Brewster-Geisz and Miller: Management of Carcharhinus plumbeus 



241 



Elasticities depend on a stable stage distribution 

 and should be compared qualitatively. 



Transition probability estimation 

 for management options 



Current conditions Fishing mortality (F) and espe- 

 cially natural mortality (M ) are difficult to estimate. 

 Owing to the uncertainty in estimates and in order 

 to simplify the model, we used an estimate of M = 0.1 

 for all stages and all projections (NMFS^ Sminkey 

 and Musick, 1996). We projected the population for- 

 ward using F = 0.20 for juveniles and older stages, 

 as estimated in the 1996 shark evaluation workshop 

 (SEW) for sandbar sharks only (NMFS'^). For neo- 

 nates a lower value of F - 0.10 was used because 

 small sharks may be, but are not as likely to be, 

 caught on the same gear as older sharks (Branstet- 

 ter and Burgess' ). Using these values of F and M and 

 f^=4.5, we iterated Equation 6 to estimate all P, and 

 G, values. We initialized the population with 1000 

 neonates. Then we estimated the initial number in 

 subsequent stages using the dO'Jr survival schedule 

 for sandbar sharks given in Sminkey and Musick 

 ( 1996). These calculations yielded an initial popula- 

 tion of 9640 sharks (Nq). 



Estimate of Fcritical ^^ defined Ff^j^jj^f^j^i^ as the 

 limiting level of fishing mortality that is sustainable, 

 i.e. the F for which r = or A = 1, where r = InA. We 

 systematically reduced F on all stages to define the 

 relationship between F and r. For our estimations, 

 Fj remained 0.10 as long as ^234 5 ^^^ >0.10. For 

 any ^2 3 4 5 <0- 1^- ^i =^2345- Thus, the fishing mor- 

 tality of neonates was never greater than the fishing 

 mortality on other stages. 



Protecting neonates and pregnant adults: an extreme 

 example We used the model to determine how effi- 

 cient protecting different stages would be in pro- 

 moting recovery of sandbar shark stocks. We asked 

 the question: If neonates and pregnant adults are 

 removed from the commercial fishery, how much 

 will F on other stages need to be reduced to arrive 

 at a sustainable population level? To address this 

 question, we modified the model to remove all mor- 

 tality on neonates (F^=0, Mj=0) and to protect all 

 pregnant adults from fishing pressure (^^=0). In 

 reality, we could not completely protect neonates 

 from mortality (i.e. Mj>0) and we could not fully pro- 

 tect pregnant adults from commercial fishing. Thus 

 the scenario represents an idealized nursery closure 

 scheme. Fishing mortality on juveniles, subadults, 

 and resting adults remained at 0.20. The fecundity 

 for pregnant adults was left at 4.5. 



Nursery closures and size limits We also ran the model 

 using more realistic scenarios. In this case F on neo- 

 nates and juveniles was 0, and F on the older stages 

 was 0.20. Natural mortality for all stages remained 

 at 0.10. This scenario is a fairly realistic size limit or 

 nursery ground closure because sandbar sharks seg- 

 regate by size. This scenario is similar to, but not as 

 strict as, the 1998 SEWs recommended size limit. 



Size limits protecting only one stage are another 

 management option available. This method can be 

 used to reduce the fishing mortality on any range of 

 sizes. In this paper, two scenarios of this type are pre- 

 sented: 1 ) a size limit which reduces the F on juve- 

 niles to 0; and 2) a size limit which reduces the F on 

 subadults to 0. Fishing mortality was equal to 0.20 

 in all other stages except neonates, where F=0.10. 

 Implementing such management actions would be 

 difficult because the gear (longlines) cannot realisti- 

 cally avoid catching only the restricted stage, but the 

 results would be indicative of the potential of these 

 mechanisms to improve stocks. 



Using the relationships (Eqs. 1-10) and vital rate 

 estimates defined above, we now proceed with an 

 analysis of the population dynamics of sandbar 

 sharks. For each scenario, we calculate the stable 

 stage distribution, the proportional reproductive 

 value for each stage, and the elasticity of A to changes 

 in each matrix parameter, and compare the popula- 

 tion growth rate and potential population reduction 

 after 20 years for each scenario. 



Results 



Current conditions 



When F = 0.20 for all stages except neonates, the 

 population decreases (Table 1). The model predicts 

 the intrinsic rate of natural increase, r, of the popu- 

 lation as r = -0.124. The population is 13% of the 

 initial abundance when projected 20 years forward. 

 Population growth rate, stable stage distribution, 

 and reproductive values are not affected by choice of 

 the actual numbers used for initial abundance. The 

 stable stage distribution is reached after 21 years in 

 this scenario. 



The largest proportion of the population (>0.56) 

 are juveniles (Table 2). The smallest proportions 

 (0.04, 0.03) are pregnant and resting adults, respec- 

 tively. Adults have much larger reproductive values 

 than prereproductive stages (Table 3). 



The pattern of model proportional sensitivity is 

 shown in Figure 2. The elasticity of A to a small 

 change in fecundity was expressed only in the preg- 



