Persistence and extinction in three species Lotka-Volterra competitive systems

Extinction in a three species Lotka-Volterra competitive system is classified in terms of the model parameters. Necessary and sufficient conditions are given for one and two species extinction; these are, alternatively, conditions for two and one species persistence. Persistence of the system is studied assuming all pairwise interactions between species are known. An intransitive species arrangement is the only case of persistence where pairwise interactions are, by themselves, sufficient to govern persistence. No persistent arrangement can contain a pair of species that interacts in an unstable manner.