  May and Leonard (SIAM J. Appl. Math 1975) introduced a three-species
Lotka-Volterra type population model that exhibits heteroclinic cycling. Rather
than producing a periodic limit cycle, the trajectory takes longer and longer
to complete each "cycle", passing closer and closer to unstable fixed points in
which one population dominates and the others approach zero. Aperiodic
heteroclinic dynamics have subsequently been studied in ecological systems
(side-blotched lizards; colicinogenic E. coli), in the immune system, in neural
information processing models ("winnerless competition"), and in models of
neural central pattern generators. Yet as May and Leonard observed
"Biologically, the behavior (produced by the model) is nonsense. Once it is
conceded that the variables represent animals, and therefore cannot fall below
unity, it is clear that the system will, after a few cycles, converge on some
single population, extinguishing the other two." Here, we explore different
ways of introducing discrete stochastic dynamics based on May and Leonard's ODE
model, with application to ecological population dynamics, and to a neuromotor
central pattern generator system. We study examples of several quantitatively
distinct asymptotic behaviors, including total extinction of all species,
extinction to a single species, and persistent cyclic dominance with finite
mean cycle length.
