Estimation of Growth and Size at First Maturity under a Multimodel Approach of Anadara tuberculosa (Sowerby, 1833) on the Southeast Coast of the Gulf of California

Abstract

The clam fishery in northwestern Mexico encompasses the mangrove cockle Anadara tuberculosa. It is extracted manually, at low tides and between the roots of mangroves. Biological samplings were carried out in Estero Las Lajitas, Sinaloa, from May 2021 to April 2022. A total of 661 A. tuberculosa organisms were analyzed, of which 126 were males, were 363 females and 172 were undifferentiated, yielding a statistically different overall sex ratio between females and males (1♀:0.3♂) (X² = 113.19; p < 0.05). The length–weight relationship showed a potential type (W = 0.0002L^3.125) (95% CI 3.027–3.222 for b). To determine the growth of the species, five models were employed: von Bertalanffy, Gompertz, Logistic, Richards, and Gompertz using an oscillatory component (GO). The Akaike Corrected Information Index for Small Samples (AIC_C) was used. The GO model yielded the lowest AIC_C (L∞ = 80.98 mm 95% CI 77.59–84.36, k = 1.02 year⁻¹ 95% CI 0.89–1.16), a low growth oscillation intensity (C = 0.03), and slower growth in August (WP = 1.67). The Logistic and Gompertz models were used to calculate the size-at-maturity (L_50%). Gompertz obtained the lowest AIC_C with L_50% = 32.53 mm (95% CI 30.67–34.31). Considering the lack of biological information and the parameters generated in the present investigation, as regards A. tuberculosa on the coast of Sinaloa, Mexico, its dissemination is essential for the adequate management of the fishery.

1. Introduction

The group of marine invertebrate mollusks comprises about 100,000 species, of which about 10,000 belong to the bivalve group, including the black clam or mule leg Anadara tuberculosa (Sowerby, 1833). They are distributed from the coasts of Baja California, Mexico to Peru [1]. They inhabit the intertidal zone, are often buried in mud, associate with mangrove ecosystems, and share a habitat with other bivalve species, such as A. multicostata and A. similis [2,3]. A. tuberculosa is characterized by protandrical sequential hermaphrodite behavior influenced by anthropogenic and environmental pressures [4,5].

The clams are caught by manual extraction and are marketed mainly to intermediaries, who distribute them on the local, national, and international markets [6,7]. Globally, in 2018, the catch of benthic mollusks, including A. tuberculosa, exceeded 664,000 t live weight. In Mexico, the total national catch in 2020 of the set of species known as “clams” was 12,333 t of live weight, more than 80% of which came from the Pacific Ocean. Sinaloa ranked third, with 2777 t [8]. These volumes of catches highlight the importance of the clams to Mexican fisheries.

The estimation of growth parameters of species is one of the most important ecological areas. For this reason, the Logistic, Gompertz and von Bertalanffy models are some of the most widely used to describe the growth of various species. The Logistic and Gompertz models are characterized by the sigmoidal shape, the second most flexible due to the inflection point, while von Bertalanffy’s model is exponentially inverted [9]. Multimodel inference (MMI) is frequently used to choose the model that best describes the individual growth of the species, highlighting the Akaike Information Criterion (AIC) [10]. One of the main topics addressed regarding the black clam (A. tuberculosa) found along the coasts of the south-central Pacific is reproductive biology and the estimation of maturity size (L_50%) [11,12,13,14]. In Mexico, few studies have been carried out in relation to A. tuberculosa on issues related to growth and maturity; among these, Félix-Pico et al. [15] estimated a maximum asymptotic length (L∞) of 81 mm and a growth rate (k) of 1.85 years⁻¹, while [6,16] estimated an L_50% of 38 and 36.5 mm, respectively, on the coasts of Baja California Sur. In Sinaloa, A. tuberculosa has been barely studied for its growth and maturity, so it is essential to generate further biological information that allows the comprehensive management of the fishery. Biological traits in organisms subject to exploitation must be known to ensure proper management. On the eastern coast of the Gulf of California, clams are negligibly managed in contrast to the western coast of the Gulf. The new knowledge generated in relation to clam species is useful for the scientific community, fishers, and fisheries administrator. On the other hand, the use of oscillatory models in studying A. tubersulosa is a novelty, because, in previous studies, only asymptotic models have been employed. The objectives of this study were: (1) to determine individual growth with a multi-model approach; (2) to estimate the average height at maturity (L_50%); (3) to record the sex ratio and (4) to determine the allometry of the species A. tuberculosa.

