NOTE Beerkircher et al : A Monte Carlo demographic analysis of Carcharhinus falciformis 



169 



Table 1 



Natural mortality (M) estimation methods used in demographic models for silky sharks {Carcharhinus falciformis). W - body 

 weight in grams, K, tg = von Bertalanffy growth parameters, and t^^ = maximum age. 



Citation 



Method 



Peterson and Wroblewski ( 1984) 



Jensen (1996) 



Chen and Watanabe ( 1989) 



M = 1.92^-0 '^5 

 M=1.6K 



M{t) = K/ll-e~' 



M(t.t^)=Wt^,,-t)]\nl(e>'' 



for i = through 11 



(Branstetter, 1987; Bonfil et al., 1993). Pups-per-litter 

 estimates were then converted into female pups per year 

 (m^) 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^niax"*"^'- Actual values of fecundity (m^.) 

 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 estima- 

 tion. Output parameters (net reproductive rate, i?,, 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 incorpo- 

 rated 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 esti- 

 mated 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 (NMFSM 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,=N,_,e-^\ 



where Af, = 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: 



ln(A^,/7V,_,)/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,. Because the 

 value of Z cannot be calculated for age with this method, 

 we assumed that total mortality was the same for ages 

 and 1. The values of Z-at-age produced by this method 

 were used in a demographic model (scenario 2) and com- 

 pared 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 selec- 

 tivity 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 (Anony- 

 mous-^) applied uniformly to all age groups. An exponential 

 function of the form y=6e'^"', where y = F, x = age, and the 

 mean F = 0.093, was fitted to the catch-at-age data assum- 

 ing 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, cal- 

 culated to produce maximum sustainable yield (MSY) for 

 large coastal sharks (Anonymous^), 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 natu- 

 ral mortality where appropriate) varying as described 

 above. The 2.5th and 97.5th percentiles of the ranked 

 output were used as approximate percentile confidence 



NMFS (National Marine Fisheries Service). 2000. Unpubl. 

 data. Southeast Fisheries Science Center Pelagic Observer 

 Program, 75 Virginia Beach Drive, Miami, FL 33149. 



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. 



