To persist or not to persist?

Ecological vector fields \dot x_i = x_if_i(x) on the non-negative cone \\bf R^n_+ 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 \partial {\bf R}^n_+ of the non-negative cone is repelling. 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 \partial {\bf R}^n_+ . In particular, this construction implies that robustly permanent vector fields are not dense in the space of permanent vector fields. Hence, verifying robust permanence is important. 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.