2. Materials and Methods

Were analyzed a total of 661 black clam (A. tuberculosa) organisms obtained via monthly random biological sampling from May 2021 to April 2022, extracted manually from the Las Lajitas estuary, Sinaloa, located between 26°5′11.76″ North Latitude and −109°22′40.80″ West Longitude (Figure 1). In the field, the total length of the shell (L) and the total weight (W) were recorded, in addition to identifying the sex and maturity of the organisms at the macroscopic level [16].

To determine and compare the sex ratio (F:M) per month, a chi-square test (X²) was used with Yates’ continuity correction [17].

$${XX}^2=\sum \frac{{\left(\left|f_1-\left.f_2\right|\right.-0.5\right)}^2}{f_2},$$

(1)

where f₁ is the observed results and f₂ is the expected results.

A linear regression analysis of the natural logarithms of L and W was performed. The intersection (a) and slope (b) of the linear regression were used to establish the potential relationship between L and W, of the form W = aL^b. The 95% confidence intervals of parameters a and b (Φ) were estimated with the standard error (ET) of the means a and b [17], IC = Φ ± ET·tn − 1(95%). If the confidence intervals of b do not give values of 3, the growth will be considered allometric; otherwise, it is considered isometric. If b is >3, it is positively allometric, and if b is <3, it is negatively allometric [18].

To identify the age groups in relation to the size distribution, separated by integers of 3 mm, a multinomial model was applied, and the mean L and standard deviation of each group were derived according to the model [19]:

$${\mathrm{F}}_{\mathrm{i}}={\sum }_{\mathrm{a}=1}^{\mathrm{n}}\left[\frac{1}{{\mathsf{\sigma }}_{\mathrm{a}}\sqrt{2\mathsf{\pi }}}{\mathrm{e}}^{\frac{{({\mathrm{x}}_{\mathrm{i}}-{\mathsf{\mu }}_{\mathrm{a}})}^2}{2{\mathsf{\sigma }}_{\mathrm{a}}^2}}\right] \mathrm{\ast } {\mathrm{P}}_{\mathrm{a}},$$

(2)

where x_i is the mean length of each length group i, ${\mathsf{\mu }}_{\mathrm{a}}$ is the mean length of the cohort a, ${\mathsf{\sigma }}_{\mathrm{a}}$ is the standard deviation of the length in the cohort a, ${\mathsf{\sigma }}_{\mathrm{a}}$ is the weight of the cohort a, and F_i is the total frequency of occurrences of the length group i across the cohort.

The maximum likelihood model [20] was used to fit the model:

$$\mathrm{L}\mathrm{L}\left\{\mathrm{X}|{\mathsf{\mu }}_{\mathrm{a}},{\mathsf{\sigma }}_{\mathrm{a}},{\mathrm{P}}_{\mathrm{a}}\right\}=-{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{F}}_{\mathrm{i}}\mathrm{L}\mathrm{n}\left(\frac{{\mathrm{F}}_{\mathrm{i}}}{\sum {\mathrm{F}}_{\mathrm{i}}}\right) \mathrm{\ast } {\left[\sum {\mathrm{f}}_{\mathrm{i}}-\sum {\mathrm{F}}_{\mathrm{i}}\right]}^2,$$

