A Monte Carlo demographic analysis of the silky shark : implications of gear selectivity

A Monte Carlo demographic analysis of the silky shark : implications of gear selectivity – Carcharhinus falciformis

Lawrence Beerkircher

Demographic analysis has recently been used as a tool to approximate the dynamics of shark populations (Cailliet, 1992; Sminkey and Musick, 1996; Au and Smith, 1997; Cortes, 1999). The widespread use of demographic models for shark species, however, is hampered by a paucity of information on vital rates, which are required as input parameters for the models. A vital rate of special importance is natural mortality, which remains unknown for most shark species and many other marine taxa. Demographic models thus must often rely on indirect methods of natural mortality estimation rather than empirically observed values specific to a given population. Several methods to estimate natural mortality have shown promise for shark demographic analysis (Cortes, 1998). Other vital rates such as fecundity, although often strongly influenced by size and age, likely also vary independently of time over the lifespan of individuals within a population. Incorporation of Monte Carlo simulation in demographic models has been used to account for some of this uncertainty in vital rate estimates and construct confidence intervals for model output (Cortes, 1999; 2002).

One of the most valuable aspects of demographic modeling is the ability to examine how populations might respond to various levels of fishing mortality (Au and Smith, 1997; Cortes, 1998; Smith et al., 1998; Simpfendorfer, 1999). To date, most authors have used empirical estimates of fishing mortality (F) and either applied them uniformly to all age groups or have produced various scenarios with F applied starting at a particular age to simulate various fisheries management schemes (Sminkey and Musick, 1996; Au and Smith, 1997; Liu and Chen, 1999). Gear selectivity and its effect on F at various ages, however, has not been incorporated into demographic models even though it is very likely that some selection occurs. Here, we produce a demographic analysis of the silky shark (Carcharhinus falciformis) off the southeastern United States. Monte Carlo methods are used to simulate variability in model input parameters, and multiple scenarios are considered, incorporating natural mortality only, and added fishing mortality components. We include in our analysis scenarios that reflect possible longline gear selectivity for silky sharks, using catch-at-age information estimated from length-frequency data provided by the U.S. pelagic longline observer program for the southeastern U.S. coast. Finally, we compare scenarios that incorporate the same mean value of fishing mortality, but with different assumptions of selectivity patterns. The results of the demographic analysis are discussed in the context of the robustness of model assumptions.

Materials and methods

Life history parameters for the silky shark demography were obtained from the best available literature sources. Von Bertalanffy parameters of [L.sub.[infinity]]= 311 cm total length, K = 0.101/yr, and [t.sub.0] = -2.718 yr were taken from Bonfil et al. (1993). To account for some of the uncertainty in the estimate of age-at-maturity ([t.sub.mat]) of 12 years given by Bonfil et al. (1993), values for [t.sub.mat] were drawn at random from a discrete probability distribution of P = 0.25, 0.5, and 0.25 for ages 11, 12, and 13, respectively. Longevity ([t.sub.max]) was similarly varied by assigning the estimate of Bonfil et al. (1993) of 22 years a P = 0.5 and linearly decreasing P by 50% for each subsequent year.

Estimates of natural mortality (M) were selected by using three methods: Peterson and Wroblewski (1984), Chen and Watanabe (1989), and Jensen (1996) (Table 1). For the Peterson and Wroblewski method, weights-at-age were determined by calculating total length-at-age from the von Bertalanffy growth function, and then converting these values to fork length and weight (in grams) by using the relationships given by Kohler et al. (1995) for the silky shark. Wet weight was used instead of dry weight to generate values of M. Annual survival ([S.sub.x]; S=[e.sup.-M], x=age) values were randomly drawn from a distribution in which the three methods of calculating mortality rates had equal probability.

