Prager: A nonequilibrium surplus-production model 



375 



Model formulation and fitting 



Basic differential equations 



Surplus-production models characterize a population 

 as an undifferentiated biomass. The number of indi- 

 viduals present or harvested plays no part in these 

 models, nor is age or size structure considered. A 

 quantity termed "surplus production" is used to char- 

 acterize population dynamics at different levels of 

 population size (measured in biomass). Surplus pro- 

 duction is the algebraic sum of three major forces: 

 recruitment, growth, and natural mortality. The ad- 

 jective "surplus" refers to the surplus of recruitment 

 and growth over natural mortality; i.e. the net pro- 

 duction. In this article, surplus production will often 

 be termed simply "production," and the models 

 termed "production models." 



In the simplest production model, the logistic or 

 Graham-Schaefer (Graham, 1935; Schaefer, 1954, 

 1957) model, a first-order differential equation de- 

 scribes the rate of change of stock biomass B t due to 

 production. In the absence of fishing, the population's 

 rate of increase or decrease is assumed to be a func- 

 tion of the current population size only: 



^L = rB t -^Bf, 

 dt * K *' 



(1) 



where B t is the population biomass at time t and r 

 and K are parameters. The right side of Equation 1 

 is simply the start of the Taylor expansion of an ar- 

 bitrary function 0(B) passing through the origin 

 (Lotka, 1924). 



Equation 1 is written in the parameterization of 

 population ecology, in which K represents the maxi- 

 mum population size, or carrying capacity, and r rep- 

 resents the stock's intrinsic rate of increase (in pro- 

 portion per unit time). In this paper, both are as- 

 sumed constant. Other parameterizations could be 

 used, and indeed a slightly different parameteriza- 

 tion is used for simplicity in the next section. 



Adding fishing mortality F t to the model, it be- 

 comes 



f-^-l* 



(2) 



This model, like many fisheries models, is much sim- 

 pler than the real world. In particular, it excludes 

 such factors as environmental variation, interspe- 

 cific effects, or the possible presence of more than 

 one stable regime. 



Time trajectories of biomass and yield 



Integration of Equation 2 with respect to time al- 

 lows modeling the biomass and yield through time. 



Before integration, simplify notation by defining 

 a t =r-F t and fi=r/K to express Equation 2 more sim- 

 ply as 



dB t 

 dt 



« t B t -ffi 



(3) 



Equation 3 can be conveniently solved for biomass 

 under the assumption that F is constant and that 

 therefore a ( is constant. This is a weak assumption, 

 for if F t varies, time can be divided into short peri- 

 ods of constant or nearly constant F and a solution 

 found for each period. Fitting would then require 

 knowing the catch and effort for each short period. 



For the period beginning at t = h and ending at 

 time t = h + 8, during which the instantaneous fish- 

 ing mortality rate is F h , the solution to Equation 3 is 



B 



h+5 



a h B h e 



a h S 



a h +pB h (e a ^ s -l) 



when a h * 0, or (4 a ) 



B 



B h 



h+5 



l+p8B h 



when a h = 0. 



(4b) 



Equation 4a is the familiar logistic equation. How- 

 ever, if a h = (i.e. if F h = r), Equation 4a is undefined 

 and Equation 4b is used in its place. 



Modeling the yield during the same period involves 

 another integration with respect to time: 



J'h+S 

 t=h 



F h B t dt, 



(5) 



where B r the biomass at instant t, is defined by Equa- 

 tions 4a and 4b; F h is the (constant) instantaneous 

 rate of fishing mortality during the time period; and 

 Y h is the yield taken during the period. Performing 

 the integration in Equation 5, 



V*=-m 



PB h (l- 



„a h S , 



■ln(l+5)SB A ) 



when a h *0, or (6a) 



when a h = 0. (6b) 



Equation 6a was apparently first given by Pella 

 (1967) (and a similar form developed by Schnute 

 [1977 J); Equations 4b and 6b seem not to have been 

 presented in fishery biology before now. 



It follows from the definition of F that the esti- 

 mated average biomass during the period is simply 

 Y h I F h . The surplus production P h during the time 

 period can be determined by mass balance: 



B 



h+S 



B h +Y h 



(7) 



When yield is equal to surplus production, the popu- 

 lation is in equilibrium. 



