Oriented graphs generated by random points on a circle

Extending the cascade model for food webs, we introduce a cyclic cascade mo del which is a random generation model of cyclic dominance relations. Put n species as n points Q(1), Q(2),..., Q(n), on a circle. If the counterclock wise way from Q(i) to Q(j) on the circle is shorter than the clockwise way, we say Q(i) dominates Q(j). Consider a tournament whose dominance relation s are generated from the points on a circle by this rule. We show that when we take n mutually independently distributed points on the circle, the pro bability of getting a regular tournament of order 2r + 1 as the largest reg ular tournament is equal to (n/2r + 1) / 2(n-1) This probability distribution is for the number of existing species after a sufficiently long period, assuming a Lotka-Volterra cyclic cascade model.