Population Dynamics of the Crocodile Shark, Pseudocarcharias kamoharai, in the Tropical Equatorial Pacific Ocean, Ecuador

Abstract

The objectives of this study were to determine the rates of natural mortality (M), fishing mortality (F), total mortality (Z), the exploitation rates (E), as well as the biological reference points (BRPs) and the annual removal rates (R) of the crocodile shark, Pseudocarcharias kamoharai, in the Ecuadorian Pacific. Thirty similar and different models were applied to determine all these rates. These equations were obtained from studies on teleost and chondrichthyan fish. The biological parameters, including age, growth, longevity, and reproduction, were obtained from the specialized literature based on the biology of P. kamoharai in Ecuadorian waters. These biological parameters were used in all the models considered here. The M estimations were 0.14 to 0.28 based on six models for chondrichthyans and osteichthyes. These values were similar to the six algorithms designed for cartilaginous fish, ranging from 0.16 to 0.35; for this reason, these mortality rates were considered low. The Z values ranged from 0.08 to 0.51; however, they were not considered given that the three estimations were less than M, and only the Z = 0.51 was considered. Given that Z = 0.51 and M = 0.24, an F = 0.27 was obtained by subtraction, indicating a low mortality by fishing. E had values between 0.21 and 0.53, which indicated overexploitation that exceeded the E_opt = 0.50 value. The obtained BRPs were F_opt = 0.10 and 0.12 and F_lim = 0.16, which showed that the optimal fishing levels (best possible capture) to achieve long-term sustainable exploitation of the stock encompass 10 to 16% of the fishing effort applied for this species. However, the F surpassed this prudential range. The annual removal percentage (R = 21%) demonstrated that 21% of the population was being removed. Based on the biology and ecology of this species, all models applied in this study showed that P. kamoharai had low natural and fishing mortality rates and moderate total mortality; its exploitation rate exceeded the fishing limits. These values and their life history traits indicated that this shark species cannot tolerate any fishing level without threatening its populations.

1. Introduction

Population dynamics is the study of factors that influence the number and density of populations in a determined time and space [1]. It describes and quantifies the continuous changes in a population. The dynamics of a fish population imply knowing the size and structure of the population, as well as the form and intensity in which it changes and renews [2]. The specific abundance for each individual cohort in time and space is determined by three basic rates (birth, growth, and death) and the demographic processes of emigration and immigration, subject to genetic, demographic, environmental, monitoring, and human-induced stochasticity. The effect of these varied sources on vital rates and demographic processes finally determines a population’s destiny. Ideally, a population dynamics model should capture the interaction of the vital rates and demographic processes with all the variability sources to provide knowledge regarding population abundance in time and space [3].

Mortality is a crucial parameter in understanding the dynamics of any population. Knowing how fast individuals are removed from it is necessary to model fish population dynamics, estimate their sustainable exploitation rates, and calculate other management parameters [4]. The interest in the study of mortalities is centered on exchange rates. Generally, it is more convenient to use instantaneous exchange rates, i.e., the rate or velocity at which individuals decrease in the population [5,6]. Mortality is measured by the number of fish that die, and this occurs in all stages in a population; it tends to decrease with age. Deaths can occur due to diverse factors, such as pollution, starvation, disease, and predation (in non-exploited stocks) and by fishing (in those exploited) [7].

The total instantaneous mortality rate is composed of the natural instantaneous mortality rate and mortality rate by fishing, namely the velocity with which organisms die in a population, while the total annual mortality rate is the fraction or percentage of the individuals that die yearly [8]. Natural mortality is related to the number of fish that die by natural causes [9], while mortality by fishing indicates the number of fish that die due to capture [2]. Another essential factor derived from mortality rates is the exploitation rate, which is the relation between the number of captured individuals and the number of individuals that die in a certain period. Moreover, the biological reference points (BRPs) are those estimated values of a population’s biomass and the mortality rate by fishing that are used in fishery management to obtain the best capture under a precautionary approach [8]. The BRPs are maximum mortality values by fishing or minimum biomass values that should not be surpassed. Otherwise, it is considered that the self-renewal capacity of a stock is in danger [10].

The crocodile shark, Pseudocarcharias kamoharai, has a wide distribution in tropical and subtropical waters of the Pacific, Atlantic, and Indian Oceans [11]. Regarding its abundance, it has an occurrence that varies from sporadic to locally abundant. Although it inhabits oceanic environments, it can also be found in coastal areas; its vertical distribution is epipelagic and mesopelagic, inhabiting from the surface up to depths of 590 m [12]. This shark species has a key role in the marine and oceanic ecosystems because it feeds on squids, octopuses, and teleosts, hence its ecological importance. In Ecuador, this species has a targeted fishery, and it is captured both artisanally and industrially for local consumption [13]. Despite its economic importance, there are no current statistics regarding its landings. A total of 37 748 t of chondrichthyans, including more than 30 species of sharks and rays, were landed during 2007–2011 [14].

