He et al A prior for steepness in stock-assessment relationships, based on an evolutionary persistence principle 



429 



would require a minimum recruitment compensation 

 to allow the species to rebound from low abundances. 

 We argue that distribution of the h parameter for any 

 species could be determined from its life history and 

 recruitment variability. Using the model, we derived 

 the prior distributions of h for fish species that have a 

 range of natural mortality, recruitment variabilities, 

 and low critical abundance (A'^,) values. 



Methods 



In calculating steepness priors, we used a simple popu- 

 lation model with a Beverton-Holt stock recruitment 

 relationship: 



N„^ = N„ 



.e-^+- 



A^„ 



-4) 



a + PN,,_ 



(1) 



where A^,,, = population size at year /; 

 M = natural mortality; 

 a and p = recruitment parameters; and 



Rill = the logarithm recruitment residual at year 

 t that follows a normal distribution of N[0, 

 a-), where a is recruitment variability (Hil- 

 born and Walters, 1992). 



This model produces recruitments that are log-normally 

 distributed, and a correction factor of " is applied to 

 R,,,. The correction factor is included because it is com- 

 monly used in stock assessments. The Beverton-Holt 

 stock recruitment relationship can be reparametrized 

 as (Mace and Doonan^) 



where T = a time far into the future; 



n = a specific value of population size; and 

 N^. = a critically low level of abundance, below 



which the population would have very high 



probability of extinction. 



At ^ = T,p{n.T\h,T\ = 1 if n>N^. and p(n,T\h,T) = oth- 

 erwise. In addition, p(n,t\h,T) satisfies the boundary 

 condition that p(N^J\h.T) = for all t. 



For times previous to T, p(n,t\h,T) satisfies the sto- 

 chastic iteration equation (Mangel and Clark, 1988; 

 Clark and Mangel, 2000) 



'-,[i 



p(n,t\h,T) = 



«e^+- 



a(h) + P(h)n 



e''',t + l\h,T 



(7) 



where £^ denotes the expectation over the stochastic 

 processes associated with recruitment. We have indi- 

 cated that the Beverton-Holt parameters depend upon 

 steepness (as Eqs. 2-5 show, they also depend upon 

 mortality M, which we hold to be a fixed value). 



Because the recruitment uncertainty is normally dis- 

 tributed and Equation 7 cannot be evaluated analyti- 

 cally, we used a discrete distribution for i?, (=7?|,|-^) 

 into K ( = 61) uniformly spaced values (r^) between -3a 

 and 3 a, so that 



Pr{R. 



expl — ^ 



^exp 



.2 \ 



(8) 



2(j' 



and 



N^ 1-^ 

 Ro 4h 



5h-l 



AhRi, 



where N^ = virgin abundance at equilibrium; 

 Rq = virgin recruitment; and 

 h = recruitment steepness. 



At equilibrium. 



(2) 



(3) 



and 



N, = N,e-^+R, 



Ro = DNo, 



(4) 



(5) 



where D - death rate and is equal to 1 - e^'^. 



We now introduce the persistence criterion. For a 

 given h, the persistence criterion is defined as 



p{nJ\h,T) = PT{N,^,>N^. forall*<s<TIAf,, , = /)}, (6) 



The iteration equation then becomes 

 pin,t\h,T} = 



k=3a 



X Pr{R,=r,] 



p\ ne-" + 



a(h) + P(h)n 



e'\t + l\h,T\\. 



{9) 



We solve Equation 9 backwards in time, starting at 

 t = T-1 (Mangel and Clark, 1988; Clark and Mangel, 

 2000) until reaching t = 1. Assuming that the popula- 

 tion starts at the deterministic steady state (?2=A^g), 

 we can then calculate the prior probability for each /; 

 value using Equation 9. Because h is bounded between 

 0.2 and 1, we used an interval of 0.005 so that a total 

 of 161 solutions of Equation 9 are needed to produce the 

 prior probability for h. We then used a logistic equation 

 to fit 161 prior probability values to produce a derived 

 h prior curve {<p{h)\: 



^h) = 



0, 



<^i~^2) ^-e.Ji, 



(10) 



1 + 



e. 



