To persist or not to persist?

Josef Hofbauer and Sebastian J Schreiber
Nonlinearity 17 (2004), 1393-1406.

Abstract .
Ecological vector fields on the non-negative cone on Rn are often used to describe the dynamics of n interacting species. These vector fields are called permanent (or uniformly persistent) if the boundary of the non-negative cone is repelling, implying the coexistence of all species.
We construct an open set of ecological vector fields containing a dense subset of permanent vector fields and containing a dense subset of vector fields with attractors on the boundary. In particular, this construction implies that robustly permanent vector fields are not dense in the space of permanent vector fields.
We illustrate this result with ecological vector fields involving five species that admit a heteroclinic cycle between two equilibria and the Hastings-Powell teacup attractor.

reprint with color pictures, also available at IOP