SIZE OF ANIMAL POPULATIONS 



197 



In order to estimate the population by this 

 method, a sampling station or group of stations is 

 established that will result in a random sample of 

 all parts of the population. Samples are dra^\^l at 

 intervals and the fish are tagged and replaced. 

 Records are kept, for each sample, of the number of 

 fish caught and the number of recaptures. 

 Schnabel (1938) provided a solution to the problem 

 of estimating the population from the resulting 

 data. 



We may let A^be the total population, as before, 

 Ti be the number of tagged fish in the lake when 

 the i'" sample is drawn, 7i j be the total number in 

 the i'* sample, consisting of ti tagged fish recap- 

 tured and Ui untagged. Schnabel finds that where 

 k samples are drawn, the method of maximum 

 likeliliood gives as an estimate of N the positive 

 real root of the k"' order equations 



k 



N-Ti 



k 



-12 ti 



which can be expanded in the form 



^..T. 



m N V~^N^N'^ 



/ 1=1 



(15) 



(16) 



By taking sufficient terms in formula 16 the root 

 maj' be approximated as closely as desired. 

 Schnabel states that 3 terms of the series are 

 usually sufficient, and that the computations 

 necessary for higher approximations are often 

 prohibitive. 



Schnabel also considers some sp(>cial cases of 

 formula 16. By writing the equation (15) in the 

 form 





(17) 



it may be seen that if Ti is negligible compared to 

 AT', the root of formula 15 is approximately 



JZniTi 



i=l 



k 



i=l 



(18) 



This is the formula which has been emploj'ed by 

 fislieries workers in practice. Its application wall 

 be clear from the example given in table 1 , the data 

 for which are from a marking experiment by 

 Krumholz (1944). 



Table 1. — Schnabel's method of computing a fish population 

 by repeated sampling and marking 



Next Schnabel points out that if Ti= T for all i 



N=T 



rr^ni 



(19) 



and states that "This formula is applicable to the 

 data of experiments in wliich the number tagged 

 is held constant after a certain point . This method 

 has the disadvantage that the data taken before 

 T become constant are not utilized." 



It may be readUy seen that if we consider the 

 sum of the samples in this last case as a single 

 largo sample, fonuula 19 is identical with formula 

 1. Thus the simple case considered earlier may 

 be regarded as a special case of the method of 

 the present section. 



Schnabel's formula 18 has been emploA^ed by 

 Ricker (1942, 1945a) to estimate fish populations 

 of lakes and ponds in Indiana. Ricker has as- 

 sumed that, in situations where this formula is 

 applicable, the fiducial limits of the Poisson distri- 

 bution appUed to Sf, would give some idea of the 

 variabUityascribable to random sampling (Ricker 

 1945b), but also states that "an estimate of error 

 obtained dii'cctW from the data themselves, for 

 both the general and the special case, is to be 

 desired." 



Underbill (1941) applied this method and 

 formula 18 to the computation of a chub-sucker 

 population of a pond in New York, and Roach 



