THEORETICAL AND MATHEMATICAL MODELS 219 



The individual parameters tend to become fused into a single quantity 

 (A = «p), a feature which was noticeable in the model considered by Holt 

 earUer. This problem becomes more serious when the mathematical form 

 of the model is not exact, for the parameter values may then suffer con- 

 siderable distortion in order that the closest fit between theory and observa- 

 tion should be achieved. In general I would stress the importance of attempt- 

 ing to assess population parameters directly wherever possible and to seek 

 confirmation by independent methods of attack. The model Gulland has 

 oudined exhibits oscillatory features (of the kind envisaged by P. P. Moran, 

 1950, Biometrics, 6, 250-8). It must be most gratifying to him to find not 

 merely that this oscillatory property apphes also to the fish population 

 studied, but particularly that it comes into operation near to the level of 

 fishing intensity estimated theoretically. 



J. A. Gulland: I do agree that it is essential to look at mathematical 

 models critically, and not just to assume that one is right because it fits. 



L. B. Slobodkin: Watt's procedure seems to be to set up a programme 

 to learn all the parameters — then to assemble them in a formula, and then 

 to make predictions from it. But if all the parameters are known the formula 

 becomes unnecessary. There must be a distinction between an empirical 

 system giving information about the world and a model in which the output 

 in terms of conclusions is only as good as the input data. 



I. A. McLaren: Models have also a considerable psychological value as 

 an aid to clear thinking. 



N. Waloff: In Gulland's equation there are terms F and M — how is 

 the mortality obtained? 



J. A. Gulland: F is an instantaneous rate. Thus F = 0-2 means that 

 fishing mortahty is about 20 per cent. 



N. Waloff: But the natural mortahty is therefore obtained only by 

 subtraction from the equation term for total mortality, and is not independ- 

 ently assessed? 



J. A. Gulland : It is very difficult to measure natural mortahty except as 

 a residual from the known total aimual figure. Tagging returns, for example 

 in North Sea herring, give some idea as to how mortality is allocated — in 

 fact the total mortality is 75 per cent and fishing 50 per cent. 



S. J. Holt: I can think of only one case where natural mortality has 

 been estimated directly, by counting shells of Pecten. And even here one 

 cannot determine the causes, but only the total time/frequency distribution 

 of dead shells. 



J. A. Gulland : In scallops one can only estimate types of mortahty 

 which cause empty shells to remain. If a seal removes whole animals, shells 

 and all, this would not be allowed for. 