(3)

where $\mathrm{L}\mathrm{L}\left\{\mathrm{X}|{\mathsf{\mu }}_{\mathrm{a}},{\mathsf{\sigma }}_{\mathrm{a}},{\mathrm{P}}_{\mathrm{a}}\right\}$ is the probability value of the parameters ${\mathsf{\mu }}_{\mathrm{a}},{\mathsf{\sigma }}_{\mathrm{a}}{\mathrm{P}}_{\mathrm{a}}$, f_i is the total observed frequency of the group i in terms of length, and F_i is the total expected frequency of the size group i according to the multinomial model.

The mean separation index [19] was determined using the following model:

$$\mathrm{I}.\mathrm{S}.=2 \mathrm{\ast } \frac{({\mathsf{\mu }}_2-{\mathsf{\mu }}_1)}{({\mathsf{\sigma }}_1+{\mathsf{\sigma }}_2)},$$

(4)

Five candidate models were used to estimate the growth of A. tuberculosa: von Bertalanffy (VBGM) [21], Gompertz [22], Logistic [18], Richards [23] and Gompertz using an oscillatory component (GO):

Von Bertalanffy (VBGM; [21]):

$$Lt=L_{\mathrm{\infty }}(1-e^{-k\left(t-t_0\right)})$$

(5)

Gompertz [22]:

$$Lt=L_{\mathrm{\infty }}\left(e^{-e^{-k\left(t-t_0\right)}}\right)$$

(6)

Logistic [18]:

$$Lt=L_{\mathrm{\infty }}/(1+e^{-k(t-t_0)})$$

(7)

Richards [23]:

$$Lt=L_{\mathrm{\infty }}(1-e^{-k(1-m)(t-t_o)}{)}^{(1/1-k)}$$

(8)

Gompertz oscillation (GO):

$$Lt=L_{\mathrm{\infty }}\left[\left(1-e^{-k\left(t-t_0\right)}\right)+\left(\frac{Ck}{2\pi }\right)sen 2\pi (t-t_s)\right],$$

(9)

where Lt is the mean length at time t, L_∞ is asymptotic length, k is the instantaneous rate of growth, t₀ is the theoretical age whereat the organism measures 0, m is a surface factor, C is the amplitude of growth oscillation, and t_s is the period of the year whereat growth has its maximum value. The Winter Point (WP = t_s + 0.5) indicates the period of the year during which growth is slower.

The maximum likelihood model (LL) [20] was used to fit the growth models using the Excel Solver function using the Generalized Reduced Gradient (GRG) algorithm [24].

$$LL=-\frac{n}{2}\left[Ln 2\pi +2 Ln\sigma +1\right],$$

(10)

where

$$\sigma =\sqrt{\sum \frac{({Lt}_{obs}-{Lt}_{esp.}{)}^2}{n}} \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h} \mathrm{a}\mathrm{d}\mathrm{d}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e} \mathrm{e}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r}.$$

(11)

The calculation of the height at first point of maturity (L_50%) was based on the logistic models of [22,25], which relate the proportions of mature and immature individuals.

Brouwer and Griffiths [25]:

$$Pmi=\frac{1}{1+e^{\left(\frac{-\left(L_i-L_{50\%}\right)}{\varphi }\right)}},$$

(12)

Gompertz [22]:

$$Pmi=e^{-e^{(-\varphi \ast \left(Li-L50\%\right))}},$$

(13)

where Pmi is the proportion of mature organisms in size group Li, L_50% is the size at first maturity and $\varphi$ is a parameter of the fit of the model.