Fecundity ([m.sub.x]), the number of female pups produced per each year of life, was calculated by combining the data of Fourmanoir (1961), Bane (1966), Gilbert and Schlernitzauer (1966), Branstetter (1987), and Bonfil et al. (1993), which yielded a mean of 10.3 (SD=2.213, n=16) pups per litter. Observations in the literature of litter sizes lower than five pups were not included because of the possibility that females might have aborted pups from one uterus during capture (Branstetter, 1987; Bonfil et al., 1993). Pups-per-litter estimates were then converted into female pups per year ([m.sub.x]) by using the embryo sex ratio of 1:1.17 (male:female) given by Bonfil et al. (1993) and a reproductive cycle of two years suggested by Branstetter (1987). All females were assumed to enter the breeding population after first age-at-maturity, and reproductive senescence was assumed to occur at age ([t.sub.max]+1). Actual values of fecundity ([m.sub.x]) used as model input were drawn at random from a normal distribution with a mean of 10.3 and a standard deviation of 2.213 calculated as described above, and then scaled to represent female pups per year.

The parameters estimated above were used to construct life tables by using standard methods, but incorporating Monte Carlo simulation for output parameter estimation. Output parameters (net reproductive rate, [R.sub.0]; mean generation length, G; and intrinsic rate of population increase, r) were calculated by standard demographic methods (Wilson and Bossert, 1971; Krebs, 1985), and r was solved iteratively with the Euler equation.

Various scenarios, each of which varied vital rates as described above, were analyzed. A base scenario incorporated natural mortality only. To attempt to incorporate gear selectivity, we compared a model that incorporated total mortality only (Z, scenario 1) estimated from a traditional catch curve with a model incorporating Z estimated from a modified catch curve method. To produce a catch curve estimate of Z, length-frequency data for silky sharks observed on longlines off the southeastern United States during 1992-98 (NMFS (1)) were used with an age-length key derived from the length-at-age data of Bonfil et al. (1993). The assumptions in this method are that in all years 1) recruitment is constant, 2) fishing mortality is constant, and 3) catchability is equal (there is no gear selectivity). The decrease in population over time is

[N.sub.t] = [N.sub.t-1][e.sup.-Za],

where [N.sub.t] = population in numbers at time t; and a = age.

An assumption of gear selectivity was then used to modify the standard catch equation. If one assumes that gear selectivity exists, then Z can no longer be constant and Z can be estimated at each age:

-Z = ln([N.sub.t]/[N.sub.t-1])/a.

To avoid positive values of Z at some ages (due to those ages not being present in the catch as predicted by the age-length key) an exponential curve was fitted to the catch-at-age data to produce the values of [N.sub.t]. Because the value of Z cannot be calculated for age 0 with this method, we assumed that total mortality was the same for ages 0 and 1. The values of Z-at-age produced by this method were used in a demographic model (scenario 2) and compared with scenario 1. For ages where estimates of Z were not reasonable (Z<M), the natural mortality estimate was used for that age.

To examine the effect of various patterns of selectivity, we also compared scenarios with the same average value of fishing mortality, but with various assumptions of selectivity patterns. These scenarios added a fishing mortality component into the base scenario. Scenario 3 incorporated the 1997 instantaneous fishing mortality (F) value of 0.093 estimated for Atlantic large coastal sharks (Anonymous (2)) applied uniformly to all age groups. An exponential function of the form y=[be.sup.(-ax)], where y = F, x = age, and the mean F = 0.093, was fitted to the catch-at-age data assuming zero fishing mortality after age 13 (scenario 4) and a positive level of fishing mortality through all age groups (scenario 5). Scenario 6 applied an F value of 0.034, calculated to produce maximum sustainable yield (MSY) for large coastal sharks (Anonymous (2)), uniformly to all age groups. Scenarios 7 and 8 were constructed in the same manner as scenarios 4 and 5, respectively, but with the mean value of F = 0.034.

Each of the scenarios were run 1000 times with vital rates (first age-at-maturity, fecundity, longevity, and natural mortality where appropriate) varying as described above. The 2.5th and 97.5th percentiles of the ranked output were used as approximate percentile confidence intervals for the parameters [R.sub.0], G, and r. Finite rates of population increase or decrease ([e.sup.r]) and approximate percentile confidence intervals were then calculated based on the r value output. All simulations were run by using Microsoft Excel spreadsheet software.


