SECT. 4] FISHERY DYNAMICS THKIR ANALYSIS AND INTERPRETATION 471 



so that if a relation between growth and population size can be established from 

 data, for example, in the simplest case, the linear relation 



Loo = a-bN, (12) 



where a and b are constants, then the two equations (11) and (12) can be 

 treated as simultaneous equations and together provide a solution for the value 

 of Lao corresponding to any particular value of F or t c . Since the value of Woo 

 corresponding to Lao can readily be determined from the relation between 

 weight and length for the species in question, it is now possible to compute 

 the relation between equilibrium catch and F or t c from equation (10), with 

 the growth rate varying according to the population density. 



Compared with the behaviour of the simple model, introducing the com- 

 pensatory effect of a density-dependence of growth causes the population 

 biomass, and hence the weight of the catch, to be less sensitive to changes in 

 either F or t c . Sometimes this may lead to quantitatively different conclusions 

 being drawn. In a fairly heavily fished stock, for example, the simple model 

 with density-independent growth might predict that some decrease in fishing 

 effort would result in an increased equilibrium catch, and vice versa; the 

 introduction of a sufficiently marked dependence of growth on density could 

 reverse this conclusion in certain circumstances, because of the retardation of 

 growth that would follow from a reduction in fishing effort, or enhancement of 

 growth from an increase in effort. Examples of the influence of the density- 

 dependence of growth on the theoretical relation between catch and fishing 

 effort, and between catch and gear selectivity, will be found in Beverton and 

 Holt (1957). 



The theoretical introduction of a relation between the natural mortality 

 coefficient M and population density in the simple population model presents 

 no special difficulties, and can be treated by methods analogous to those 

 described above for growth. In practice, however, virtually nothing is known 

 about the variation, if any, of the natural mortality rate with density in any 

 major fish population, primarily because of the difficulty of measuring the 

 natural mortality rate with the necessary accuracy. Unlike the density- 

 dependence of growth, in which it can be safely accepted that growth will 

 decrease fairly steadily as density increases, and vice versa, even this assumption 

 may not necessarily be true for the natural mortality rate. It is, perhaps, some- 

 thing of a reassurance to find that in many of the major fish populations of 

 temperate waters in which the natural mortality rate has been measured, it is 

 fairly small, usually less than 30% per year ; and some of this is almost certainly 

 due to causes of death such as senility and adverse environmental conditions 

 which are unlikely to be much influenced by population density, if at all. 

 Natural mortality rates of important species of fishes and invertebrates of 

 tropical regions (e.g. tuna, anchovies and shrimp) have, however, been found 

 to be a good deal higher than this. It has to be admitted that knowledge of all 

 aspects of the mortality in fish populations due to causes other than fishing is 



