Hayes; A biological reference point based on the Leslie matrix 



77 



population, S(j)=e-'^" '+''"" where F(i) is the age-spe- 

 cific instantaneous rate of fishing mortahty. The Les- 

 he matrix for the population under exploitation (L^) is 

 thus formed in exactly the same manner as for the un- 

 exploited population except that annual survival rates 

 are decreased through fishing mortality. 



It is important to emphasize that in this analysis, fe- 

 cundity and natural mortahty (including age-0 survival 

 or R/SSB ) are assumed to be constant. As such, the de- 

 termination of a reference point based on an analysis of 

 L^ is valid over the range of stock sizes for which these 

 age-specific fecundity and mortality rates apply. 



Methods of analyzing the Leslie matrix are well es- 

 tablished (e.g. Keyfitz, 1977; Caswell, 1989). Proper- 

 ties of the Leslie matrix under certain regularity con- 

 ditions include the following: 



1 The Leslie matrix has at least one positive root 

 (eigenvalue); 



2 The largest of these roots (the dominant eigenvalue 

 or A) determines the population growth rate, except 

 in cases where the population is inherently cyclical 

 and the largest roots are of equal magnitude; 



3 If A > 1 the population will increase; 



If A = 1 the population will remain steady; 

 If A < 1 the population will decrease. 



(Pielou, 1974; Caswell, 1989; Getz and Haight, 1989). 

 Of particular importance to this reference point is the 

 dominant eigenvalue which, in the deterministic case, 

 is sufficient to determine the long-term trend in popu- 

 lation abundance (Keyfitz, 1977; Cohen et al., 1983). 

 Given these properties, the following assertion for the 

 deterministic case can be made: 



(see Rothschild and Fogarty (1989) for cautions on 

 this practice, however). Because data on spawning 

 stock biomass are more commonly presented in fish- 

 ery assessments than data on fecundity, I will pres- 

 ent the model using age-specific fecundity equivalents 

 (i.e. spawning biomass of individual fish) rather than 

 fecundity. In this representation, the survival rate of 

 age-0 fish is expressed as recruits per unit of spawn- 

 ing stock biomass ( R/SSB ) rather than as actual sur- 

 vival rate from egg to age 1, and SSB is used in place 

 of egg production. It is easily shown that the use of 

 R/SSB and SSB in the Leslie matrix is algebraically 

 equivalent to using fecundity and survival from egg to 

 age 1. 



Given the mapping of the vital rates [E{i), Sii)] of 

 the unexploited population into a Leslie matrix, it is 

 straightforward to represent the dynamics of the pop- 

 ulation under exploitation. Observe that for the un- 

 exploited population S(i)=e~'^"', where Mii) is the in- 

 stantaneous natural mortality rate. For an exploited 



1 A population under exploitation can maintain it- 

 self at or above a given level of abundance only if 

 the dominant eigenvalue of L^ (i.e. A^) > 1. 



From this assertion arises the proposed reference 

 point: F^f (for F steady, after Quinn and Szarzi, 1993) 

 is a fishing mortality pattern where A^ = 1. Note that 

 F^i is actually a vector comprising two components: an 

 overall level of fishing mortality (often termed fully re- 

 cruited F) and the relative fishing mortality between 

 age classes (often referred to as the partial recruit- 

 ment vector or selection pattern), and that F^ = (fully 

 recruited F) x (selection on age ;). By convention, I 

 will use the fully recruited F as an index of the over- 

 all level of fishing mortality, but stress that specifi- 

 cation of the partial recruitment function is also nec- 

 essary to determine the impact of harvesting on a 

 population. Also note that there is an infinite set of 

 fishing mortality patterns for which the condition that 

 A =1 is satisfied. For a given partial recruitment fiinc- 



