1. The Social Use of Space 21 



And normalizing, it is found that: 



C = ^ — rh'-'-'" dr (22) 



Where r/a = R, the cumulative probability of capture, Cum, as a func- 

 tion of the radius R from home becomes : 



Cumii = 1 - f — + /? + 1 j e-« (23) 



Equation (23) above will be called Curve II as shown in Table III and 

 Fig. 8. It may be compared to a similar cumulative probability curve for 

 the bivariate normal distribution, which will be called Curve I, and which 

 has the form: 



Cumi = 1 - exp (-rV2) (24) 



Curve II, Eci. (23), may be compared to Cui've I, Eq. (24), by con- 

 verting R into units of r, where r, the radius from the center of the home 

 range, is measured in the a units of the bivariate normal distribution, pro- 

 vided the constant a. oi R = r/a is known. This conversion was arbitrarily 

 accomplished as follows: It can be shown that the cumulative probability 

 of "capture" of Eq. (23), when expressed in terms of a and r, has the form : 



Cum = 1 - ^-— (r2 + 2ar + 2a2)e-'-/a (25) 



2a:- 



By Eq. (24), Curve I, when r = 1.2, Cum = 0.513. Therefore, by succes- 

 sive approximations, utilizing Eq. (25), it was found that when a = 0.44, 

 Cumii = 0.511 atr = 1.2. Therefore, 0.44 is the a conversion factor applied 

 to ^ = r/a, so that Curves I and II may be compared in terms of the bivariate 

 normal home range cr distance. 



The values for these two curves, as shown in Table III, are shown in 

 Fig. 8. Note that up to about 2<t radius these two curves are so nearly 

 identical that they are either likely to approximate actual field data equally 

 well. There is considerably more "tail'' to Curve II, but since so few ob- 

 servations occur in the greater than 2a range, it will still be difficult to 

 decide which of these two curves most nearly approximates actual field 

 data for the longer recapture radii. 



However, the objective was to determine how well phenomena observed 

 in the use of one-dimensional space could lead to a curve approximating 

 the bivariate normal distribution. One of the assumptions was that the 

 wandering responsive phase was proportional to radius from home. In 

 the analysis of wanderings, vacillations at ends of trips in the one-dimen- 



