APPLICATION OF MATHEMATICAL MODELS TO FISH POPULATIONS 211 



able departures between reality and the mathematical model using constant 

 parameters. However, changes in the other parameters, or at least growth, 

 are rather easier to handle both in theory and application. Changes in growth 

 are probably most readily dealt with as growth is easily measurable for most 

 fish, especially when the age of individuals can be determined; further, by 

 using increments put on by fish of different ages in the same year, rather 

 than the complete growth pattern of a single fish during its whole life-span, 

 a growth curve appertaining to any single year can be readily built up. Given 

 a series of pairs of observations of stock abundance and growth an empirical 

 relation between them is not difficult to determine, except for any difficulties 

 with concurrent changes in environmental conditions. Some theoretical 

 forms which the relation between growth and density should take have been 

 put forward. For instance Beverton & Holt (1957) have suggested that in the 

 von Bertalanffy form of the growth equation the changes should be confined 

 to the parameter Loo, the limiting length offish, while the parameter K, the 

 rate at which this limit is approached, remains constant. They analysed 

 Raitt's (1939) data on the North Sea haddock, and showed a definite density 

 dependent effect. However, while the increase in length during the year 

 shows a significant negative correlation with density, the data do not show 

 whether this is due to changes in L^ or K, or both (see Fig. 2). Plotted in 

 Fig. 2 against the number density in the year concerned, for the haddock in 

 Raitt's western area, are the increments in length during the second, third 

 and fourth years of hfe, and the values of K and L^ obtained by fitting the 

 Bertalanffy curve to these increments. 



The handling of the mathematical expressions is probably easier if it is 

 assumed that changes occur only inL^o, and usually the observed relationship 

 can be adequately represented by a linear relation, i.e. we can write Loo 

 = L- aP. 



The effect can then be rcctdily incorporated into the earlier constant 

 parameter model; an approximate value of Loo may be used to compute the 

 population under any desired conditions, hence giving a value of P and 

 from the equation above a better value of Loo- This can be used instead of 

 the original value of Loo to give a further value of P, and hence by such 

 iterative steps the best estimate of Loo and P. In fact the relation found above 

 for haddock related Loo to number density, which is independent of Loo, 

 except possibly where it may affect the age at which the fish becomes 

 vulnerable to the gear, so that no iteration would be necessary. 



Density dependent natural mortality is less easily studied, for it is very 



difficult to get any rehable estimate of natural mortality, and those that are 



obtained usually refer to the average over a period. Thus changes in the 



mortality cannot easily be related to changes in density either empirically, or 



15 



