FISHERY BULLETIN: VOL. 70, NO. 3 



minimum size or stage of development and that 

 the species' abundance does not change appre- 

 ciably with size, or that corrections can be made 

 for these two factors. Assume that the species' 

 swimming ability is known, and proportional to 

 size, while its reaction distance remains constant, 

 or nearly constant, for all sizes sampled. Then 

 if the catch were sorted into size class intervals 

 (ideally, into components, in the operational 

 sense) and counted, the size classes could each 

 be assigned a corresponding escape speed, in- 

 creasing with size; plotting these speed-frequen- 

 cy values as points on either graph of Figure 3 

 would produce a curve which resembles one of 

 the family of curves shown on those graphs. 

 Proper matching of observed data to the the- 

 oretical curves would yield unique values of both 

 Pc and Xq/R (and therefore Xo, since R is known) 

 for each component or class interval of that spe- 

 cies. Once Pc is known, the absolute population 

 density can be calculated from equation (3). 

 Finally, values of Xo, or Xo/R, provide valuable 

 information as to the relative merits of different 

 sampler designs. 



The following examples illustrate this use of 

 avoidance theory. In the first example, the pop- 

 ulation density structure is known, so that Pc 

 can be calculated directly and only Xo remains 

 to be determined from avoidance theory. In the 

 second example, two different samplers are com- 

 pared to obtain estimates of Pc and Xo for both, 

 from the theory, even though the population 

 structure is not known. The third example il- 

 lustrates information which can be obtained 

 when one sampler is towed at two different 

 ranges of speed. Finally, catch data from a 

 single net, towed at one speed, are considered. 



Murphy and Clutter (1972) present length- 

 frequency data for Hawaiian anchovy, the en- 

 graulid, Stolephorus purpureus, caught with 

 their plankton purse seine and a 1-m net. Their 

 data for paired daylight tows are reproduced on 

 Figure 4. The uppermost curve on this figure 

 shows length-frequency data from the purse 

 seine, assumed to represent the population struc- 

 ture as a function of size. I approximate these 

 data by means of the straight line shown on this 

 figure, in effect assuming that the population 



declines exponentially as the fish grow to larger 

 sizes. That is, Nl = No exp (—kL), where 

 Nl is the population density at length L, No is 

 a fictitious population density at zero length ob- 

 tained by extrapolation, and A; is a constant. The 

 lowest curve on Figure 4, marked Cl, shows the 

 catch (C) in each 1-mm length class (L). 



To remove the effects of changes in popula- 

 tion density, set No = 1.0 by dividing all values 

 of Nl by the population density at No. On semi- 

 log graph paper this can easily be accomplished 

 simply by aligning iVo with 1.0 on the ordinate 

 scale, after the length-frequency values have 

 been plotted on tracing paper using the three- 

 cycle semilog scale of Figure 3. Next, divide 

 each value of Cl by the corresponding value of 

 Nl to obtain the middle curve of Figure 4, 

 labeled Cl/Nl. This division can be performed 

 graphically by moving each value of Cl vertically 

 upward a distance equal to the vertical distance 

 between Nl at the corresponding length, L, and 

 the horizontal line Pc = 1.0. This procedure 

 yields the length-frequency curve which would 

 have been obtained if the population density in 

 each class interval had been the same, i.e., 



Nl = A^o. 



The above procedure amounts to dividing 

 catch per unit volume by the population per unit 

 volume, which according to equation (2) yields 



Catch/unit volume 



No. of organisms/unit volume ~ 



Probability of capture (if losses = 0). 



Thus the curve labeled Cl/Nl on Figure 4 is 

 also a curve showing Pc for each class interval, 

 providing that catch per unit volume has been 

 correctly related to the population per unit vol- 

 ume in each class interval and that mesh losses 

 are negligible. 



Mesh losses appear to be negligible for animals 

 larger than the two smallest class intervals, 

 which will not be used in the analysis. Since Ue is 

 not known, length for each class interval is con- 

 verted to escape velocity by a'ssuming that 

 Ue — lOL where Ue is in cm/sec and L is in cm. 

 Each value thus obtained is divided by U, nom- 

 inally 76 cm/sec. Resulting values of Ue/U are 

 shown in the upper abscissa of Figure 4. The 



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