The lack of ecological studies related to the population dynamics of chondrichthyans, specifically for P. kamoharai in Ecuador, prevents data availability regarding mortality and exploitation rates, as well as BRPs that support a baseline for elaborating fishery management plans that are adequate for the species. One of the reasons that the study of fish population dynamics is essential lies in understanding how mortality, growth, and recruitment interact to affect population abundance. All of this information is required for adequate fishery management [15].

Based on the fishing patterns observed for sharks in Ecuadorian waters and because there are no studies regarding their population dynamics, the hypothesis to be tested was as follows: if the exploitation rate was equal to the recommended (0.50) to maintain populations in a state of equilibrium, then overfishing would be prevented. This study aimed to determine the natural, fishing and total mortality rates and exploitation rates of P. kamoharai in the Ecuadorian Pacific, along with the BRPs and annual removal rates.

2. Materials and Methods

2.1. Data Source

The fishing port of Santa Rosa de Salinas (02°12′56″ S, 80°57′26″ W), province of Santa Elena, Ecuadorian coast, was visited for two years from June 2012–May 2013 and June 2017–May 2018 (Figure 1). Four hundred-eight individuals of P. kamoharai were caught by artisanal fishery and analyzed. The total length (TL) in cm, total mass (TM) in g, sex, vertebrae of the cervical zone, and gonads of each individual were sampled.

The size and total mass of each shark were considered to investigate the size and total mass structures of this shark population in this region of the Pacific Ocean. The vertebrae were used to study age composition, growth parameters, and longevity. The gonads were used to analyze the reproductive biology of this species (fecundity, size at sexual maturity, etc.). The size and body mass structures and length–body mass relationship of P. kamoharai, as well as age, growth, longevity, and reproductive biology, were described in previous research by the authors [16,17,18]. The biological parameters included instantaneous growth rate, longevity, and fecundity. These biological parameters were necessary for the development of the empirical methods for estimating all the mortality rates used in this investigation.

2.2. Data Analysis and Models

Empirical methods of mortality estimates were used given that there were no capture data for the species. The instantaneous natural mortality rate (M) was calculated with six methods commonly employed for bony and cartilaginous fish. Thus, the equations of Alverson and Carney [19], Rickhter and Efanov [20], Pauly [21], and three Jensen models [22] for osteichthyes were calculated. With the aim to compare them, another six equations destined for chondrichthyans were applied according to Frisk et al. [23], Then et al. [24], Zhou et al. [25], and the specialized literature.

The Alverson and Carney [19] algorithm is based on growth parameters of the Von Bertalaffy model and adds an element—the critical age (t*)—which is the time (years) in which the individuals of a population reach their maximum biomass. The t* was calculated with the following algorithm:

t* = 0.75 ∗ L_∞(1)

The L_∞ = asymptotic length (18 years) was obtained from the growth parameters reported by Calle-Morán et al. [17]. Afterwards, the critical size ($\omega$) was calculated through the ratio between t* and L_∞:

(2)$\omega ={\displaystyle \frac{t^{\ast }}{L_{\infty }}}$

and then M was calculated through the following equation:

(3)$M={\displaystyle \frac{b\ast k(1-\omega )}{\omega }}$

The b value (2.70) of the length–body mass relationship and the k = 0.15 were estimated by Calle-Morán et al. [16,17].

The Rickhter and Efanov method [20] expresses the relationship between M and the mean age at sexual maturity (T_mat, 50% criteria):

(4)$M=\left({\displaystyle \frac{1.52}{{T_{mat}}^{0.72}}}\right)-0.16$

Given that the size at sexual maturity (L₅₀) was obtained from Calle-Morán and Aragón-Noriega [18], an average was used from both sexes and verified with the following MCVB inverted equation:

(5)$T_{mat}=t_0-{\displaystyle \frac{1}{k}}\ast ln\left(1-{\displaystyle \frac{L_{50}}{L_{\infty }}}\right)$

The Pauly equation [21] indicated a relationship between M, the growth parameters of the de von Bertalanffy model, and the habitat’s average annual temperature (T, °C) over the database compiled for 175 fish stocks. The parameters for both sexes and an average annual temperature of 15 °C were considered for the application of this method, considering that P. kamoharai is an epipelagic and mesopelagic species whose distribution goes from the surface until 590 m depth. The equation was the following:

