Heteroclinic cycles in the chemostat models and the winnerless competition principle

Winnerless competition principle (WLC) is a type of competition that does not have a winner; all species take turns (or switch) to win. In the phase space, it appears as a stable heteroclinic contour connecting single-species equilibria. In ecology, May and Leonard [R.M. May, W.J. Leonard, Nonlinear aspects of competition between three species, SIAM J. Appl. Math. 29 (1975) 243–253] were the first to discover the behavior in their famous paper that the competition of three species experiences a special type of WLC competition, the rock-paper-scissors competition. Recently, WLC concepts are used for the design in neural network dynamics. In this manuscript, it is shown that WLC can also appear in the chemostat model. We consider a chemostat model of n species of microorganisms competing for k essential and growth-limiting nutrients. Sufficient conditions for a stable heteroclinic cycle connecting single-species equilibria in the limit sets are given. The heteroclinic cycle can be constructed so that the equilibria are connected in the following order: E 1 → E 2 → E 3 → ⋯ → E n → E 1 in which E i ʹs are the ith species equilibria. This heteroclinic cycle describes the rock-paper-scissors winnerless competition; all of the n species take turns to win, there is no final winner.