VARIABILITY IN TRAWL CATCHES 



157 



the fish of a given species is not tenable. Two 

 theoretical distributions, developed from models 

 postulating heterogeneity or aggregation, are 

 shown to apply to the observed data with suf- 

 ficiently high probability to retain them as work- 

 ing hypotheses. One of these, the negative 

 binomial distribution, describes satisfactorily the 

 distribution per tow of individuals of a species. 

 The other, the logarithmic series distribution, is 

 shown to describe the distribution of individuals 

 and species. 



The occurrence of species in successive samples 

 is shown to have a Poisson distribution, and this, 

 in theory, is sufficient to account for the loga- 

 rithmic distribution of species and individuals ob- 

 served. It is pointed out that various authors 

 have shown that the occurrence of numbers of a 

 particular species drawn in samples from a loga- 

 rithmic distribution will be of the negative-binomial 

 form. The observed distributions are therefore 

 self-consistent. 



A high variance associated with the mean 

 catch per tow is shown to be a necessary con- 

 sequence of the basic heterogeneous distribution 

 of the numbers of fish. The agreement of ob- 

 served variances with theoretical variances is 

 sufficiently good to indicate that the variance due 

 to variations in the sampling technique is of 

 relatively minor importance. 



It is shown that the variance may be reduced 

 and the amount of information increased by 

 decreasing the size of the observed mean, and it is 

 suggested that this may be accomplished by 

 reducing the length of tow and by reducing the 

 size of the trawl net. 



Appropriate transformations for use of the 

 census-trawl data in analysis of variance and in 

 other statistical procedures are indicated. 



APPENDIX A 



A model for fish population distribution 



The following model is an attempt to postulate 

 simple conditions which would lead to the mathe- 

 matical distributions observed in the trawl-sam- 

 pling studies. It is a restatement, in terms of 

 species and numbers of fish, of mathematical rela- 

 tions which have been noted in somewhat more 

 general terms by Fisher (1941, 1943), Kendall 

 (1948), Quenouille (1949), and Anscombe (1950). 

 We have borrowed freely from these authors. 



Let us postulate that S species are distributed 

 at random within the area of observation, each 

 species moving independently of others, so that 

 the distribution is Poisson with the probability of 

 observing exactly n species in any one sample 



P(n species) = 



"m" 



n\ 



(1) 



Let us postulate further that the number of 

 individuals represented by each species is not 

 random but rather that the individuals of each 

 species tend to aggregate so that the mean m 

 varies from sample to sample in a Eulerian dis- 

 tribution. The probability of observing a sample 

 of size r is, then, 



P(r individuals) 



(fe+r-1)! p' 



(k-l)\r\ (l-r-p)*-*-'' 



■(2) 



which is the standard form of the negative bi- 

 nominal with index k and parameter p/l+p=x 

 (Fisher 1941; Quenouille 1949). Letting k tend 

 to infinity and excluding the first term as unobserv- 

 able, a logarithmic series distribution is obtained 

 (Fisher 1943) where the probability of observing r 

 individuals of one species is 



P(r individuals of one species) = a x'/r. . .(3) 

 or the coefficient of V in 



— a log, (1— xt) 



1 



where 



x=— 7— and a= 

 l+3» 



'log, (1— *)' 



(4) 



The following relations result (Quenouille 1949): 

 P (r individuals) =P (n species) X P(r indi- 

 viduals in n species) (5) 



From equation 4 the probability of observing r 

 individuals in n species is the coefficient of F in 



[-a log. (1-xOY 



Rewriting equation 5 we have 

 P(r individuals) 



coefficient of V in S — rr-X[-alog. (l-xt)] n 



n=0 n - 



=coefficient of V in exp [—m—am log, (1— xt)] 

 coefficient of F in {\-xt)-" m e~ m 



^-^ m{ ^m xT ^ c&{l - x) ' a==e - (6) 



