Brewster-Geisz and Miller Management of Carcharhlnus plumbeus 



239 



lina (Castro, 1993). In the model, females alternate 

 between pregnant and resting adult stages, spend- 

 ing one year in each. Thus, the stage durations used 

 in the model were the following: 1 year for neonates; 

 6 years for juveniles; 8 years for subadults; 1 year for 

 pregnant females; and 1 year for resting females. 



Model development 



The approach we present below is based on the gen- 

 eral framework presented by Caswell ( 1989). Details 

 on the general background of the approach can be 

 found in Caswell (1989). In all equations, matrices 

 and vectors are shown in boldface type, parameters 

 in italic type. 



The model is a postbreeding census, follows only 

 females, and uses a yearly time step. The total 

 number of sharks in the population at time t can be 

 expressed in vector form as N^. Each element of N, 

 represents the number of sharks in the appropriate 

 stage at time t. There are three possible transitions 

 for each individual in each stage: the probability of 

 surviving and growing to the next stage, G,; the prob- 

 ability of surviving yet remaining in the same stage, 

 P,; or the probability of dying, 1-G— P,. The individual 

 transition probabilities G, and P, may range between 

 and 1. The sum of G, and P, is further constrained 

 such that when a stage is not subject to mortality, G, 

 + Pj = 1. One other parameter, stage-specific fecun- 

 dity, is required to estimate the number of young 

 females produced per breeding female per year. 



The vital rates governing the dynamics of the 

 shark population can be expressed mathematically 

 in a 5 X 5 transition matrix, A. The fundamental 

 equation to estimate the stage-structure in the pop- 

 ulation at any time t is given by 



N, = A' - N„ 



(1) 



where N^ = a vector of the number of individuals in 

 each stage at time /; and 

 the transition matrix A for sandbar 

 sharks given by 



A = 



G^xf^ 

 Gi P„ 

 





 

 



G2 ^3 

 













 G, 







G. 







(2) 



For large values of t, AN, = AN,=N,^j, where the 

 scalar A is the finite rate of population increase. Fur- 

 ther, InA = r, the intrinsic rate of increase. 



The columns of the matrix represent the fates of 

 individuals in each stage. For example, surviving neo- 



nates can grow only to the juvenile stage (G,) where- 

 as surviving juveniles can either remain a juvenile 

 (P2) or survive and grow into a subadult iG^)- Sur- 

 viving pregnant adults can give birth and become 

 resting adults (G^). Surviving resting adults (Gr,) can 

 grow only into a pregnant female. The rows repre- 

 sent the origins of individuals in each stage. Neo- 

 nates arise from pregnant adults who survive (G^x/^j) 

 whereas juveniles arise from neonates surviving and 

 growing into juveniles (Gj) or from juveniles surviv- 

 ing and remaining juveniles (Pg). Pregnant adults 

 can arise from subadults surviving and growing into 

 a pregnant female (G3) or from resting adults sur- 

 viving and becoming pregnant adults (Gr,). Resting 

 adults can arise only from pregnant adults that sur- 

 vive (G4). 



The transition probabilities, P, and G,, can be cal- 

 culated from estimates of the probability that during 

 a single time step an individual of stage /' survives, 

 a,, and an individual of stage / grows, y^. In this way 

 G,, the probability of surviving and gi'owing to the 

 next stage is given by 



G, = oj. 



Consequently P,, the probability of surviving, 

 not growing to the next stage is given by 



P, = a,(l-y,). 



(3) 



but 



(4) 



The probability of survival, a^, over a single time step 

 can be expressed as 



a, = e-^i. 



(5) 



Following traditional fisheries models, total mortal- 

 ity (Z,) is calculated by using the equation Z, = F^ + 

 M,, where F, is the rate of fishing mortality and M, is 

 the I'ate of natural mortality at stage /. 



Estimates of 7, can be obtained in several different 

 ways. Caswell (1989) recommended assuming con- 

 stant stage duration for all individuals in the stage 

 when only a relatively crude estimate of survival 

 over broad age ranges is available. For this approach, 

 individuals entering the stage have an equal prob- 

 ability of survival as individuals nearing the end of 

 the stage. Employing this assumption yields an esti- 

 mate of Y, as 



Y, 



(CT,/A„,„) 



-((T, A 



J.-i 



<^- ^,n„ 



.1-1 



(6) 



where T 



the expected stage duration of a single 

 stage; and 