(6)$Log10\left(M\right)=-0.0066-0.2790\ast Log10\left(L\infty \right)+0.6543\ast Log10\left(k\right)+0.4634\ast Log10\left(T\right)$

The Jensen equation [22] sustains a relationship between M and life history parameters, according to Beverton and Holt. Thus, the optimal compensation variables between reproduction and survival were used with the following formula:

(7)$M=\left({\displaystyle \frac{1.65}{T_{mat}}}\right)$

Similarly, Jensen indicates a simple theoretical relationship between M and the k parameter from the de Von Bertalanffy growth model:

M = 1.50 ∗ k(8)

Afterwards, the author analyzed the data from the Pauly equation [21] and made a readjustment and modification in the equation as follows:

M = 1.60 ∗ k(9)

Models were developed exclusively for chondrichthyans based on previous models, with some modifications for this taxonomic group to compare the previously described methods commonly used in bony and cartilaginous fish. The two algorithms by Frisk et al. [23] were used. They were obtained through the regression analysis from data of 30 elasmobranch species from nine families:

Ln(M) = 0.42 ∗ ln(k) − 0.83(10)

(11)$M={\displaystyle \frac{1}{0.44\ast t_{mat}+1.87}}$

Another simpler way to obtain M is from the specialized literature. Thus, an average of natural mortalities from the order Lamniformes was employed, as there is no specific one for P. kamoharai. The formulas by Then et al. [24] relate M with longevity and the MCVB parameters:

M = 4.899 ∗ t_max − ^0.916 (12)

M = 4.118 ∗ k 0.73 ∗ L_∞ − 0.33(13)

The Zhou et al. [25] equation is a modification of the formula by Jensen [19], which in turn used the equation by Roff [26] as its base, the same one that maintains that the age at maturity is highly correlated with the age at which the MCVB inflection point in total body mass occurs. In this version, the t₀ (−2.05) parameter was also included:

(14)$M={\displaystyle \frac{1.65}{t_{mat}+t_0}}$

Similarly, the traditional methods used for osteichthyes and chondrichthyans were compared using the Student’s statistical test (t).

The instantaneous rate of natural mortality (Z) was estimated using four methods for comparative means: Beverton and Holt [27,28], Ssentengo and Larkin [29], and Hoenig [30]. The equation by Beverton and Holt [27] indicates a relationship between Z, the MCVB k parameter, and the size of the captured fish from the average catch size, using the following equation:

(15)$Z={\displaystyle \frac{k(L_{\infty }-\mathsf{Ḹ})}{\mathsf{Ḹ}-L^{\prime }}}$

where L′ = length of the smallest animals fully represented in capture samplings and Ḹ = mean capture length estimated from the larger L′ lengths.

The formula by Beverton and Holt [28] is a modification of the latter, given that it replaces L’ by the length of the first capture (L_c). Given that there is no L_c for P. kamoharai, a larger length was employed for the L₅₀ of females and males (92 cm TL) [18]:

(16)$Z={\displaystyle \frac{k(L_{\infty }-\mathsf{Ḹ})}{\mathsf{Ḹ}-L_c}}$

The algorithm by Ssentengo and Larkin [29] considers that the Z comes from the relationship between the number of captured organisms, the MCBV parameters, and the median lengths such as Ḹ and L′. Therefore,

(17)$Z={\displaystyle \frac{n\ast k}{\left(n+1\right)\ast \mathit{Ln}\left(\frac{L_{\infty }-L^{\prime }}{L_{\infty }-\mathsf{Ḹ}}\right)}}$

The formula by Hoenig [30] suggests a relationship between Z and longevity (t_max), since most long-lived species die at a slower rate than those that live shorter periods. The equation is based on the Z estimation from 84 osteichthyes stocks. However, it is also employed for chondrichthyans:

Ln(Z) = 1.46 − 1.01 ∗ Ln (t_max)(18)

The instantaneous mortality rate by fishing (F) was calculated with the algorithm from Pauly [31], which indicates that once the M and Z are determined, it is easier to obtain it by the difference of both:

F = Z − M(19)

The annual natural mortality rates (H_M), mortality by fishing (H_F), and total mortality (H_Z) were estimated to investigate the fraction of the population that dies using the equations by Ricker [32]:

H_M = 1 − e^−M/year(20)

H_F = 1 − e^−F/year(21)

H_Z = 1 − e^−Z/year(22)

The exploitation rate (E) was calculated according to the methods by Beverton and Holt [33] and Cushing [34]. The Berverton and Holt [33] equation indicates the proportion of a cohort or population that died by fishing pressure:

(23)$\mathrm{E}={\displaystyle \frac{F}{Z}}$

The Cushing algorithm [34] adds another element for the estimation of Z, given that it considers the product with the difference in the survival of the population:

(24)$E={\displaystyle \frac{F}{Z}}\ast (1-e^{-z})$

Here, an average was obtained between the obtained values to establish if the resource was subject to overfishing, assuming that the optimal E (E_opt) value is equal to 0.50. Using E ≈ 0.50 as an optimal value for the exploitation rate is based on the fact that the maximum sustainable yield is optimized when F ≈ M [5].

