NOTE Restrepo: Estimating fishing mortality rates for Menippe mercenana 



413 



where 'iNq is the population size at the beginning of 

 the time-period (t = 0). This solution can be substituted 

 into Equation (lb) to solve it since, without claw re- 

 generation, ''Nt is independent of "Nt (i.e., by assump- 

 tion, there is no transfer from the nonharvestable into 

 the harvestable population): 



d"Nt 

 dt 



"M"Nt-HFShNo e-<F *'")'. 



(3) 



This is a first-order linear differential equation that 

 can be solved with the integrating factor 



g/"Mdt _ g"Mt 



Multiplying (3) throughout by this factor gives 



dt 



= FS^Nn e-<F+'''^-"'^". 



Integrating and letting "Nt = "No at t = gives the 

 solution to Equation (3): 



"Nt 



(4) 



FSi^No (i_e-(f'+''M-"M)t) e-"Mt 

 F-t-hM-"M 



-t- "No e -"■«'. 



The djrnamics explained by Equations (2) and (4) de- 

 pend on six parameters (^Nq, "Nq, ''M, "M, S, and F). 

 The main usefulness of these two equations lies in 

 simulation modeling (e.g., for yield-per-recruit anal- 

 yses) in which parameters are given as inputs rather 

 than estimated from fitting the equations to data. 

 However, as shown below, the number of parameters 

 can be reduced to three by taking the ratio R = "Nt/ 

 ^'Nt . Note that the ratio of the two population types 

 is largely independent of the level of sampling inten- 

 sity, provided that the availabilities of harvestable 

 and unharvestable crabs to the sampling gear do not 

 change. Obtaining estimates of either "Nt or ''Nf alone 

 would be a more difficult task which could involve tag- 

 ging or detailed survey statistics (see Seber 1982 for 

 a discussion on the estimation of abundance). Dividing 

 Equation (4) by Equation (2) gives 



Figure 1 



Expected trends in the ratio R (number of nonharvestable: 

 harvestable stone crabs Menippe mercenaHa) for three values 

 of b in Eq. (5). Lower curve: b is negative and the trend ex- 

 hibits convex curvature. Middle curve: b approaches zero and 

 the trend is a straight line. Upper curve: b is positive and the 

 trend is concave-upwards. 



b = F-i-'^M-"M, and 

 FS 



c = 



F-i-hM-"M 



Rt 



a e 



bt 



- c, 



(5) 



Equation (5) shows that in a closed population and 

 in the absence of claw regeneration, the ratio of non- 

 harvestable to harvestable crabs should change expo- 

 nentially with a concave, convex, or straight trend 

 depending on the value of b. Consider a hypothetical 

 population in which hNo = 1000, "No = 0.0, S = 0.5, F = 

 0.2 per month, and 'iM = 0.2 per month. When "M = 

 0.6, b= -0.2 , and R increases with convex curvature 

 (Fig. 1, filled squares). When "M approaches 0.4, b 

 approaches zero, and R increases linearly (Fig. 1, 

 crosses). When "M = 0.2 ("M = 'iM), b is positive and the 

 trend in R is a concave-upwards curve (Fig. 1, aster- 

 isks). In practice, some of the model's assumptions may 

 not be always met. For instance, if "M = 0.2 per month 

 and 250 crabs recruit to the harvestable stock at the 

 beginning of months 5, 6, and 7, then the trend in R 

 decreases starting in month 5, while fishing is still 

 ongoing (Fig. 2). 



where R^ = 



''Nt' 



a = 



FS 



"No 



F-i-hM-"M hN ' 



Application of tfie model to a data set 



No studies have been carried out in which the data 

 necessary for the model have been collected. For this 

 reason the estimates presented below are meant to 



