Permanent coexistence in general models of three interacting species

We address the question of the long term coexistence of three interacting species whose dynamics are governed by the ordinary differential equations xi = xifi(i = 1,2,3). In order for any theory in this area to be useful in practice, it must utilize as little information as possible concerning the forms of the fi, in view of the great difficulty of determining these experimentally. Here we obtain, under rather general conditions on the equations, a criterion for judging whether the species will coexist in a biologically realistic manner. This criterion depends only on the behaviour near the one or two species equilibria of the two dimensional subsystems, the behaviour there being relatively easy to examine experimentally. We show that with the exception of one class of cases, which is a generalization of a classical example of May and Leonard [21], invasibility at each such equilibrium suitably interpreted is both necessary and sufficient for a strong form of coexistence to hold. In the exceptional case, a single additional condition at the equilibria is enough to ensure coexistence.