FLETCHER: TIME-DEPENDENT SOLUTIONS AND EFFICIENT PARAMETERS 



productivity in the Graham system, and they rep- 

 resent the whole extent of available control over 

 the graph of Equation (1). Coordinate p of the 

 critical point has the fixed value B^/2, and the 

 graph of Equation (1) has a fixed curvature of 

 second degi'ee. 



Wherefore, productivity Equation (1), cast di- 

 rectly in terms of analytical parameters m and 

 B^, takes on the form 



p=4m[|-]-4m[|-]^ ,la, 



and intrinsic rate k, as it turns out, bears a propor- 

 tionality dependence on maximum productivity 

 and maximum biomass in the relationship 



k = 



i>oo L ^maxj 



And with the substitution of Equation (la) into 

 Equation (5), the formula for the net productivity 

 of a Graham stock becomes 



B = 4m 



["LJ-^^-ll:] 



FB. 



(6) 



In the integrated, equilibrium versions of the 

 Graham system, maximum latent productivity m 

 becomes maximum sustainable yield (MSY), 

 hence parameter m may be directly interpreted as 

 MSY in any optimization procedure on Equation 

 (6). 



If we restrict the time-dependence of F to abrupt 

 changes so that any solution of Equation (6) cor- 

 responds on its interval of validity, however brief, 

 to some constant value of F, then the time- 

 dependence of B in Equation (6) becomes 



productivity (Equation (6)) and the biomass solu- 

 tion (Equation (7)) for cases where 



F < 



4m 



Boo 



As indicated in the figure, root fi* becomes the 

 adjustment level to which biomass trajectory B(t) 

 will trend when F is less than critical quantity 

 AmlBy^ (and obviously, Bit) trends to By- in Equa- 

 tion ( 7 ) when F is zero) . The system is governed by 

 the positive branch of Equation (6) when Y <P (in 

 which case, 5 > 0), and by the negative branch of 



• • • 



Equation (6) when Y > P (in which case, B < 0). 

 But this partitioning of F into subranges for nega- 

 tive or positive B is a density-dependent process. 

 Although we must have F < 4m /B^ foJ* positive 

 B*, the values of F on that range that drive the 

 stock either up or down will depend on initial stock 

 size B,,. To insure, for arbitrary B^, that F < P in 

 Equation (6), mortality F must have a value such 

 that 



< F < 



4m 



Bno 



[ - tl 



in which case B(t) increases from initial value B^ 

 towards a higher adjustment level B.;,. But for any 

 value of F on the interval 



4m 



Boo 



Br 



Be 



< F < 



4m 



Boo 



then Y > P and B(t) decreases from B„ towards a 

 lower adjustment level B.: . 



Figure 3 illustrates the relationship between 

 net productivity (Equation (6)) and the biomass 

 solution (Equation (7)) when 



Bit) = 



B. 



l + Cne-<*'"/^^-^^>' 



B. 



r FBool 



(7) 



Be 



and with initial time tQ set arbitrarily to zero, the 

 integration constant in Equation (7) becomes 



Co = 



B, -B( 



b7~ 



Figure 2 illustrates the relationship between net 



F > 



4m 



Boo 



in which case the adjustment level of biomass cor- 

 responds to the zero root of Equation (6). As indi- 

 cated by the figure, any mortality F so great as to 

 equal or exceed the quantity AmlBy., if main- 

 tained, will fish a Graham stock to extinction. 



Since Equation (6) governs the relationship be- 

 tween transient biomass and nonequilibrium re- 

 moval, we look to its solution (Equation (7)) for 

 time delays between equilibria. But the asympto- 

 tic behavior of Equation (7) is a minor analytical 

 annoyance to be circumvented here. Let us 



379 