Furthermore, the BRPs were obtained through the methods by Patterson [35] and Zhou et al. [36]. The Patterson equation [35] allows the estimation of the fishing mortality rate of maximum sustainable yield (F_opt). It indicates a direct relationship between F and half the value of M. It is estimated using the following equation:

F_lim = 0.5 ∗ M(25)

Similarly, the limit fishing mortality rate (F_lim) can be obtained, indicating the relationship with the 67% M value, which constitutes a tolerance range between F_lim and F_opt. This value was obtained as follows:

(26)$F_{\mathit{lim}}={\displaystyle \frac{2}{3}}\ast M$

The algorithm by Zhou et al. [25] also helps to calculate the F_opt based on the meta-analysis of osteichthyes and chondrichthyan stocks. After analyzing the relationship between F, the growth parameters, and M, the authors concluded that the latter considerably influences F. The following applies for cartilaginous fish:

F_opt = 0.41 ∗ M(27)

The maximum sustainable yield size (L_opt) was estimated using two techniques, Beverton [37] and Froese and Binohlan [38]. Beverton [37] sustains that the L_opt is the size at which individuals of a non-exploited population will reach their maximum biomass, which is analogous to t* (described before). For this means, the L_∞ was employed:

(28)$Lopt=L_{\infty }\ast \left[{\displaystyle \frac{b}{b+\left(\frac{M}{k}\right)}}\right]$

In turn, the formula by Froese and Binohlan [38] applies the natural logarithm (base 10) as well as L_∞:

Log₁₀(L_opt) = 1.0421 ∗ Log₁₀ (L_∞) − 0.2742(29)

The L_opt, L₅₀, and L_∞ were graphed in a size–frequency histogram by combined sexes to obtain a reference of the captured organisms based on these three parameters. The size data were taken from Calle-Morán et al. [16]. Likewise, the annual removal percentage (R, %) was estimated through the equation by Ricker [32]:

(30)$R\left(\%\right)=\left[{\displaystyle \frac{F}{Z}}\ast (1-e-z)\right]\ast 100$

3. Results

3.1. Natural Mortality Rates

The M estimation based on the first six applied models for chondrichthyans and osteichthyes ranged from 0.14 to 0.28 ($\overline{\mathrm{x}}$ = 0.24; ±0.05), similar to the six algorithms designed for cartilaginous fish from 0.16 to 0.35 ($\overline{\mathrm{x}}$ = 0.24; ±0.07). Thus, this mortality was considered low. The H_M estimations were between 0.13 and 0.24 ($\overline{\mathrm{x}}$ = 0.21; ±0.04) for the applied methods for both taxonomic groups, which were the same to those destined for chondrichthyans, varying from 0.15 to 0.29 ($\overline{\mathrm{x}}$ = 0.21 ± 0.05). This means that an average of 21% of the population dies by natural causes every year (Figure 2 and Figure 3). Additionally, no significant differences were registered between both M estimation sets (t = −0.05; p > 0.05) and H_M (t = −0.04; p > 0.05). Therefore, they were grouped and treated as a single group (Table 1).

3.2. Total Mortality Rates

The Z presented values from 0.08 to 0.51. Three estimations were not considered because they were lower than the M. Only the method by Beverton and Holt [28] of Z = 0.51 was considered. Similarly, given that the estimated H_Z quantities were between 0.08 and 0.40, three of them were not considered, and the latter equation was considered (H_Z = 0.40). Hence, in total, 40% of the population dies each year in this region of the Pacific Ocean (Table 2).

3.3. Fishing Mortality Rates

Given that Z = 0.51 and M = 0.24, an F = 0.27 was obtained by difference, indicating a low mortality rate by fishing. This F was equal to M. The H_F = 0.24, indicating that 24% of the crocodile sharks die by fishing each year.

3.4. Exploitation Rate

The E values from 0.21 to 0.53 ($\overline{\mathrm{x}}$ = 0.37; ±0.22) indicated overexploitation of P. kamoharai, as the E_opt = 0.50 value was exceeded (Figure 4).