The models were fitted using the maximum likelihood method:

$$-\mathrm{L}\mathrm{L}={\sum }_{\mathrm{i}=1}^{\mathrm{n}}\left[{\mathrm{m}}_{\mathrm{i}}\ln \left(\frac{{\mathrm{p}}_{\mathrm{i}}}{{1-\mathrm{p}}_{\mathrm{i}}}\right)+{\mathrm{n}}_{\mathrm{i}}\ln \left(1-{\mathrm{p}}_{\mathrm{i}}\right)+\ln \left(\frac{{\mathrm{n}}_{\mathrm{i}}}{{\mathrm{m}}_{\mathrm{i}}}\right)\right],$$

(14)

where n = total number of organisms in class p_i and m = number of mature organisms in class i.

The confidence intervals for the growth model and L_50% parameters were obtained using likelihood profiles and the chi-square distribution [26]. The confidence interval was defined with the following inequality:

$$2(L\left(Y|\theta \right)-L\left({Y|\theta }_{best}\right))<{x^2}_{\mathrm{1,1}-\propto },$$

(15)

where $L\left({Y|}_{best}\right)$ is the log likelihood of the most likely value of $\theta$, and ${x^2}_{\mathrm{1,1}-\propto }$ is the value of $x^2$, with a degree of confidence freedom of 1 − α. Thus, the 95% confidence interval encompasses all values that are twice the difference between the log likelihood of a given value and the log likelihood of the greater value, which is greater than 3.84 [9].

The choice of the growth model and L_50% was made based on the information theory approach using the Akaike Information Index Corrected for Small Samples (AIC_C).

AIC_C = AIC + (2k (k + 1))/(n − k − 1),

(16)

where LL is the maximum likelihood and k is the number of parameters estimated in each case, $AIC=2\mathrm{\ast }(k-LL)$.

The model with the lowest Akaike index (AIC_cmin) was chosen as the best model [10]. AIC_C differences were calculated for all candidate models used:

$$∆i={AIC}_{C,i}-{AIC}_{Cmin},$$

(17)

To quantify the plausibility of each model, given the data and the set of candidate models, the “Akaike weight” of each model (W_i) was calculated.

$$w_i=\frac{\mathrm{e}\mathrm{x}\mathrm{p}\left(-0.5∆i\right)}{{\sum }_{k=1}^5\mathrm{e}\mathrm{x}\mathrm{p}\left(-0.5∆i\right)}$$

(18)

3. Results

The sizes of all black clams (A. tuberculosa) collected were distributed between 31 and 76 mm L and a W of 6 to 151 g (Figure 2). The overall sex ratio was statistically different between females and males (1:0.3) (X² = 113.19; p < 0.05). Only in 4 of the 12 months analyzed was there no significant difference between sexes (1:1) (p > 0.05) (

Table 1. Sexuality proportions of Anadara tuberculous in the Las Lajitas Estuary, Sinaloa (1:0.3) (X² = 113.19; p < 0.05).

Year	Month	Females	Males	Proportion	X2	Probability
				Female–Male
2021	May	60	0	1:0.0	58.0	0.000
	June	25	14	1:0.6	2.6	0.109
	July	25	13	1:0.5	3.2	0.074
	August	18	6	1:0.3	5.0	0.025
	September	25	7	1:0.3	9.0	0.003
	October	35	6	1:0.2	19.1	0.000
	November	21	26	1:1.2	0.3	0.560
	December	21	0	1:0.0	19.0	0.000
2022	January	31	12	1:0.4	7.5	0.006
	February	41	17	1:0.4	9.1	0.003
	March	26	13	1:0.5	3.7	0.055
	April	35	12	1:0.3	10.3	0.001
	TOTAL	363	126	1:0.3	113.9	0.000

).

Figure 3 shows the relationship L and W, the potential model yielded the equation W = 0.0002L^3.125, with 95% CI of 0.0001–0.0002 for a and 3.027–3.222 for b establishing a positive allometric growth (t = 63.25).

A maximum of four cohorts (July) and a minimum of one cohort (August and December) were identified. By performing modal monitoring over time, it was determined that organisms captured in May with sizes of 50 mm reached sizes of 66 mm in September of the same year, while organisms that were 72 mm in July reached sizes of 74 mm in November 2021 (Figure 4).

