Brewster-Geisz and Miller: Management of Carcharhinus plumbeus 



237 



rebuild the large coastal stocks to the optimum yield 

 level. In October 1998, the NMFS released a draft 

 FMP for Atlantic tunas, swordfish, and sharks. The 

 measures in this FMP were designed to halt over- 

 fishing and to rebuild these stocks. Management 

 options under consideration for large coastal sharks 

 included restrictions on effort, size limits, and area 

 closures that were focused on nursery gi-ounds. 



Many traditional approaches that could be used to 

 compare management options, such as surplus produc- 

 tion models or age- and size-structured approaches, 

 rely on catch or effort data, or both. However, because 

 logbook reporting in the shark fishery was not man- 

 datory until 1993, fishery-dependent time series have 

 been insufficiently long to permit reliable application 

 of these approaches. Yet a comparison of the efficacy 

 of the potential management options is still required. 

 The paucity of fishery-dependent data suggests that 

 demographic approaches, such as life-table or stage- 

 based analyses, may be the appropriate tools to 

 explore the potential response of shark populations 

 to management actions. 



Life-table analysis is a common age-structured 

 demographic approach with a long history in pop- 

 ulation dynamics (Kingsland, 1985). It is a matrix 

 projection approach that estimates the contribution 

 of each age class to future generations. Sminkey 

 and Musick (1996) applied a life-table approach 

 to sandbar sharks. From the intrinsic rate of nat- 

 ural increase, r, predicted by the model, they con- 

 cluded that the population could not sustain the 

 observed rates of fishing mortality. Heppell et al. 

 (1999) developed matrix-based life tables for leopard 

 {Triakis semifasciata) and angel (Squatina califor- 

 nica) sharks. Heppell et al. calculated the elasticity 

 or proportional sensitivity of the population growth 

 rate to changes in survival and fecundity and con- 

 cluded that the two species differ in the degree to 

 which each can compensate for changes in exploi- 

 tation. Simpfendorfer (1999) developed a life table 

 for the dusky shark {Carcharhinus obscurus). He 

 concluded that in the absence of exploitation, dusky 

 shark populations in southwestern Australia would 

 increase at 4.3% annually. Analysis also indicated 

 that current patterns of exploitation were sustain- 

 able. However, there are problems in application of 

 life-table analysis to long-lived marine species. The 

 intrinsic rate of natural increase predicted is depen- 

 dent on the products of survival and fecundity for all 

 ages and the estimated generation time. Thus, life 

 tables require estimates of the schedules of mortal- 

 ity (survival) and fecundity over the entire age range 

 (Gotelli, 1995). Consequently, in a long-lived species 

 such as the sandbar shark, small errors in para- 

 meter estimates can become magnified. 



Stage-based modeling is a matrix-based demo- 

 graphic approach that considers aggregate stages 

 (defined in terms of size or life history stages) that 

 represent functional biological units (Gotelli, 1995). 

 It too has a long history in the field of ecological pop- 

 ulation dynamics (Kingsland, 1985). A stage-based 

 model can be formed by collapsing a life table into 

 discrete stages. Thus, unlike the life-table analysis 

 that requires estimates for every year the organism 

 lives, the stage-based model requires only estimates 

 for each stage. Therefore, the realism of a many- 

 staged model can be balanced with the precision of 

 a simpler model when parameter estimation error is 

 of concern. As with life tables, stage-based projection 

 models can easily be solved analytically to permit 

 formal sensitivity analysis (Caswell, 1989). Ander- 

 son ( 1990), and Hoenig and Gruber (1990) have sug- 

 gested that stage-based models may provide a more 

 realistic view of the dynamics of some populations. 



The population dynamics of several marine spe- 

 cies, including sandbar sharks (Cortes, 1999), sea 

 turtles (Grouse et al., 1987; Crowder et al., 1994), 

 blue crabs (Miller and Houde''), sardines (Lo et al., 

 1995), and anchovies (Pertierra et al., 1997) have 

 been explored by using stage-based models. Cortes 

 (1999) developed a stochastic stage-based model for 

 sandbar sharks in the western North Atlantic. He 

 used the model to explore the implications of three 

 different harvest strategies on population viability 

 when fecundity varies. He concluded that in the 

 absence of exploitation, the sandbar shark popula- 

 tion should increase slowly by about 1.3% annually. 

 Additionally, Cortes concluded that all three pat- 

 terns of exploitation would cause declines in popula- 

 tion abundance. Cortes' model and results indicate 

 the utility of stage-based models in exploring poten- 

 tial management alternatives for sandbar sharks. 



Here, we develop a deterministic stage-based model 

 for sandbar shark populations. The model includes 

 the two-year reproductive cycle of fertile and rest- 

 ing periods known to occur in sandbar sharks, but 

 which were not included in Cortes's (1999) original 

 description. We chose to use a deterministic frame- 

 work to permit a formal elasticity (proportional sen- 

 sitivity) analysis of the basic model. Stages to which 

 the population dynamics are most sensitive can be 

 interpreted either as being stages at which manage- 

 ment action is likely to have the most impact or as 

 stages at which parameter estimates have to be most 

 precise because of impacts of potential environmen- 

 tal stochasticity. We use the model to examine the 

 expected change in population growth resulting from 



5 Miller, T. J., and E. D. Houde. 1998. Blue crab targeting. 

 U.S. EPA Chesapeake Bay Program Report, Annapolis, MD, 167 p. 