3.5. Biological Reference Points

The obtained BRPs were F_opt = 0.10 and 0.12 ($\overline{\mathrm{x}}$ = 0.11; ± 0.01), and F_lim = 0.16, which indicated that the optimal fishing levels (best possible capture) to reach or attain a sustainable long-term exploitation of the stock range from 10 to 16% of the applied fishing effort for this species. However, the calculated F = 0.27 exceeds these values, suggesting a risk in the stock’s self-renewal capacity.

Moreover, the estimated L_opt were between 74.3 and 77 cm in TL ($\overline{\mathrm{x}}$ = 75.70 ± 1.90), the size at which the population reached its maximum biomass (Figure 5). The annual percentage removal (R = 21%) demonstrated that 21% of the population is being removed.

4. Discussion

4.1. Natural Mortality Rates

The natural mortality rate of P. kamoharai obtained from the life history parameters in this study was low. This agrees, to a certain degree, with the vital rate patterns established by Beverton and Holt [28] and Hoenig [30], which claim that short-lived fish have a high growth coefficient, a low asymptotic length, short longevity, high fecundity, and high natural mortality. In contrast, long-lived species have a lower growth rate, higher asymptotic length, extended longevity, lower fecundity, and lower natural mortality. Therefore, this shark species would present the life history characteristics of both groups. Crocodile sharks are short-lived fish (14 years old) with a low growth rate (k = 0.14), low asymptotic length, short longevity (18 years) [17], low fecundity (four offspring) [18], and low natural mortality. Based on their features—such as breeding frequency, litter size, size at birth, and growth rate—sharks, rays, and chimeras have different alternative types of life history patterns, which allows the cartilaginous fishes to offset their natural mortality rates [39]. Cartilaginous fishes are highly diverse and show numerous alternative life-history styles, which are multiple methods of exploiting the available niches [40]. The latter author classified chondrichthyans based on the following criteria: body size, marine habitat, mobility, type of feeding, fecundity, fertilization, kind of fetal development, growth, etc. Furthermore, biological parameters, such as body size, reproduction (size at maturity, litter and offspring size, maternal size), age, and growth (growth completion rate, longevity) for sharks explain how different species may cope with juvenile mortality [41].

The natural mortality rate obtained by the methods initially developed for bony fish and later applied to cartilaginous fish yielded the same results, so they were grouped into one. This is congruent with the findings of Dureuil et al. [42], who estimated the methods used in 16 species of bony and 2 cartilaginous fish. They did not find any difference; thus, using any technique was feasible. However, this differs from the theory proposed by Zhou et al. [25], who demonstrated that the methods developed for osteichthyes and applied in a group of 15 stocks of 13 shark species in the Western Pacific Ocean (north, center, and south zones) presented higher biases than those applied explicitly for elasmobranchs. Also, the latter authors highlighted the benefits of the empirical methods, given that they are easier to apply, have no cost, and can be adapted to any group of organisms. Furthermore, they recommend using various models to evaluate such population parameters, given that they help decrease the bias of applying only one technique.

The natural mortality rate estimated for P. kamoharai was low, similar to other species of cartilaginous fish in general. These obtained values were higher than those reported for some species of Lamniform sharks, such as the pelagic thresher, Alopias pelagicus; bigeye thresher, A. superciliosus; common thresher, A. vulpinus; shortfin mako shark, Isurus oxyrinchus; and the porbeagle, Lamna nasus (M = 0.12–0.21) [25]. The M is low for these shark species because their growth rate values are also low (k = 0.05–0.20) [43,44,45,46,47,48]. Based on Beverton and Holt [28] and Hoenig [30], long-lived species have a low growth rate and lower natural mortality. Likewise, the M of the studied species is similar to the leopard shark, Triakis semifasciata; some carcharhinids; and the shovelnose guitarfish, Pseudobatos productus (M = 0.20–0.29) [25,49,50] (Table 3).

The M is one of the main life history parameters, which is, in turn, derived from other parameters, such as the growth rate, asymptotic length, longevity, size and age of sexual maturity, and critical size, among others. The M must be used to develop most of the reference point methods in fishery management, thus its importance within fish population dynamics [25].

The M values of P. kamoharai fluctuated between 0.14 and 0.35. This differs from the values obtained for other species, such as the dusky shark, Carcharhinus obscurus, in southwestern Australia, where five algorithms were employed, and the results were similar, from 0.08 to 0.09 [54]. In the Australian sharpnose shark, Rhizoprionodon taylori, in northern Australia, seven models were applied, and a range was obtained between 0.56 and 1.65 [55].

