Thompson" Management advice from a simple dynamic pool model 



553 



models might support Rules I and II, it remains to be 

 seen whether these rules are inconsistent only for 

 isolated special cases, or are actually incompatible with 

 a major class of models. 



Review of simple dynamic pool models 



One place to start in the search for models that might 

 be compatible with Rules I and II is within the family 

 of simple dynamic pool models. As distinguished from 

 surplus production models such as those of Schaefer 

 (1954) and Pella and Tomlinson (1969), dynamic pool 

 models describe stock dynamics in terms of the indi- 

 vidual processes of recruitment, growth, and mortal- 

 ity, and incorporate age structure at least implicitly 

 (e.g., Pitcher and Hart 1982). Within the broad class 

 of dynamic pool models, a model will be referred to here 

 as "simple" if it reflects the following assumptions: (A) 

 Cohort dynamics are of continuous- time form, (B) vital 

 rates are constant with respect to time and age, (C) 

 fish mature and recruit to the fishery continuously and 

 at the same invariant ("knife-edge") age, (D) mean 

 body weight-at-age is determined by age alone, (E) the 

 stock (or population) consists of the pool of recruited 

 individuals, (F) maximum age is infinite, (G) the stock 

 is in an equilibrium state determined by the fishing 

 mortality rate, and (H) recruitment is determined by 

 stock biomass alone. Within the framework provided 

 by these assumptions, particular models are distin- 

 guished by the forms assigned to the weight-at-age and 

 stock-recruitment functions. 



Assumptions (A-C) imply that simple dynamic pool 

 models conform to the following pair of equations: 



dn(F, a) 

 da 



= -n(F, a)Z, 



and 



n(F, a) = n(F, a^) e-Z(a-a,), 



(1) 



(2) 



where a = age, n(F, a) is the stationary population 

 distribution (in numbers) by ages a when the stock is 

 exploited at a fishing mortality rate of F, Z is the in- 

 stantaneous rate of total mortality (F -i- M), and a^ is 

 the age of recruitment. 



Equation (1) gives the instantaneous rate of change, 

 by age, of the distribution n. When integrated with Z 

 constant (Assumption B), Equation (1) gives Equation 

 (2), numbers as a function of age. Assumption (D) im- 

 plies that Equation (2) can be cast in terms of biomass 

 by multiplying both sides of the equation by the weight- 

 at-age function w(a): 



b(F, a) = W(a) n(F, a^) e-^^'^-'^l 



(3) 



where b(F, a) is the stationary population distribution 

 (in biomass) by ages a when the stock is exploited at 

 a fishing mortality rate of F. 



Assumptions (B), (C), (E), and (F) imply that total 

 equilibrium stock numbers can be obtained by inte- 

 grating Equation (2) from a = ar to a = oo, giving 



N(F) 



n(F, a^) 



(4) 



where N(F) represents total equilibrium numbers when 

 the stock is exploited at a fishing mortality rate of F. 

 Likewise, equilibrium stock biomass is obtained by 

 integrating Equation (3) from a = a,. to a = °°: 



X 



B(F) = n(F, a,.) f w(a) e -Zf^'-ar) da. 



(5) 



In the case where a = ar, Assumptions (G) and (H) 

 imply that the left-hand side of Equation (3), recruit- 

 ment biomass, is a deterministic fimction of equilibrium 

 stock biomass r(B(F)): 



b(F. a,) = r(B(F)). 



(6) 



Average weight of individuals in the stock W(F) can 

 be written 



;: 



w(a) e-^'^^-^r) da 



W(F) 



= Z 





e-Z(a-a,) da 



w(a) e"^*^"'''' da. 



(7) 



Equation (5) can then be rewritten 



B(F) 



W(F) n(F, a,) 



(8) 



For the case of a pristine stock (F = 0), Equations (4) 

 and (8) imply that equilibrium stock size (in terms of 



