102 John B. Calhoun 



response, and (c) the amount of space in which the movement of A'' in- 

 dividuals takes place. Random distribution of positions of individuals at 

 any moment in time is assumed. We choose to ignore a small correction 

 factor arising from the fact that all individuals move. Velocities of all in- 

 dividuals are initially assumed to be a constant. Furthermore, we assume 

 that all individuals are identical. Thus our concern is not which individuals 

 meet, but rather which state, responsive or refractory, the contacting in- 

 dividuals happen to be in. 



N = Number of animals forming the group. 



d = The diameter of interaction for each animal, that is, that distance 

 between the centers of two individuals at which a physical or 

 psychological collision or contact occurs. In the simplest case 

 animals may be considered equivalent to billiard balls. Then d 

 represents the diameter of the ball, the individual. See Section 

 XIII A, 1 for further elaborations. 



Assume an animal moving in some direction on the plane in a population, 

 A^ — 1, of other animals. 



Each of these other individuals presents a target of dimension d, normal 

 to the X direction. The expectation that the incoming animal will make a 

 collision while moving a distance A.r (in time t) is the ratio, 



d(N - 1)A.T 



of surface covered by the targets to the total surface, where A is the area 

 available to the animals. 



It should be emphasized that the unit of time must be sufficiently large 

 so that the number of collisions in that time interval is large enough to 

 justify using the statistical law of large numbers in the derivation. For 

 similar reasons, it must be assumed that the mean free path of the in- 

 dividuals must be large in comparison with the target diameter. 



Since the velocity v may be considered ec^ual to Ax/t, the average num- 

 ber of contacts w,-, per individual in time /. is 



djN - l)vt 



Tie = (o2j 



A 



For present purposes we are concerned only with the average ric in t and 

 not in the variability in contacts in t. The frequency of contacts by a given 

 individual will be : 



^. . - . rf(iV-i)_. (33) 



{ A 