The three methods used to estimate natural mortality yielded annual survivorship (S) values ranging from 0.657 to 0.904 (Table 2). The method of Peterson and Wroblewski produced S values of 0.760 (age 0) to 0.904 (age 22), slightly higher than the values produced by the method of Chen and Watanabe (0.657 to 0.874, ages 0-11; 0.890, ages 12+). The method of Jensen produced the lowest S value (0.851) for ages 7 and above, and the highest survivorship in the first 4 years of life.

The catch-at-age histogram indicated that few silky sharks over the age of 7 years were caught in the southeastern pelagic longline fishery (Fig. 1). The catch curve estimated a Z value of 0.329 ([r.sup.2]=0.838, Fig. 2). The Z-at-age values calculated from the modified catch curve method predicted values of Z = 0 for ages 12 and above (scenario 2; Table 3). Values of F-at-age used in scenarios 3-8 (Table 3) were as high as 0.287 (scenario 4, age 0).


The results of the base scenario (Table 4) indicated that the silky shark population, in absence of fishing mortality, would increase at a median rate of 4.9 %/yr. The approximate percentile lower confidence limit also showed positive population growth ([e.sup.r] = 1.027). Incorporating the Z obtained from the catch curve into the model (scenario 1) resulted in the greatest rate of population decline of any of the scenarios considered ([e.sup.r] = 0.895). However, the use of the Z-at-age values from the modified catch curve method (scenario 2) produced a positive population increase of 1.8 %/yr.

With estimated F for large coastal sharks in 1997 applied to all age groups (scenario 3), the population would decline at a rate of 4.4 %/yr. Application of exponentially decreasing fishing mortality resulted in a more optimistic median [e.sup.r] value when the mean F was confined to ages 0-12 (scenario 4), but when mean F was applied to all ages (scenario 5), the output was more pessimistic than that for the constant F = 0.093 model. With F estimated for maximum sustainable yield (scenario 6), the silky shark population would slowly increase at a rate of 1.5 %/yr. Incorporation of variable F-at-age in scenarios 7 and 8 resulted in [e.sup.r] values slightly higher and lower, respectively, than the constant F scenario.


The base scenario results given in our study match well those from other demographic models for silky sharks. Using a demographic model that incorporated density-dependence, Smith et al. (1998) found an r value of 0.043 for silky sharks. Cortes (2002) found an average r value of 0.055 for silky sharks from the southern Gulf of Mexico using density-independent matrix simulation models. These r values place the silky shark toward the middle of the range of productivity values calculated for a mix of species (Smith et al., 1998; Cortes, 2002).

The pessimistic results of the model that incorporated Z calculated from the catch curve are consistent with the stock assessments that indicate large coastal sharks are overfished and suggest, in particular, that exploitation rates in the pelagic longline fishery may not be sustainable for silky sharks. However, incorporating gear selectivity with the modified catch curve method altered the results, yielding one of the highest rates of population growth of any of the scenarios. If the assumptions about the relationship between the catch curve and selectivity are reasonable, the results indicate that incorporation of gear selectivity into demographic analysis is important.

The importance of considering selectivity as explored by the fishing mortality scenarios was less clear. Scenario 3 indicated that a recently estimated level of fishing mortality would produce declines in the population, whereas scenario 6 indicated that the fishing mortality level producing maximum sustainable yield would result in a slightly increasing population. When exponentially decreasing F-at-age was incorporated (scenarios 4, 5, 7, and 8), the results differed–but much less so than those of scenarios 1 and 2–from those obtained in the constant F scenarios (3 and 6). All the fishing mortality scenarios (3-8) used mean values of F estimated through density-dependent modeling and were used here only as comparative base values.

