FISHERY BULLETIN: VOL. 76. NO. 3 



0<n < -i 





S^ 



T 



b^>0 



Y — Fb^ 

 A 



6, 



5- 



n-i\ 



r>p 



MSV 



n >-/' 



E>oo 





Y/77 



' MSV 



Figure l. — Time-dependent response of the Pella-Tomlinson system to parametric variations of exponent n and mortality F. The 

 upper diagram summarizes system response when n falls on the range < /! < 1 . The adjustment level of biomass is never zero for this 

 range of n however great the value ofF, and mortality F ms-, has no absolute constraints; such a stock cannot be fished to extinction. 

 The lower diagram summarizes system behavior when n falls on the range n > I. Mortality F ^sv 's then constrained to the interval 

 indicated by the diagram. WhenF exceeds the critical value ym/By^, then the stock, over sufficient time, trends to extinction. 



f 



(lY = 



ym 



and for any such equilibrium interval t, the inte- 

 grated yield rate iY^h) takes on the parametric 

 formulation 



= 7m 



(11) 



with maximum latent productivity m of the time- 

 dependent system becoming the maximum sus- 

 tainable yield rate (the MSY) of the equilibrium 

 system. With B^ as the parametric variable in 

 Equation (11), a zero left endpoint exists for Y^.Ij 

 when /? > 1 and F = ymlB^. Should F exceed the 

 critical value ymlBy- when exponent n >1, no 

 equilibrium state exists; such conditions in the 



518 



time-dependent system correspond to extinction 

 trends. But when n has any value on the range 

 0<n<l, no left endpoint of Equation (11) exists; a 

 positive equilibrium level of biomass and a non- 

 zero yield rate may be defined for any value of F, 

 however great. 



The equilibrium biomass levelp where MSY (or 

 m) occurs can be regulated in Equation (11) by 

 relationship (4). And once designated in (4), the 

 corresponding value of n determines the value of 

 coefficient y, as given by Equation (3). Either of 

 the parametric sets [m, p, B-j. } or {m, n, B-^ } 

 (augmented by the auxiliary parameters^ andB* 

 orF andfi^) will constitute a complete, indepen- 

 dent set of controls for equilibrium yield in the 

 Pella-Tomlinson system. Collectively, the 

 parameters control the behavior of equilibrium 

 model Equation (11) but the influence of any one 

 parameter remains independent of the remaining 



