6 John B. Calhoun 



where : 



s = unbiased estimate of the home range sigma for all the animals 



in a sample 

 St = unbiased estimate of the home range sigma for any particular 

 animal 

 Ki = number of captures of ith animal 

 n = number of animals 



n 



N = total captures = ^ Ki 



ij = jih observation of the iih. animal 



A detailed analysis of the home range for 25 male harvest mice 

 (Reithrodontomys) on which there were 10 to 24 captures each indicated 

 that there was a significant variation of sigma among animals. In other 

 words, some animals had significantly larger home ranges than others. 

 Therefore, in order to compare the observed recapture radii with the 

 theoretical (Table II in Calhoun and Casby, 1958), each recapture radius 

 was normalized into a standard measure denoted by Z in which the home 

 range sigma for each animal was assigned a value of 1.0 and all recapture 

 radii expressed as a proportion of this. 



As may be seen from Fig. 1, the theoretical closely approximated the 

 observed. Although this detailed analysis has been applied only to this one 

 species, it shall be assumed for the purpose of developing further formula- 

 tion that the bivariate normal distribution function adequately describes 

 fixed home ranges of other species. 



Comparison of observed and theoretical distribution of home range 

 radii required viewing home range as a probability of capture which changes 

 with radial distance from the home range center. Bands of equal width 

 increase in area with radial distance from the home range center, while 

 probability of capture per unit area decreases with increase in radial dis- 

 tance. Interaction of these two factors results in more captures at about 

 one sigma from the home range center than at any other distance (Fig. 1). 



However, the ecologically important aspect of the bivariate normal dis- 

 tribution as an expression of home range is the relative probability that an 

 animal will be in a unit of area with respect to the radial distance of that 

 unit area from the home range center (Fig. 2). For any given sigma char- 

 acterizing a particular species, its density function in terms of area curve 

 may be obtained by multiplying the relative sigma value on the abscissa 

 by the observed sigma and dividing the density function values on the 

 ordinate by the square of the observed sigma. 