Stock assessments of northwestern Atlantic shark stocks are based on species groups rather than individual species. These assessments use data from fisheries of various gear types, which may not correctly represent the life history or the pattern of exploitation of an individual species. In that respect, Carcharhinus falciformis may be an excellent example. Off the coast of the southeastern United States, pelagic longlines may be the dominant cause of silky shark fishing mortality, whereas bottom longlines rarely capture, and drift gillnets do not capture, this species (Parrack et al., 1993; Trent et al., 1997). Examination of length frequencies from the pelagic longline fishery observer database suggests that age 8+ silky sharks are rarely caught. Therefore, the assumption of constant fishing mortality throughout all age groups for this species is probably invalid, at least for the pelagic fishery off the southeastern United States. If individual species management of silky sharks is ever to become a reality, the patterns of selectivity of longline gear are very important.

It seems logical that the larger the shark, the higher the probability it can bite through the monofilament gangions used by U.S. pelagic longliners and escape. Because size is correlated with age, there must be a functional relationship that describes the probability of retention and capture for a given age. Accurate elucidation of such gear selectivity patterns requires knowledge of the actual size-at-age characteristics of the population that can only be derived from sampling studies in which methods that are not size-selective are used, or that are at least less size-selective than longline gear. These types of comparative data are almost wholly lacking for sharks in the pelagic environment. Indirect methods of estimating selectivity for shark species by comparing catches from gillnets of various mesh sizes have shown promise (Kirkwood and Walker, 1986; Simpfendorfer and Unsworth, 1998). Previous indirect studies of selectivity in hook and line gear (Cortez-Zaragoza et al., 1989; Ralston, 1990) cannot be applied in this case because selectivity in those studies was described as a function of hook size rather than the actual breaking strength of the line, which is the speculated mechanism of selection in the present study. Walker (1992) incorporated hook selectivity into an age-structured model, but he assumed that all recruited members of a stock had an equal value of catchability with longline gear. We believe that an assumption of equal catchability is incorrect for silky sharks, and sharks in general, hooked on monofilament pelagic longline gear.

Determining actual patterns of selectivity and resulting estimates of fishing mortality-at-age, however, was beyond the scope of our analysis, which was to suggest that patterns of gear selectivity can affect results from demographic models. We chose a simple modification of a traditional catch curve and an intuitive, although arbitrary, assumption of selectivity patterns for silky sharks on pelagic longlines to explore this effect. Arbitrarily derived selectivity schedules were employed previously to examine the potential effects of gear selection in yield-per-recruit models (Goodyear, 1996). Clearly this approach, as applied here, has limitations. For example, the curve fitted to catch-at-age data used to calculate Z values estimated zero total mortality at ages 12+, whereas the catch-at-age data showed a few captures of ages 12+ (Fig. 1). No mortality is assumed for individual animals that escape the gear, although it is possible that some animals may die. Despite shortcomings in the methods to estimate F-at-age values used in our study, better estimates of fishing mortality will require more robust, comparative selectivity data.

Like many other studies of shark populations, the results reported in our study must be considered preliminary because of the lack of validated age and growth information and the uncertainty in the estimates of longevity and fecundity for the silky shark. It is difficult to determine how representative the yon Bertalanffy parameters (Bonfil et al., 1993), which were derived from silky sharks in the Campeche Bank in the southern Gulf of Mexico, are for silky shark populations from the entire southeastern U.S. Atlantic. Although tagging data (Kohler et al., 1998) indicate movement of silky sharks between the Gulf of Mexico and the U.S. Atlantic coast, the small number of tagged and recaptured individuals makes firm conclusions about whether these are the same or separate stocks very difficult. The age and growth results of Bonfil et al. (1993) from the southern Gulf of Mexico differed from a study on silky sharks by Branstetter (1987) in the northern Gulf. It is possible, however, that these variations may be a result of methodological or sampling differences because Branstetter’s (1987) data came from pelagic longline operations and had fewer specimens over 250 cm TL, compared to Bonfil et al.’s (1993) data which came mainly from specimens caught in gillnet fisheries. Even if migration of silky sharks occurs between the northern Gulf of Mexico, southern Gulf of Mexico, and southeastern U.S. Atlantic coastal region, the populations of sharks in these areas might have significant life history differences. Given the absence of age and growth studies from Atlantic silky sharks, the Gulf of Mexico studies must be regarded as the best available age and growth information for this species. Similarly, fecundity and longevity estimates are relatively uncertain. Especially important in fecundity estimates is information regarding age-specific birth rate and the possibility of reproductive senescence, which can affect demographic results.

