240 



Fishery Bulletin 98(2) 



A = the finite rate ofpopulation increase given 

 by the dominant eigenvector of A (lnA=r, 

 the intrinsic rate of increase). 



We began with an initial value of A,„„ = 1. We then 

 iterated A until the value of A given by an eigenanal- 

 ysis of the matrix A (see below), equaled the value 

 of A„j„ used in Equation 6. Together, Equations 1-6 

 described above allow estimates of G, and P, within 

 A to be defined. 



Fecundity must also be defined. The fecundity 

 term, /j, is simply the expected number of female 

 offspring produced by a female in stage i. For sand- 

 bar sharks only one stage is reproductively active 

 and thus the only fecundity term in the matrix A is 

 f^ = 4.5. This estimate is based on an equal sex ratio, 

 9 pups per brood, and one brood per year. However, 

 as the model is a postbreeding census, the fecundity 

 has to be discounted by the probability that a preg- 

 nant female will survive the gestation year to pup. 

 Thus the realized fecundity term used in the model 

 is G4 X f^. All parameters within the matrix A are 

 now defined. 



One feature of stage-based projection models that 

 motivated their use was that they allowed us to solve 

 A analytically in order to calculate important demo- 

 graphic features and find the sensitivity of the model 

 to parameter estimates. The two demographic fea- 

 tures that can be calculated from A are the stable 

 stage distribution and the reproductive value of each 

 stage. Once the stable stage distribution has been 

 reached, the relative proportion of individuals in 

 each stage remains constant over time. The I'epro- 

 ductive value is the relative number of offspring 

 that are yet to be born by individuals in a given 

 age (Gotelli, 1995). This value depends on individu- 

 als surviving to maturity and reproducing. Thus, the 

 youngest stages should have the lowest reproductive 

 values because individuals in those stages must sur- 

 vive and reach maturity before they can reproduce. 

 Both features can be calculated from an eigenanaly- 

 sis of A. For any |/!xn| matrix one may define up to 

 n scalar values (Aj ,, ) and /! -associated right and left 

 vectors such that 



Aw = Aw 



vAT - Av 



(7) 



(8) 



where A^ = the transpose of A; 

 A = the eigenvalue; and 

 w and V = the right and left eigenvectors of A. 



interpretation is simplified for the sandbar shark 

 transition matrix. A, because it is non-negative, irre- 

 ducible, and primitive. Thus, we are guaranteed that 

 there is a single, dominant eigenvalue, Aj, that is 

 real, positive, and strictly greater than all other pos- 

 sible As. This dominant eigenvalue, Aj will eventu- 

 ally describe the population rate of increase and In 

 Aj = r, the intrinsic rate of increase of the popula- 

 tion. Moreover, the right and left eigenvectors associ- 

 ated with Aj will be strictly positive. The population 

 structure will eventually become proportional to a 

 single stable stage distribution, given by Wj. Finally, 

 there will be a single vector, Vj, associated with 

 Aj, that expresses the relative contributions of each 

 stage to the future population — a vector of reproduc- 

 tive values. Reproductive values are standardized so 

 that the reproductive value of an individual in the 

 first stage is one. 



We were interested in calculating the change in A 

 following a change in vital rates expressing a tran- 

 sition from stage / to any other stage (including 

 remaining in i ) that may have been caused by man- 

 agement activities. This change reflects the sensitiv- 

 ity of A to the transition probability. If entries in the 

 transition matrix A are represented as a, „ it can be 



'J 



shown that 



3A 

 da,, 



(9) 



i>,v) 



where <w,v> = the scalar product of the two vectors. 



Simply stated, the sensitivity of the population 

 growth rate to changes in any vital rate is the prod- 

 uct of the reproductive value of stage / and the 

 proportional level of stagey in the stable stage dis- 

 tribution. 



Because transition probabilities are censored 

 parameters, varying only between and 1, and 

 fecundity is noncensored, it is more helpful to report 

 the elasticity of A. This is defined as the proportional 

 change in A for proportional changes in a . Elasitici- 

 ties are calculated as 



3A 



A da. 



(10) 



Importantly, elasticities are additive, such that the 

 sum of elasticities for each stage defines the pro- 

 portional contribution of a to overall population 

 growth, A. as 



The sandbar shark transition matrix has five pos- 

 sible eigenvalues and eigenvect9rs. However, our 



II'„ 



