4 John B. Calhoun 



John Gilbert. However, I assume full responsibility for any errors, in- 

 adequate presentation, or overextension from their initial guidance. 



I have found this effort a rewarding one for the development of insight 

 into complex social systems, and I can only hope that in some small meas- 

 ure it may serve as a bridge for others in their design of experiments or in 

 their further theorizing. 



II. The Bivariate Normal Type of Home Range 



Home range denotes the area covered by an individual in its day-to- 

 day activities. Field studies of many species of mammals have revealed 

 that each individual customarily stays within a restricted area for long 

 periods. The individual utilizes the center of such an area most inten- 

 sively. With increasing radial distance from this home range center (HRC) 

 the relative frequency of visitation per unit of area decreases. Calhoun 

 and Casby (1958) found that the bivariate normal distribution function 

 adequately describes home range. The following is a summary of their 

 analyses. 



In home range studies, "density function" is a mathematical expression 

 representing the probability of an animal being present in some arbitrarily 

 small area. Three assumptions are made : 



The home range is fixed. In other words, the statistics of the home range 

 are stationary or time independent. 



There is a true center of activity although the apparent center, the mean 

 coordinate point of capture, of activity may deviate from it. 



The probability of an animal being in a unit of area decreases with in- 

 creasing distances from the true center of activity. This and the second 

 assumption suggest a bivariate normal distribution of the density function : 



f{x, 7j) dxdy = — — - exp [- (a:^ + if)/2a^'] dxdy (1) 



where o- is the standard deviation of the distances in the x and y direction 

 and is assumed to be equal for both, and x and y are measured from their 

 respective means. This density function may be used to represent the 

 percentage of time spent in the area dxdy located at the Cartesian coordi- 

 nates X, y, or in polar coordinates: 



/(/•, 6) rdedr = — — exp (-r'^/'Ia^) rdddr (2) 



zcrV 



Here, the area rdddr is determined by r. 



The density function in terms of the Cartesian coordinates is more 



