FISHERY BULLETIN: VOL. 71, NO. 4 



hatching is assumed, so f^ represents the 

 mean hatching time. 



Individual fecundity (eggs per female) is 

 usually correlated with the size of the individ- 

 ual. Many plots of fecundity on length appear 

 to be concave upward, which indicates that 

 fecundity is closely proportional to the weight 

 of the individual. In order to be more useful 

 generally, however, fecundity in GXPOPS is 

 represented by a vector of mean viable eggs per 

 female at hatching by age class, X/, such that 

 the total number of larvae hatched, L^, is 

 given by 



Lh-Pt', .^^i^fi^ith- (25) 



where jd-^ and /J2 are the density-independent 

 and density-dependent larval mortality coef- 

 ficients. Equation (26) integrates to 



Lj = [M2/i"i (e^l - 1) + e^l ILj _ i]-l (27) 



for the survival of larvae from month j - 1 to j. 

 Collecting the constants we have 



Lj= lliai + a2/Lj_ i). 



or 



t,-l 



tr 



Lh + t,= ^li(^l 5 ^^ + ^2/^^) (28) 



Recruitment 



Any equilibrium point achieved with the 

 population model as now formulated is likely 

 to be unstable, such that with a sustained in- 

 crease or decrease in mortality the population 

 will decrease to extinction or increase to in- 

 finite size. Most successful natural populations 

 are believed to achieve equilibrium through any 

 of a number of homeostatic mechanisms asso- 

 ciated with density-dependent reproduction or 

 mortality. One such mechanism already men- 

 tioned is a decrease in copulation rate at high 

 densities. Others include a lack or destruction 

 of oviposition sites at high densities as in insects 

 or salmonid fishes, an overutilization or com- 

 petition for a fixed food supply, cannibalism, 

 predator-prey interactions, etc. The usual mode 

 of population regulation assumed in fisheries 

 studies is through density-dependent early 

 stage (or larval) mortality. Treatises on this 

 subject can be found in Ricker (1954, 1958) 

 and Beverton and Holt (1957). 



Two models have been widely used in fishery 

 population dynamics; GXPOPS allows the 

 selection of one or the other. The first model, 

 owing to Beverton and Holt (1957), states that 

 the simplest assumption one can make is that 

 the larval mortality coefficient can be expressed 

 as a simple linear function of larval population 

 size 



for tf. months of larval existence. Equation 

 (28) is a concave downward function that in- 

 creases monotonically with L: for a constant 

 period of larval existence and it approaches an 



asymptote of 



V 



r=0 



a 



^ja-, at an increasing 



rate as 0C2 increases. 



Thesecond model, owingtoRicker(1954, 1958), 

 simply assumes that the density-dependent co- 

 efficient, IJL2, may operate only until some criti- 

 cal size is reached and that the time to attain 

 this critical size may be proportional to the 

 size of the larval population at hatching, L^, 

 which gives 



dLfldt = -ini + H2Lh)Li 



or on integrating 



^h + tf- ^h^ 



-(Ml^r + A^2^/j) 



(29) 



dLildt = -{ill + 1x2 Lf) Lf 



(26) 



for tf. months of larval existence. Equation 

 (29) is also concave downwards, but monotoni- 

 cally increases to a maximum at L^ = l/iU2, 

 and then monotonically decreases approaching 

 zero as L^ becomes infinite. 



Both equations (28) and (29) allow a popula- 

 tion to achieve stability over a range of sustained 

 mortality rates. Equation (29), additionally, 

 will produce oscillations in the population 

 (Beverton and Holt, 1957), the criteria for 

 which are given by Paulik and Greenough (1966). 



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