60 John B. Calhoun 



situation where home ranges contract as individuals become more 

 subordinate. 



How much beta and gamma members may be expected to contract 

 their home ranges requires consideration of phenomena treated in Section 

 XIV, E, titled "Velocity and Home Range." There, it is shown that in 

 terms of o-„ units of distance, the home range sigmas for alpha, beta, and 

 gamma individuals become, respectively, 1.0, 0.9575, and 0.6457. Since 

 an optimum uniform utilization of the environment requires a 2a interval 

 between home range centers, the distance between alpha and beta home 

 range centers wall become 1.9575o-a, and 1.6032o-a between beta and gamma 

 home range centers. 



The periphery of each constellation (Fig. 27) may be arbitrarily desig- 

 nated as that radius from the center of the alpha member's home range 

 extending to one aa beyond the center of each gamma individual's home 

 range. This radius is 3.1478a-a. And since the home range centers of the 

 alpha members of adjoining constellations are 6.92o-a apart, it is obvious 

 that between constellations there lies what may be called an interconstella- 

 tion matrix, receiving very little usage from the dominant species forming 

 constellations. Furthermore, this interconstellation matrix must markedly 

 reduce communication between members of adjoining constellations. How 

 extensive this reduction becomes must be viewed against the requirement 

 of a 2a inter-HRC interval for an optimum state. Yet, the HRC of any 

 beta or gamma member of one constellation will lie on the average 4a units 

 of distance away from the nearest beta or gamma members of an adjoining 

 constellation with reference to their own reduced home ranges. 



On the average, such constellations will consist of twelve individuals, 1 

 alpha, 6 betas, and 5 gammas. Of these, the gammas with smallest home 

 ranges form 0.417 of the population. These are the individuals who, after 

 removal of their dominant alpha and beta associates, will enlarge their 

 home ranges. Actually, some gammas will be trapped before the 15th day 

 and some alphas and betas will be taken after it. Furthermore, Fig. 25 

 clearly shows that a few more gammas would have been taken after the 

 30th day of trapping. Therefore, the observed proportion of the population 

 consisting of gamma individuals, estimated at about 36%, reasonably well 

 approximates the theoretical expected of 41.7%- 



To date, this approximation stands as the only direct supporting proof 

 (see Section XI) that populations of dominant species of small mammal 

 communities do, in fact, tend to form such constellations. If they really 

 do, we may anticipate that there has been evolution relative to physiology 

 such that its optimum state is most compatible with interactions among 

 individuals transpiring in groups with a mean size of twelve adults. In 



