FISHERY BULLETIN: VOL 76. NO. 3 



B = J in 



B 



B 



7 m 



m-'' 



(1) 



with the quantity y wholly a function of n in the 

 relationship 



7 



n-1 



(2) 



B'ti, the solution ofB, represents the stock size at 

 time ^ while Y(t), the solution of Y , represents the 

 cumulative catch of the stock. Parameter 5 ^^ is the 

 maximum stock size of the unexploited popula- 

 tion, while m is the maximum productivity in the 

 productivity function or the maximum sustain- 

 able yield (MSY) in the complete exploitation 

 model. Exponent n controls the location of the 

 inflexion point in the latent productivity function 

 of the stock. Therefore, parameters fi^- "'^ arid n 

 are nonnegative. With this new formulation of the 

 system equations, the sign reversals of the 

 coefficients at the turning point /? = 1 are now 

 automatic. Also the parameters are expressed in 

 more meaningful terms for the fishery scientist, 

 and some aspects of parameter estimation are 

 simplified thereby. 



By presuming that f(t) units of effort operate on 

 the population over time increment dt, the yield 

 rate is often put into the instantaneous form 



Y = qf{t)B{t) 



(3) 



where q is the catchability coefficient. Equations 

 ( 1 ) and ( 3 » constitute a coupled system of nonlinear 

 differential equations. The system, as it is formu- 

 lated in Equations (1) and (3), represents the 

 continuous-time model. In practice, though, for 

 each finite time interval rover which yield statis- 

 tics are integrated, fishing effort is usually as- 

 sumed to be constant. It follows that fit) must be a 

 step function that describes the effort as being 

 constant over each time interval t with abrupt 

 changes at the end of each period. Then the effort 

 required, over one time interval, to maintain 

 maximum productivity is given by 



MSY 



7 m 



n 



(4) 



and at MSY the yield per unit of effort ^/msy is 

 obtained by dividing m by /"msy- 



DETERMINISTIC MODEL AND 

 STOCHASTIC REPLICATES 



The Pella-Tomlinson system has the five 

 parameters /?; , firo, n,q, andBo- The fifth parame- 

 ter, initial population size Bq, is needed to specify 

 a particular solution of Equation ( 1 ). By taking the 

 following arbitrary values for the parameters, 



we constructed an example of a fishery over the 

 course of 20 yr with/?/) increasing within the first 

 10 yr and stabilizing thereafter (Table 1). 



We also constructed 20 stochastic replicates of 

 the deterministic catch history. In all the stochas- 

 tic versions we assume additive proportional error 

 terms e, (with / the annual index), consisting of 20 

 sets of 20 values each of normally distributed, 

 independent random variables with expected 

 means of zero and standard deviations ( a) of 0.025, 

 0.050, 0.075, 0.100, 0.125, 0.150, 0.175. 0.200 (12 

 replicates), and 0.250. Although Fox (1975) takes 

 a similar approach, we recognize the fact that se- 

 rial correlation of errors is likely to exist in 

 natural data. As put by J. J. Pella in a personal 

 communication, "If yield is above average in one 

 year because the population is above average, it 

 will probably be above average in the following 

 year." But at this stage of the analysis, the explicit 

 consideration of serially correlated errors would 

 only complicate the estimation problem unneces- 



TabLE 1. — Simulation of a logistic stock under exploitation (de- 

 terministic model). 



524 