The model with the lowest AIC_C was GO, with a maximum asymptotic length (L∞) of 80.98 mm, a k = 1.02 year⁻¹ and a W_i% of 64.25%; however, the Gompertz and Logistic models provide valuable information, with a W_i% of 24.19 and 11.56%, and with an L∞ of 75.89 and 74.99 mm, respectively. The intensity of growth oscillation was low (C = 0.03) with continuous growth (

Table 2. Growth parameters, confidence intervals (95% CI), AIC_C and Wi% values of the growth models for Anadara tuberculosa in Estero Las Lajitas, Sinaloa.

	BVM	Gompertz	Logistic	Richards	Gompertz Oscillatory
L∞	57.49 (53.68–61.31)	75.89 (72.35–79.42)	74.99 (71.38–78.59)	57.49 (53.67–61.31)	80.98 (77.59–84.36)
k	6.28 (3.71–14.22)	1.21 (1.03–1.41)	1.52 (1.23–1.88)	6.29 (3.71–14.23)	1.02 (0.89–1.16)
t0	0.00 (0.00–0.00)	0.01 (0.00–0.12)	0.31 (0.19–0.42)	0 (0.00–0.00)	0.05 (0.00–0.14)
m	-	-	-	0 (0.00–0.00)	-
C	-	-	-	-	0.03 (0.00–0.05)
ts	-	-	-	-	1.17 (0.04–2.31)
AICC	168.88	152.81	154.29	171.84	150.85
Wi%	0.01	24.19	11.56	0.00	64.25

; Figure 5). The time of year when growth was slowest was in August (WP = 1.67).

An L_50% of 32.53 and 36.5 mm was estimated using the Gompertz (1825) and Logistic models, the first of which had the lowest AIC_C, with 96.2 vs. 99.42 (

Table 3. Parameters of first maturity size (L_50%), confidence intervals (95%CI), AIC_C and W_i% values for Anadara tuberculosa in Estero Las Lajitas, Sinaloa.

	Logistic	Gompertz
L50%	36.53 (34.66–38.35)	32.53 (30.67–34.31)
φ	9.91 (8.59–11.56)	0.085 (0.075–0.097)
AICc	99.43	96.22
Wi%	16.74	83.26

and Figure 6).

4. Discussion

The new knowledge generated in the present study in relation to the cockle A. tubersulosa is very novel for the scientific community, fishers, and fishery administrators, because the biological traits of the species were manly measured on the western coast of the Gulf of California. The sizes obtained in the present research were similar to those reported by Puentes [27] on Colombian coasts, and lower than those reported by Flores, Licandeo and Vega et al. [28,29,30] on the coasts of Colombia, Ecuador and Panama, respectively, with sizes greater than 92 mm. However, the values here are higher than that those established by Vega and Moreno et al. [31,32,33,34], reaching maximum sizes of 70.83 mm on the central coasts of the Pacific Ocean.

The total sex ratio was significantly different from expected—1:0.3. This finding differs from what was reported by Pérez-Medina [16] in Baja California Sur, Mexico, [3,31,35] in Costa Rica, and [7] in Peru, where a sex ratio of 1:1 was reported, while it is similar to that reported by Ordinola et al. [33] for the coasts of Peru, who derived a different ratio than was expected (1:1), favoring females (1.5:1). In this context, Panta-Vélez et al. [14] mentioned that the protandry hermaphroditism of the species may be associated with the imbalance in sexual proportion, since the reproductive activity of A. tuberculosa as regards sexual maturity is not coherent with that measured in other areas of the Ecuadorian Pacific. In addition, the emergence of protandry hermaphroditism and changes in sex ratio indicate that the population may respond to anthropogenic and environmental pressures [5,12]. The numerical superiority between females and males may also be caused by the physicochemical properties of the water, which favor the development of females or interfere with the development of males, as mentioned by Guilbert [36]. It should be noted that the area in which the sampling was carried out is affected by discharge from rural and agricultural settlements, which could affect the natural physicochemical properties of the region.