4.2. Total Mortality Rates

The Z estimated in this study for P. kamoharai is within the calculated values for other chondrichthyans but higher than various species, including the whale shark, Rhincodon typus; threshers; carcharhinids; the tiger shark, Galeocerdo cuvier; and the speckled guitarfish, Pseudobatos glaucostigma (Z = 0.14–0.39) [25,50,54]. In contrast, it was lower than in P. productus in the Gulf of California, Mexican Pacific Ocean (Z = 0.68) [50] (Table 3). The Z found for C. falciformis in Indonesia (2.79–2.86) by Hidayattuloh [56] and Chodrijah et al. [57] were five times higher than those found in this study. These variations could be explained by the differences between life history parameters and patterns since larger species of elasmobranchs have lower k, potential population increases (r’), and lower mortality rates. In contrast, the smaller species of chondrichthyans have a higher k and a higher mortality rate.

Information regarding the state of the population of sharks, rays, and chimeras and how they respond to the increasing total mortality is essential to making the best management decisions for captured and endangered species. Hence, the specialized literature provides a lot of information regarding this topic. In addition, long-term time series (captures, fishing effort, abundance changes) are required to develop and apply dynamic fishing models that help implement extensive management related to the biology and ecology of species. This information has been scarce for chondrichthyans, given that certain species lack commercial value or are discarded. Thus, population models that mainly depend on life history parameters provide valuable information for adequate management [4].

4.3. Fishing Mortality Rates

The F estimated for P. kamoharai was low but equivalent to the M. In this study, the need for landing data limits this estimation. Thus, this value must be taken cautiously, given that it does not specify how many individuals die from this cause. Despite the latter, this F value was higher than in other shark and ray species analyzed in the Pacific and Indian Oceans. Still, it was lower than in P. productus (Table 3). Likewise, the F found for C. falciformis in Indonesia was 1.83–2.30 [56,57], between 7 and 9 times higher than in this study.

On the one hand, based on Simpfendorfer [4], the M can always be considered as a constant factor that similarly affects all fish permanently. Conversely, F is a variable factor given that the fishing effort can be higher in one particular year and lower in another; the same applies to the exploited capture size and the fishing zone, among other factors. Nonetheless, the circumstances under which F can vary should be considered when studying fish population dynamics because the fishing regime affects captures and the population. According to such an adopted regime, the fish population can suffer significant changes in the number and total biomass of its members, in its age and size structures, its growth velocity, and even in its reproductive capacity; e.g., one of the easier effects to observe is the reduction in the size or mean age of the fish in capture and population.

An essential aspect of fishery management was cited by [58], who recommends that the F must be reduced to half of the M value and that the capture size should be increased to avoid the cartilaginous fish population decline. However, the mortality estimation for the sharptooth houndshark, Triakis megalopterus, on the coasts of southeastern South Africa noted that the fishing levels must be much lower than previously cited, ergo, around 0.02M, so that moderate capture levels at a more extended scale can be endured [52].

4.4. Exploitation Rate

According to the maximum sustainable yield, the E values indicated an overexploitation of P. kamoharai. If the species’ exploitation level is close to or exceeds the limit reference point (E = 0.50), then more than 50% of the existing biomass is being extracted. However, for F = M, the E_opt should be 0.33 [59]. In this case, the E obtained for the studied species indicated that it exceeded the E_opt.

The E of P. kamoharai was lower than that found for C. falciformis, with 0.64 and 0.82 [56,57]. The variability of the found results for M, F, and Z in the latter species is probably due to the oceanographic conditions of each sampling site [60], given that each environment has its characteristics. Thus, different habitats and organisms develop distinct body forms and sizes. Despite the estimation differences, F was higher than M in all cases, and the exploitation rates were also high. Therefore, the E = 0.50 value indicates stock utilization at a maximum level according to the criteria by Sparre and Venema [61]. Given that E > 0.50, the stock is being overexploited, which is the case for these two carcharhinid species [53] and the crocodile shark in the Ecuadorian Pacific.

Furthermore, the information regarding life history patterns can provide a considerable vision of the response by shark populations to the exploitation of such resources. For instance, different life history strategies offer answers that are different to the fishery of two shark species of similar size in the Triakidae family, the gummy shark, Mustelus antarcticus, and the school shark, Galeorhinus galeus, which are fishing targets in Australia. M. antarcticus matures relatively early, lives 16 years, and has very high population growth. The fishery for this species has been evaluated as sustainable. In contrast, G. galeus matures late, grows slowly, lives 60 years, and has a much lower population growth rate. This species has been constantly overexploited despite being subject to similar fishing pressures [62,63].