As with other demographic analyses, theoretical estimates of natural mortality (M) were used in our study. It has been shown that the output of demographic models of shark populations is particularly sensitive to changes in M, especially during the first few years of life (Sminkey and Musick, 1996; Liu and Chen, 1999; Cortes, 2002). Moreover, results may vary considerably in analyses that use different methods of M estimation (Simpfendorfer, 1999). In recognition of these limitations, three different methods were used here to estimate M. The methods of Chen and Watanabe (1989) and Jensen (1996) are based on a relationship between M and growth rates. Because the age and growth information for silky sharks is unvalidated, the von Bertalanffy growth parameters used in our study may not adequately describe individuals from the population analyzed. Finally, the method of Peterson and Wroblewski (1984) involves the assumption that M is due primarily to predation. Although this may be true for the younger ages of large coastal sharks like the silky shark, it may not be true for the larger, older sharks. Additionally, wet weight was used instead of dry weight, as prescribed in the original equation of Peterson and Wroblewski (1984), because the estimates of M obtained by using wet weight seem more believable for sharks (Cortes, 2002). Despite the difficulties associated with all three methods of estimating natural mortality, these methods, together with catch curves and tag-recapture studies, are the only methods available for estimating M. The use of Monte Carlo simulation to randomly vary the estimates of natural mortality can account for some of the uncertainty associated with these methods (Cortes, 1999; 2002).

Conclusions from this analysis of the silky shark population off the southeastern U.S. coast are similar to those for other large shark species. Even under scenarios assuming natural mortality only, the population would grow at a fairly slow rate, and even moderate levels of fishing mortality might produce population declines. Inclusion of age-dependent patterns of fishing mortality into the analyses, however, appear to substantially affect model results. Several key elements that need to be clarified to obtain a more conclusive analysis for silky sharks include validated age and growth information, life history parameters such as longevity, fecundity, and natural mortality, and estimates of age-dependent fishing mortality.

Table 1

Natural mortality (M) estimation methods used in demographic

models for silky sharks (Carcharhinus falciformis). W = body

weight in grams, K, [t.sub.0] = von Bertalanffy growth

parameters, and [t.sub.max] = maximum age.

Citation Method

Peterson and Wroblewski (1984) M = 1.92 [W.sub.-0.25]

Jensen (1996) M = 1.6K





Table 2

Annual survivorship (S) estimates calculated from three

studies based on best available life history information for

the silky shark (Carcharhinus falciformis).

Peterson and Chen and

Age (yr) Wroblewski Watanabe Jensen

0 0.760 0.657 0.851

1 0.799 0.724 0.851

2 0.824 0.766 0.851

3 0.840 0.794 0.851

4 0.853 0.815 0.851

5 0.862 0.830 0.851

6 0.869 0.842 0.851

7 0.875 0.851 0.851

8 0.880 0.858 0.851

9 0.884 0.865 0.851

10 0.887 0.870 0.851

11 0.890 0.874 0.851

12 0.892 0.890 0.851

13 0.894 0.890 0.851

14 0.896 0.890 0.851

15 0.898 0.890 0.851

16 0.899 0.890 0.851

17 0.900 0.890 0.851

18 0.901 0.890 0.851

19 0.902 0.890 0.851

20 0.903 0.890 0.851

21 0.904 0.890 0.851

22 0.904 0.890 0.851

Table 3

Values of total mortality (Z) or fishing mortality (F)

used in demographic models for the silky shark

(Carcharhinus falciformis). For scenario 1, Z was used

instead of estimates of natural mortality (M). In

scenario 2, Z was used only when Z>M. In scenarios

3-8, F was added to natural mortality.