Parameter b of the L-W ratio indicates that the relative growth of A. tuberculosa is of a positive allometric type; this behavior is usually expected for the species, as reported by Alemán et al. [7] for females and males (b = 3.19 and 3.13); however, when combining both sexes and deriving a representational population with heights lower than the minimum reported height (31 mm), the value of b was estimated at 2.46, which implies negatively allometric growth, that is, the growth is greater in terms of height than weight. This same behavior was reported by Sotelo-González et al. [37] for L. grandis; however, they found a greater distribution of between 31 and 142 mm. The alternation between positive and negative allometry may be associated with environmental variations or reproductive aspects, as has been reported for A. tuberculosa and other species [15,38].

The growth of A. tuberculosa in Estero Las Lajitas was constant, reaching an estimated maximum asymptotic length similar to that estimated by Félix-Pico et al. [15] on Mexican coasts; however, this value was lower than those found in studies carried out in areas influenced by the tropical–equatorial zone, where sizes of L∞ between 63 and 93 mm have been reported [12,27,29,39,40]. However, the growth estimates made in the aforementioned area were derived using the VBGM via statistical programs such as ELEFAN I [41] and FiSAT [42], and this is not the best model for describing the information generated in the present research. Thus, the size of the first catch in Mexico of A. tuberculosa (60 mm) suggests an age of just over one year (1.2 years), according to the growth parameters estimated using the model with greater plausibility (GO), which value is slightly below that established by Lucero et al. [29] (1.6 years). However, the results obtained in the present research allow us to establish that the age at which L∞ is reached (80.98 mm) is 3.3 years, similar to that calculated by the same author (3.6 years).

Along the Pacific coasts, estimates of L_50% have been made; here, A. tuberculosa has shown great potential as a breeder, which has allowed this area to remain an important fishery [29,32,34,43]. In this context, the estimated L_50% (32.5 and 36.5 mm) is lower than that reported by Baqueiro [6], who determined that the onset of sexual activity of the species was from a size of 38 mm, while Pérez-Medina [16] established an L_50% of 36.5 mm, both on the Mexican Pacific coast. Regarding other latitudes, the here-estimated L_50% was higher than those calculated by Franco [44], with an L_50% of 30 mm, and Borda and Portilla [45], with 25 mm, The latter measurement coincides with a decrease in catches and a large negative anomaly in the Southern Oscillation Index (SOI), which prevailed in the area from March 1997 to April 1998, both cases being in Colombia. These values are below the estimates given by [11,12,13,14], who reported L_50% sizes between 39.1 and 47.4 mm. The reproductive behavior of species of the genus Anadara allows us to affirm that they spawn all year round, with defined peaks over four months. In this sense, some authors mention that these events may be influenced by chlorophyll concentration, the rainy season, salinity, and temperatures [2,45,46,47,48], while Pérez-Medina [16] did not find a clear relationship between temperature and the phases of the reproductive cycle, which factor might also explain the intensity of the oscillations in growth. The estimation of biological fishing parameters as performed in this research will facilitate adequate fishing management on the coasts of Sinaloa, in addition to generating biological references for this species on Mexican coasts.

5. Conclusions

A novelty in the present study was the use of the individual growth model proposed by Gompertz [22], but with an oscillatory component—GO. The oscillatory approach applied in individual growth models is not a common practice in fishery biology, but it is important for studies seeking the best predictive curve that represents individual growth in any species under fishery usufruct jurisdiction. In the present study, the GO yielded the best performance. Our conclusion here is that some biological processes related to environmental variability are interfering with the growth, but a monotonic growth curve is not found. Our suggestion for fisheries undertaking researchers is to not limit the growth studies to the model most commonly used in stock assessment studies.