by incomplete or poor data.


The authors thank their sponsor Consejo Nacional de Ciencia y Tecnologia of Mexico for support received throughout the project (contract 106905), and for the PhD fellowship received by S.S.G.-P. The authors are indebted to CIBNOR scientific technicians Carlos Pacheco Ayub, Enrique Calvillo Espinoza, and Horacio Bervera Leon. They also appreciate the comments of two anonymous reviewers and extend their thanks to them.


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(1) Centro cle Investigaciones Biological del Noroeste S.C. (CIBNOR), Av. Instituto Politecnico Nacional 195, Col. Playa Palo cle Santa Rita Stir, CP 23096, La Paz, BCS, Mexico; (2) Posgrado en Ciencias Marinas y Costeras, Departamento de Biologia Marina de la Universidad Autonoma de Baja California Sur, Carretera al Sur km 5.5, CP 23080, La Paz, BCS, Mexico

* Corresponding author. E-mail:

DOI: 10.2983/035.034.0114

Estimates of natural mortality reported for adults of Panopea species
in different fishing grounds.

Species        Zone                      Natural mortality/y

Panopea        Hood Canal, WA            0.016 [+ or -] 0.009
  generosa     Puget Sound, WA           0.006 [+ or -] 0.005
               Southern British          0.35
                 Columbia, Canada        0.027
               West coast of Baja        0.617
                 California, Mexico      0.046
Panopea        Central Gulf of           0.546
  globosa        California, Mexico      0.171
Panopea        Golden Bay, New Zealand   0.02
  zelandica                              0.135
               Kennedy Bay and           0.07, 0.03
               Harbour, New Zealand      0.12, 0.05
                                         0.11, 0.05
Panopea        Gulf of San Mafias,       0.105-0.177
  abbreviata     Argentina               0.212-0.233

Species        Zone                      Method *

Panopea        Hood Canal, WA            a
  generosa     Puget Sound, WA           a
               Southern British          b
                 Columbia, Canada        b
               West coast of Baja        c
                 California, Mexico      d
Panopea        Central Gulf of           c
  globosa        California, Mexico      d
Panopea        Golden Bay, New Zealand   b
  zelandica                              d
               Kennedy Bay and           b
               Harbour, New Zealand      d
Panopea        Gulf of San Mafias,       b
  abbreviata     Argentina               d

Species        Zone                      Reference

Panopea        Hood Canal, WA            Bradbury et al. (2000)
  generosa     Puget Sound, WA           Blewett and Bradbury (2005)
               Southern British          Sloan and Robinson (1984)
                 Columbia, Canada        Calderon-Aguilera et
               West coast of Baja          al. (2010)
                 California, Mexico

Panopea        Central Gulf of           Cortez-Lucero et al. (2011)
  globosa        California, Mexico

Panopea        Golden Bay, New Zealand   Breen et al. (1991)
               Kennedy Bay and           Gribben and Creese (2005)
               Harbour, New Zealand

Panopea        Gulf of San Mafias,       Morsan et al. (2010)
  abbreviata     Argentina

* a, tagging (Bradbury et al. 2000); b, catch curve (Breen et al.
1991); c, Pauly's method (Pauly 1980); d, longevity method (Hoenig
1983); e, Chapman-Robson method (Chapman & Robson 1960). Where there
are two numbers separated by a comma in the Natural mortality/y
column, the first is for Kennedy Bay and the second is for Wellington
Harbour, New Zealand.


Estimates of natural mortality for seven gnomonic times divisions of
the geoduck Panopea globosa.

                 Gnomonic time divisions *

                    1         2         3          4

Estimate         537.43    230.33     134.36     33.59
Average          535.60    142.95      30.12      6.35
SE                10.309     2.751      0.580     0.122
Bias              -1.83    -87.37    -104.24    -27.24
Bias (%)          -0.34    -37.93     -77.58    -81.11
Lower interval   517.22    224.93     133.22     33.35
Upper interval   557.63    235.72     135.49     33.83

                 Gnomonic time divisions *

                    5         6         7

Estimate           2.55      2.21     0.0469
Average            1.34      0.28     0.0593
SE                 0.026     0.005    0.0011
Bias              -1.21     -1.93      0.01
Bias (%)         -47.60    -87.25     26.26
Lower interval     2.50      2.20     0.0447
Upper interval     2.60      2.22     0.0492

* I, from egg to trochophore larvae; 2, early larvae; 3, late larvae;
4. early juvenile; 5, juvenile; 6, late juvenile; 7, preadult to
adult. Descriptive statistics and confidence intervals (P < 0.05) of
the natural mortality corresponding to each gnomonic time division
were estimated by simulation.
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Author:Gonzalez-Pelaez, Sergio Scarry; Morales-Bojorquez, Enrique; Lluch-Cota, Daniel Bernardo; Bautista-Ro
Publication:Journal of Shellfish Research
Article Type:Report
Date:Apr 1, 2015
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