1 2 3 4

Age (yr) Z Z F F

0 0.329 0.486 0.093 0.287

1 0.329 0.486 0.093 0.210

2 0.329 0.243 0.093 0.164

3 0.329 0.162 0.093 0.132

4 0.329 0.121 0.093 0.107

5 0.329 0.097 0.093 0.087

6 0.329 0.081 0.093 0.069

7 0.329 0.069 0.093 0.054

8 0.329 0.061 0.093 0.041

9 0.329 0.054 0.093 0.029

10 0.329 0.049 0.093 0.019

11 0.329 0.044 0.093 0.009

12 0.329 0.000 0.093 0.09 x


13 0.329 0.000 0.093 0.000

14 0.329 0.000 0.093 0.000

15 0.329 0.000 0.093 0.000

16 0.329 0.000 0.093 0.000

17 0.329 0.000 0.093 0.000

18 0.329 0.000 0.093 0.000

19 0.329 0.000 0.093 0.000

20 0.329 0.000 0.093 0.000

21 0.329 0.000 0.093 0.000

22 0.329 0.000 0.093 0.000

23 0.329 0.000 0.093 0.000

24 0.329 0.000 0.093 0.000

25 0.329 0.000 0.093 0.000

26 0.329 0.000 0.093 0.000

27 0.329 0.000 0.093 0.000

28 0.329 0.000 0.093 0.000

29 0.329 0.000 0.093 0.000


5 6 7 8

Age (yr) F F F F

0 0.276 0.034 0.105 0.068

1 0.225 0.034 0.077 0.058

2 0.195 0.034 0.060 0.053

3 0.174 0.034 0.048 0.049

4 0.158 0.034 0.039 0.046

5 0.144 0.034 0.032 0.043

6 0.133 0.034 0.025 0.041

7 0.123 0.034 0.020 0.040

8 0.114 0.034 0.015 0.038

9 0.107 0.034 0.011 0.037

10 0.100 0.034 0.007 0.035

11 0.093 0.034 0.003 0.034

12 0.087 0.034 0.09 x 0.033


13 0.082 0.034 0.000 0.032

14 0.077 0.034 0.000 0.031

15 0.072 0.034 0.000 0.030

16 0.068 0.034 0.000 0.029

17 0.063 0.034 0.000 0.029

18 0.059 0.034 0.000 0.028

19 0.056 0.034 0.000 0.027

20 0.052 0.034 0.000 0.026

21 0.049 0.034 0.000 0.026

22 0.045 0.034 0.000 0.025

23 0.042 0.034 0.000 0.025

24 0.039 0.034 0.000 0.024

25 0.036 0.034 0.000 0.024

26 0.034 0.034 0.000 0.023

27 0.031 0.034 0.000 0.023

28 0.028 0.034 0.000 0.022

29 0.026 0.034 0.000 0.022

Table 4

Demographic output ([R.sub.0]=net reproductive rate;

G=mean generation length; r=intrinsic rate of population

growth; [e.sup.r]=finite rate of population growth) and

lower (LCL) and upper (UCL) approximate confidence limits

for the silky shark (Carcharhinus falciformis) off the

southeastern U.S. coast. The base scenario incorporated

natural mortality (M) only; scenario 1 incorporated a

constant value of total mortality (Z), scenario 2

incorporated various values of Z-at-age, and scenarios 3-8

incorporated the mean value of fishing mortality (F) shown

in the second column.

