4.4 Fitting the VCCW Model
5.4.1 What is the effect of protandry on the relationship between
between gametic output, SSB and yield?
Beverton and Holt’s (1957) assertion that SSB could be used as a proxy for larval production was based on the assumption that SSB is directly proportional to egg production. It is immediately apparent from figure 5.11 that for the population studied here the assumption of direct proportionality does not hold. The failure of this relationship is due to a breakdown in the assumptions of constant sex ratio and constant relative fecundity. If the breakdown in proportionality between E and SSB was solely due to a failure in the assumption of constant sex ratio then the sex-specific SSB would be expected to be directly proportional to population egg production. As can be seen from figure 5.12 there was a non-linear relationship between female sex-specific SSB and population egg production. Therefore the failure for SSB to accurately predict E is not solely due to a failure in the assumption of a constant sex ratio, and must be at least partially due to non-linear relationship between fecundity and weight.
Does protandry have any effect on the relationship between E, SSB and yield? Or is the breakdown in direct proportionality simply due to variable constant relative fecundity? If protandry had no effect, then female SSB would be expected to be directly proportional to the population SSB. The plot of proportional reduction in SSB against proportional reduction reduction in sex-specific SSB and egg production (Figure 5.11) indicates that this was not the case. Therefore the disparity between the reduction in egg production against reduction in SSB was due to a failure in both the assumption of constant sex ratio and a failure in the assumption of constant rel- ative fecundity. The nature of this disparity can be further examined. The distance between the 1:1 relationship and egg production (labelled x in Figure 5.11 b) indi- cates the full disparity between SSB and egg production. This can be separated into the disparity due to failure in the assumption of constant relative fecundity (labelled z in Figure 5.11 b) and the failure of the assumption of constant sex ratio (labelled y in Figure 5.11 b). The discrepancy between the 1:1 line and the female SSB line
(y in Figure 5.11 b) clearly illustrates the effects of protandry.
Returning to the initial selection of the length-fecundity relationship: had the ex- ponential relationship been rejected and the linear weight-fecundity relationship de- scribed by a cubic power law length-fecundity relationship been accepted, the as- sumption of constant relative fecundity would still hold. In this case there would be no discrepancy between egg production and sex-specific SSB. Therefore the propor- tional drop in egg production would be wholly described by the proportional drop in sex-specific SSB, and thus the female sex-specific SSB line in figure 5.11 would describe the response of egg production to a decline in population SSB.
Protandry causes a decoupling of sex-specific SSB and whole population SSB; this is because females are preferentially concentrated in the larger size classes and males in the smaller size classes. Thus when a fishery selectively exploits larger size classes females will be preferentially exploited and female SSB will decline faster than whole population SSB. It can be seen that the effect of protandry on the relationship be- tween SSB and sex-specific SSB varies between the different populations at different exposure levels. The main point to be drawn from this is that the effect of protandry on the response of a stock to exploitation will vary depending on the exact local sex- specific population structure. Thus the effects of protandry cannot be corrected for with a single simple conversion factor.
The failure of the assumptions of constant sex ratio and constant relative fecundity breaks the direct proportionality between E and SSB. Therefore some parts of the population will have a different E/SSB ratio than other parts of the population. Once this variation has been established the fishery manager could try and deter- mine which parts of the population have the lowest E/SSB ratio as exploiting this part of the population will generate the maximum yield for the minimum drop in egg production. Unfortunately, and in complete contrast to the aims of a fishery manager, with protandry and a size selective fishery the fishery selectively exploits the section of the population with the highest E/SSB ratio first. Thus the reduction in egg production for a given yield is maximised.
Figure 5.11: Proportional reduction in E and sex-specific SSB against proportional reduction in whole population SSB as a result of fishery pressure. For: a)transect A, sheltered; b) transect B, intermediate; and c) transect C, exposed. The arrows in part b) indicate; x- the full discrepancy between E and SSB, y- the discrepancy due to failure of the assumption of constant sex ration, and z- the discrepancy due to failure of the assumption of constant relative fecundity. The dotted line indicates the theoretical 1:1 relationship.
Figure 5.12: Decline in sex-specific potential fecundity plotted against decline in sex-specific SSB due to fishery pressure for: a)transect A, sheltered; b) transect B, intermediate; and c) transect C, exposed. The dotted line indicates the expected 1:1 relationship.