KLETCHKR RESTRl'CTrKlNG OF PKLLA-TOMLINSON SYSTEM 



B 



= ,2^/1 



(4) 



When /; takes any value between zero and unity, 

 coordinate/; falls on the range between zero and 

 B^ie\ when n takes any value greater than unity, 

 coordinate p falls on the range between B-_rje and 

 Bx- Wherefore, with \m,p,By:] as the parametric 

 set for Equation (2), p and B x determine n ; with 

 [m . n . B yz] as the parametric set, n and fix deter- 

 minep. In the complete exploitation model, maxi- 

 mum productivity m becomes maximum sustain- 

 able yield (MSY) and biomass level p becomes the 

 equilibrium level (the ''B opt ") where MSY occurs. 

 For any stock-production system, we may enter 

 exploitation into the productivity formulation by 

 the direct difference P - Y, with Y signifying the 

 rate of biomass removal attributed to exploitation. 

 Therefore, in writing 



B = P-Y, 



(5) 



we interprets as being the resultant productivity 

 that nets to the stock for its growth. When removal 

 rate Y exceeds latent productivity P. then net pro- 

 ductivity B<0 and the stock declines; whenP ex- 

 ceeds Y. thenB >0 and the stock increases. Should 



• • • 



Y = P, then B = Q and biomass trajectory Bit) 

 exhibits an extremum, which is the necessary 

 condition for equilibrium fishing. Yield rate Y cus- 

 tomarily takes the form 



With initial time /„ set at zero, the integration 

 constant C in Equation (8) becomes 



^0^ 



- B. 



The quantity B ;, , when positive in Equation (8), 

 becomestheadjustment level such thatBf^^ -► B* 

 over time. When, for certain ranges of n and F, 

 quantity B*<0, then the zero root of Equation (7) 

 applies and B(t) -► 0. When mortality F takes 

 the value 



MSY 



■(^) 



■ym 



bZ 



(9) 



irrespective of the value of parameter n, then 

 B(t) -►p and Y ->m (which are the conditions, in 

 the equilibrium limit, for maximum sustainable 

 yield). In terms of the parameter set { m, p, B^^ }, 

 Equation (9) becomes, simply, 



F. 



MSY 



m 

 P 



Figure 1 gives a summary of the general con- 

 straints on the time-dependent system; for a more 

 detailed treatment of system behavior, see Flet- 

 cher (1978). 



THE RESTRUCTURED 

 EQUILIBRIUM SYSTEM 



Yit) = F{t) ' B{t) 



(6) 



with the assumption that all fish of the fishable 

 stock share equal probabilities of capture. By ad- 

 mitting Equations (2) and (6) into Equation (5), 

 the differential equation that governs net produc- 

 tivity in the restructured system becomes 



B = 7m 



B 



B 



ym 



©" 



FB (!) 



and over any time interval, however brief, that 

 mortality F might be presumed to have a fixed 

 value, biomass variable B in Equations (6) and (7) 

 has the general time-dependent solution 



B(0 



= (b,i-" + Cexp ((7m/B„ 

 * [ym-FB^) 



By Equations (2) and (5), the time-varying rate 

 of yield in the reformulated Pella-Tomlinson sys- 

 tem takes the form 



y = ,„ (y _ ,„ (y _ s. ,10, 



and when, for given F and n, governing Equation 

 (7) exhibits a positive root, then B(t) -► B^ and 

 B -► in Equation (10), and yield rate Y, over 

 sufficient time, approaches a constant value. In 

 the steady-state tor "equilibrium") limit, yield 

 then accumulates as 



F)(l 



-n)t)y"'-''\ 



(8) 



1 /(!-«) 



B. 



517 