Mean F

Scenario (or Z) [R.sub.0] LCL UCL G

base none 2.161 1.547 2.959 16.115

1 0.329 0.195 0.127 0.284 14.380

2 0.065 2.615 2.083 3.158 16.091

3 0.093 0.508 0.345 0.724 15.190

4 0.093 0.652 0.477 0.886 16.113

5 0.093 0.262 0.175 0.375 15.427

6 0.034 1.266 0.898 1.747 15.755

7 0.034 1.392 0.991 1.920 16.105

8 0.034 1.130 0.783 1.539 15.815

Scenario LCL UCL r LCL

base 14.942 17.570 0.048 0.027

1 13.266 15.484 -0.111 -0.130

2 14.965 17.582 0.018 0.001

3 14.042 16.544 -0.044 -0.066

4 14.946 17.546 -0.026 -0.045

5 14.235 16.857 -0.084 -0.105

6 14.605 17.176 0.015 -0.006

7 14.937 17.565 0.021 -0.001

8 14.623 17.261 0.008 -0.015

Scenario UCL [e.sup.r] LCL UCL

base 0.071 1.049 1.027 1.074

1 -0.091 0.895 0.878 O.913

2 0.032 1.018 1.001 1.033

3 -0.022 0.957 0.936 0.978

4 -0.008 0.974 0.956 0.992

5 -0.066 0.919 0.900 0.936

6 0.037 1.015 0.994 1.038

7 0.043 1.021 0.999 1.044

8 0.029 1.008 0.985 1.029


We thank Ramon Bonfil for providing his silky shark length and age data. We also thank Dennis Lee and Cheryl Brown of the National Marine Fisheries Service Pelagic Observer Program for catch-at-length data. The comments of three anonymous reviewers greatly improved this paper.

(1) NMFS (National Marine Fisheries Service). 2000. Unpubl. data. Southeast Fisheries Science Center Pelagic Observer Program, 75 Virginia Beach Drive, Miami, FL 33149.

(2) Anonymous. 1998. Report of the shark evaluation workshop, 109 p. Panama City Laboratory, National Marine Fisheries Service, 3500 Delwood Beach Rd., Panama City, FL 32408.

Literature cited

Au, D. W., and S. E. Smith. 1997. A demographic method with population density compensation for estimating productivity and yield per recruit of the leopard shark (Triakis semifasciata). Can. J. Fish. Aquat. Sci. 54:415-420.

Bane, G. W. 1966. Observations on the silky shark, Carcharhinus falciformis, in the Gulf of Guinea. Copeia 1966:354-357.

Bonfil, R., R. Mena, and D. de Anda. 1993. Biological parameters of commercially exploited silky sharks, Carcharhinus falciformis, from the Campeche Bank, Mexico. U.S. Dep. Commer., NOAA Tech. Rep. NMFS 115:73-86.

Branstetter, S. 1987. Age, growth, and reproductive biology of the silky shark, Carcharhinus falciformis, and the scalloped hammerhead, Sphyrna lewini, from the northwestern Gulf of Mexico. Environ. Biol. Fishes 19(3):161-173.

Cailliet, G. M. 1992. Demography of the Central California population of the leopard shark (Triakis semifasciata). Aust. J. Mar. Freshwater Res. 43:183-193.

Chen, S., and S. Watanabe. 1989. Age dependence of natural mortality coefficient in fish population dynamics. Nippon Suisan Gakkaishi 55(2): 205-208.

Cortes, E. 1998. Demographic analysis as an aid in shark stock assessment and management. Fish. Res. 39:199-208.

1999. A stochastic stage-based population model of the sandbar shark in the western North Atlantic. In Life in the slow lane: ecology and conservation of long-lived marine animals (J. A. Musick, ed.), p. 115-136. Am. Fish. Soc. Bethesda, MD.

2002. Incorporating uncertainty into demographic modeling: application to shark populations and their conservation. Conserv. Biol. 16(4):1048-1062.

Cortez-Zaragoza, E., P. Dazell, and D. Pauly. 1989. Hook selectivity of yellowfin tuna (Thunnus albacares) caught off Darigayos Cove, La Union, Philippines. J. Appl. Ichthyol. 1:12-17.

