APPLICATION OF MATHEMATICAL MODELS TO FISH POPULATIONS 209 



for example, the catch in weight during a small time interval at time t is 

 equal to 



^=FX Nt X Wt 

 i.e. dt 



dY = FNoe-(^+^^) (^-^«) e-^^ dt 



and the total yield can be obtained by summation (integration) over the 

 whole time during which the year-class is fished. If necessary, because one 

 or other of the parameters is variable, a more precise formulation can be 

 obtained by sphtting this whole period into smaller periods, each with its 

 appropriate parameters. Similarly modifications may be introduced for the 

 analysis of seasonal fisheries, for instance Ricker (1958) has dealt with several 

 forms which the calculations may take, depending on how the fishery season 

 overlaps with the seasons of growth, recruitment and (presumed) natural 

 mortality. The concepts used are the same as already discussed and the 

 mathematics only slightly more complex. 



DENSITY DEPENDENT GROWTH AND MORTALITY 



In the simpler model of this type it is assumed that the only effects of fishing, 

 both direct and indirect, are on the fishing mortality F, as determined by the 

 fishing effort/, and the range of ages during which fish are liable to be caught, 

 as determined by the selectivity of the gear (e.g. mesh size of the trawl). 

 Exphcitly, the growth, natural mortality and recruitment are assumed to be 

 constant, even though changes in fishing may result in large changes in the 

 size of the population. This implies that the mechanism whereby the popula- 

 tion is kept within reasonable limits, so that it neither expands without limit 

 nor becomes extinct, lies outside that part of the population so far considered. 

 More exactly, the controlling element, when the numbers of each year-class 

 are determined, occurs in the early stages. 



In fact the size of stock must have some effect on the value of the para- 

 meters, most probably reducing the growth (e.g. due to pressure on the food 

 supply), with increasing population and increasing the natural mortahty (due 

 to disease, though some effects of.predation could, at least immediately, 

 reduce the mortality). These effects therefore act against the changes in the 

 population, damping out oscillations, including those due to changes in 

 fishing. Increased parent stock may however increase the number of subse- 

 quent recruits; this will exaggerate changes, perhaps very greatly; increased 

 recruits giving a further increase in present stock, and further increase in 

 recruitment, etc. Thus density dependent changes in recruitment are likely 

 to be the most important, in the sense of giving the largest and least predict- 



