SIZE OF ANIMAL POPULATIONS 



199 



ESTIMATION OF A CHANGING 

 POPULATION 



Some fishes, such as salmon spawning in a given 

 stream or lake, do not always form a single, homo- 

 geneous, completely mixed population. There 

 may be a tendency for the fish which migrate to 

 the spawning grounds earliest to complete their 

 spawning and die earliest; there results a positive 

 correlation between time of migration past a 

 point below the spawning grounds and the time of 

 appearance on, and of death at, the spawning 

 grounds. If, now, we are tagging fish below the 

 spawning grounds, or even on these grounds, and 

 later sampling for tag ratios, the "mi>dng" of the 

 fish between tagging and sampling is not complete, 

 and this may need to be taken into consideration 

 in our estimation of the population. Similar 

 situations may occm- among other migratory 

 animals. 



When there exists such a correlation between 

 time of tagging and time of subsequent sampling, 

 the samples drawn during any particular part of 

 the season do not represent all parts of the popula- 

 tion equallj^; the sample is not a random sample of 

 the whole population. The possible effects of 

 this on our estimates by formula 1 are easily seen. 

 If, as has already been pointed out, all parts of the 

 population have the same tag ratio, if the tags are 

 "evenly distributed," it will make no difference 

 whether the subsequent saniples represent the 

 various parts of the population equally or not. 

 Likewasc, if the population is "evenly" sampled 

 after tagging, that is, if the probability of a given 

 fish being included in the sample is not a function 

 of the time of sampling (and, therefore, not a 

 function of the tinae of tagging), any uneven dis- 

 tribution of tags by time of migration will have 

 no effect. If, on the other hand, the probability 

 of a fish being tagged (the tag ratio) varies with 

 the time of tagging, and the probability of being 

 included in the subsequent sample varies with the 

 time of sampling, and there also exists a correla- 

 tion between time of tagging and time of sampling, 

 it is obvious that the tag ratio in the total sample 

 for the season will differ from that of the popula- 

 tion to some extent, depending on the magnitudes 

 of these factors. 



Presented here is a method of estimating the 



population by which these errors may be reduced 

 when the tagging is done by means of numbered 

 tags, so that the relation between time of tagging 

 and time of recovcrj'' may be estimated. I am 

 indebted to Dr. S. Lee Crump of the Iowa State 

 College Statistical Laboratory for much assistance 

 with the mathematics involved. 



If our tagged fish have been marked by num- 

 bered tags, we know both the date of tagging and 

 date of recovery for each one recovered. This 

 makes it possible to tabulate the recoveries by 

 time of tagging and time of recovery, using as a 

 time interval a convenient period of days. Our 

 notations for the elements involved in the discus- 

 sion of this section, in addition to those introduced 

 before, are as follows: 



Let 



A^„ = the total number of fish passing the point 

 of tagging during the a'" period of tagging. 

 (a=l,2, 3, ... a). 



ra = tho number of these fish which are lagged 

 dm-mg the a'* tagging period. 



n„t=tha number of fish out of the A'a that are 

 subsequently recovered during the ;"' recovery 

 period. 



Tai=the number of fish out of the Ta that die 

 and are thus available to be recovered during the 

 i'" recovery period. 



m„<=the number of tagged fish tagged during 

 the a"" period of tagging and recovered during the 

 i'" period of recovery (i=l, 2, 3, . . . s). 



m'a,= the number of untagged fish passing the 

 point of tagging during the a'" tagging period and 

 recovered during the i'" recovery period. 



The following summation conventions are cni- 

 ployed: 



i a a i 



^m'ai = m'a- '^m'a, = m'.i 



i a 



Obviously, 



Also let: 



ma--\-m'a. = na. 



m.i + m'.t=(^i 