Fourmanoir, P. 1961. Requins de la cote ouest de Madagascar. Ser. Oceanog. 4:3-81.

Gilbert, P., and D. A. Schlernitzauer. 1966. The placenta and gravid uterus of Carcharhinus falciformis. Copeia 1966:451-457.

Goodyear, C. P. 1996. Variability of fishing mortality by age: consequences for maximum sustainable yield. N. Am. J. Fish. Manage. 16:8-13.

Jensen, A. L. 1996. Beverton and Holt life history invariants result from optimal trade-off of reproduction and survival. Can. J. Fish. Aquat. Sci. 53:820-822.

Kirkwood, G. P., and T. I. Walker. 1986. Gill net mesh selectivities for gummy shark Mustelus antarcticus Gunther, taken in south-eastern Australian waters. Aust. J. Mar. Freshwater Res. 37:689-697.

Kohler, N. E, J. G. Casey, and P. A. Turner. 1995. Length-weight relationships for 13 species of sharks from the western North Atlantic. Fish. Bull. 93:412-418.

1998. NMFS cooperative shark tagging program, 1962-93: an atlas of shark tag and recapture data. Mar. Fish. Rev. 60(2):1-79.

Krebs, C. J. 1985. Ecology: the experimental analysis of distribution and abundance, 800 p. Harper and Row, New York, NY.

Liu, K. M., and C. T. Chen. 1999. Demographic analysis of the scalloped hammerhead, Sphyrna lewini, in the Northwestern Pacific. Fish. Sci. 65(2):218-223.

Parrack, M. L., J. I. Castro, and J. E. Powers. 1993. The United States Atlantic coastal shark fishery. Int. Comm. Conserv. Atl. Tuna, Collect. Vol. Sci. Pap. 126: 406-408.

Peterson, I., and J. S. Wroblewski. 1984. Mortality rate of fishes in the pelagic ecosystem. Can. J. Fish. Aquat. Sci. 41:1117-1120.

Ralston, S. 1990. Size selection of snappers (Lutjanidae) by hook and line gear. Can. J. Fish. Aquat. Sci. 47:696-700.

Simpfendorfer, C. A. 1999. Mortality estimates and demographic analysis for the Australian sharpnose shark, Rhizoprionodon taylori, from northern Australia. Fish. Bull. 97:978-986.

Simpfendorfer, C. A., and P. Unsworth. 1998. Gill-net mesh selectivity of dusky sharks (Carcharhinus obscurus) and whiskery sharks (Furgaleus macki) from south-western Australia. Aust. J. Mar. Freshwater Res. 49:713-718.

Sminkey, T.R., and J.A. Musick. 1996. Demographic analysis of the sandbar shark, Carcharhinus plumbeus, in the western North Atlantic. Fish. Bull. 94:341-347.

Smith, S. E., D. W. Au, and C. Snow. 1998. Intrinsic rebound potentials of 26 species of Pacific sharks. Aust. J. Mar. Freshwater Res. 49:663-678.

Trent, L., D. E. Parshley, and J. K. Carlson. 1997. Catch and bycatch in the shark drift gillnet fishery off Georgia and East Florida. Mar. Fish. Rev. 59(1):19-28.

Walker, T. I. 1992. Fishery simulation model for sharks applied to the gummy shark, Mustelus antarcticus Guenther, from southern Australian waters. Aust. J. Mar. Freshwater Res. 43: 195-212.

Wilson, E. O., and W. H. Bossert. 1971. A primer of population biology, 192 p. Sinauer Associates, Sunderland, MA.

Manuscript accepted 10 July 2002.

Lawrence Beerkircher

Mahmood Shivji

Guy Harvey Research Institute

Oceanographic Center

Nova Southeastern University

8000 N. Ocean Drive

Dania Beach, Florida 33004

E-mail address (for L. Beerkircher): beerkirc@ocean.nova.edu

Enric Cortes

National Marine Fisheries Service

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Panama City, Florida 32408

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