4.5. Biological Reference Points

The BRPs of P. kamoharai indicated that the F_opt and F_lim ranged from 10 to 16%. Given that the F = 0.27, there is a risk that the stock will not renew; hence, if this exploitation rhythm continues for this resource, it would be vulnerable to overfishing. It is worth highlighting that this F value must be taken cautiously, given that the captures for the two study years are unknown.

In recent years, approaches to fisheries with limited data have become increasingly common in fishery evaluations. A particular research focus has been developed from management reference points based on the information on life history patterns. Generally, based on Zhou et al. [36], there are two types of reference points: those based on biomass, which determine if a stock is overexploited, and those based on the F, which help determine if overfishing is occurring. The purpose of the reference points is to act as components of a fishery management system. To apply them in fishing management decisions, an indicator corresponding to the estimated stock is needed for comparison. Therefore, the BRPs based on fishing, such as the F_opt, are only helpful when the F is available. It should be noted that due to the lack of reliable catch time series, the use of traditional stock assessment models to estimate F has been hampered, thus making data-limited management approaches more commonly used for analyses in elasmobranchs.

In summary, P. kamoharai is a species with life history patterns (slow growth rate, late sexual maturity, low fecundity, and long gestation), which result in short longevities and low mortality rates. All these population parameters make this species vulnerable to overfishing, making it susceptible to fishing without being able to withstand low, moderate, or high fishing levels.

5. Conclusions

The instantaneous natural mortality rate and the instantaneous natural mortality rate by fishing of Pseudocarcharias kamoharai in Pacific Ecuadorian waters were low, around 20%. The total instantaneous mortality rate was moderate (51%). The exploitation rate exceeded the natural mortality by double and indicated that this species was at an overexploitation level according to the maximum sustainable yield. The BRPs, such as the mortality rates by fishing and by limit fishing, determined that the optimal fishing levels to achieve long-term sustainable exploitation of the stock are only between 10 and 16%. However, the estimated mortality rate by fishing exceeded such limits. Therefore, the resource could not withstand the fishing pressures it was subjected to, which placed its populations at risk.

Research on updated landings, life history parameters and patterns, population dynamics, and demography is indispensable in Ecuador because all these studies support the management and conservation of this shark species and other cartilaginous fishes. Furthermore, using these types of studies that focus on fishery-limited data is essential to rapidly evaluating a population’s actual state.

Author Contributions

Conceptualization, M.D.C.-M. and E.A.A.-N.; methodology, M.D.C.-M., E.A.A.-N. and A.R.H.-T.; software, E.M.-E.; validation, J.T.-Á., J.F.A.-G. and J.P.-A.; formal analysis, M.D.C.-M.; investigation, M.D.C.-M. and A.R.H.-T.; data curation, E.M.-E.; writing—original draft preparation, M.D.C.-M. and A.R.H.-T.; writing—review and editing, E.A.A.-N., E.M.-E., J.T.-Á., J.F.A.-G. and J.P.-A.; visualization, E.A.A.-N.; supervision, E.A.A.-N., E.M.-E., J.T.-Á., J.F.A.-G. and J.P.-A. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The data analyzed in this research are part of the first author’s doctoral thesis.

Acknowledgments

The first author thanks CONHACYT (Mexico) for the scholarship received during his Ph.D. program at the FACIMAR, UAS; this study was part of his doctoral thesis.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

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Figures and Tables

Figure 1. Geographic location of Santa Rosa of Salinas within the coastal profile of Continental Ecuador, Tropical Ecuadorian Pacific Ocean.

Figure 2. Values of the instantaneous mortality rate (M) and annual natural mortality rate (HM) of the crocodile shark, Pseudocarcharias kamoharai, in Pacific Ecuadorian waters estimated through diverse methods commonly used for osteichthyes and chondrichthyans. (1) Alverson and Carney [19], (2) Rickhter and Efanov [20], (3) Pauly [21], and (4–6) the three models by Jensen [22]. The dotted line represents the mean M and the continuous line the mean HM.

Figure 3. Estimations of the instantaneous rate of natural mortality (M) and the annual natural mortality rate (HM) of the crocodile shark, Pseudocarcharias kamoharai, in Ecuadorian waters based on distinct models used specifically for cartilaginous fish. (1–2) Frisk et al. [23], (3–4) Then et al. [24], (5) Zhao et al. [25], and (6) from the literature related to sharks from the order Lamniformes. The yellow dotted line represents the mean M, and the blue continuous line is the mean HM.

