Thompson, Bayesian approach to management advice from a simple dynamic pool model 



565 



1 m 



and 



P = (— - ll(l-m) 



(18) and (19) 



For a given set of K", m, and v' values, Figure 2 shows the risk (depicted by the area under a particular curve) 

 associated with three possible F' values. 



Fishing mortality at maximum expected log-sustainable yield 



Substituting Equations (18) and (19) into Equation (13) and solving for F' gives the value that minimizes risk. 

 Because of the form used for the loss function, this process is equivalent to finding the level of F' that maximizes 

 the expected value of the logarithm of sustainable yield. It is thus convenient to refer to this value as F'melsy 

 (for "maximum expected log sustainable yield"), which for this particular model can be written 



[(m + 2) K"-2] v' - (m+1) K" + 1 + Vks v'^ - k, v' + ko 



F'melsy = r 7, r - 1- 



2 m(l-v) 



(20) 



where k. = (m + 2)2 K"~ + (12m -8) K" + 4, 



ki = (2m2 + 6m + 4) K"2 + (18m -8) K" + 4, and 

 ko = (m+l)2 K"- + (6m-2) K" + 1. 



Figure 3 illustrates how F'melsy varies with K", m, and v'. A few special cases are of particular interest. For 

 example, when q is known with certainty, i.e., m = q and v' = 0. Equation (20) reduces to Equation (2). Equation 

 (2) is thus the "certainty equivalent" solution (Ludwig and Walters 1982). The ratio between Fmelsy ^nd Fmsy 

 is illustrated in Figure 4. Differences in K" tend to have less influence on this ratio than differences in either 

 m or v'. 



Other important special cases of Equation (20) include the limits as K" approaches zero and infinity, which are 

 shown respectively below: 



lim F'melsy 



K"-0 



l-m(l-v') - 2v' 

 m(l-v') 



l-m(l-v') - 2v' 



and hm F melsy = :; " "• 



K'^oo l + m(l-v ) - 2v 



(21) and (22) 



0.0 0.2 0.4 0.6 0.8 10 



Expected value of q (m) 



Figure 3 



Values of F'j,j.lsy resulting from different combinations of 

 parameter levels. F'melsy tends to decrease as K", m, or v' 

 increases. 