Figure 4. The estimated exploitation rates of Pseudocarcharias kamoharai in Ecuadorian waters calculated using three methods. Methods employed were Beverton and Holt [33] and Cushing [34]. The dotted line represents the optimal exploitation rate according to the maximum sustainable yield. The continuous line is the average value obtained for the species.

Figure 5. Size–frequency distribution of Pseudocarcharias kamoharai and its reference population parameters. The black vertical line represents the optimal capture average size according to the maximum sustainable yield (Lopt), the grey line is the sexual maturity size of the population, and the dotted line is the asymptotic length of the individuals.

Table 1

Grouped values of the methods used to estimate the instantaneous mortality rate (M) and annual natural mortality rate (H_M) of Pseudocarcharias kamoharai in the Eastern Tropical Pacific Ocean. Superscript letters a, b, and c next to each reference represent different equations within the same study.

Methods	Instantaneous Natural Mortality Rate (M)	Annual Natural Mortality Rate (HM)
Alverson and Carney [19]	0.14	0.13
Rickhter and Efanov [20]	0.26	0.23
Pauly [21]	0.28	0.24
Jensen [22] a	0.23	0.20
Jensen [22] b	0.24	0.21
Jensen [22] c	0.28	0.24
Frisk et al. [23] a	0.20	0.18
Frisk et al. [23] b	0.22	0.20
Then et al. [24] a	0.35	0.29
Then et al. [24] b	0.21	0.19
Zhou et al. [25]	0.28	0.25
Specialized literature	0.16	0.15
$\overline{\mathrm{x}}$	0.24	0.21
±	0.06	0.05

Table 2

Estimated values of the instantaneous total mortality rate (Z) and the annual natural mortality rate (H_Z) of Pseudocarcharias kamoharai in the Ecuadorian Pacific.

Methods	Instantaneous Total Mortality Rate (Z)	Annual Natural Mortality Rate (HZ)
Beverton and Holt [27]	0.08	0.08
Beverton and Holt [28]	0.51	0.40
Ssentengo and Larkin [29]	0.15	0.14
Hoenig [30]	0.23	0.21

Table 3

Estimated values for the instantaneous natural mortality rates (M), mortality rates by fishing (F), and total mortality rates (Z) of several shark and ray species, including Pseudocarcharias kamoharai from the Ecuadorian Pacific. The Z was estimated as the sum between M and F. The taxonomic arrangement of the species was based on Van der Laan et al. [51].

Order	Family	Species	M	F	Z	Location	Reference
Orectolobiformes	Rhincodontidae	Rhincodon typus	0.09	0.05	0.14	Western and Central Pacific Ocean	[25]
Lamniformes	Pseudocarchariidae	Pseudocarcharias kamoharai	0.24	0.27	0.51	Eastern Tropical Pacific Ocean	This study
	Alopiidae	Alopias pelagicus	0.17	0.03	0.20	Western and Central Pacific Ocean	[25]
		Alopias superciliosus	0.17	0.06	0.23	Western and Central Pacific Ocean	[25]
		Alopias vulpinus	0.21	0.08	0.29	Western and Central Pacific Ocean	[25]
	Lamnidae	Isurus oxyrinchus	0.14	0.04	0.18	Western and Central Pacific Ocean	[25]
		Lamna nasus	0.12	0.04	0.16	Western and Central Pacific Ocean	[25]
Carcharhiniformes	Triakidae	Triakis megalopterus	0.14	-	-	Western Indian Ocean	[52]
		Triakis semifasciata	0.24	0.08	0.32	Northeast Pacific Ocean	[49]
	Carcharhinidae	Carcharhinus falciformis	0.13	0.04	0.17	Western and Central Pacific Ocean	[25]
		Carcharhinus longimanus	0.16	0.09	0.25	Western and Central Pacific Ocean	[25]
		Prionace glauca	0.20	0.11	0.31	Western and Central Pacific Ocean	[25]
	Galeocerdonidae	Galeocerdo cuvier	0.16	0.17	0.33	Eastern Indian Ocean	[53]
	Sphyrnidae	Sphyrna lewini	0.15	0.06	0.21	Western and Central Pacific Ocean	[25]
		Sphyrna mokarran	0.14	0.08	0.22	Western and Central Pacific Ocean	[25]
		Sphyrna zygaena	0.16	0.04	0.20	Western and Central Pacific Ocean	[25]
		Eusphyra blochii	0.20	0.11	0.31	Western and Central Pacific Ocean	[25]
Rhinopristiformes	Rhinobatidae	Pseudobatos glaucostigma	0.17	0.19	0.36	Northeast Pacific Ocean	[50]
		Pseudobatos productus	0.29	0.39	0.68	Northeast Pacific Ocean	[50]

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