Kolmogorov Vector Fields with Robustly Permanent Subsystems

The following results are proven. All subsystems of a dissipative Kolmogorov vector field ˙ xi = xifi x are robustly permanent if and only if the external Lyapunov exponents are positive for every ergodic probability measure μ with support in the boundary of the nonnegative orthant. If the vector field is also totally competitive, its carrying simplex is C1. Applying these results to dissipative Lotka–Volterra systems, robust permanence of all subsystems is equivalent to every equilibrium x∗ satisfying fi x∗ > 0 whenever x∗i = 0. If in addition the Lotka–Volterra system is totally competitive, then its carrying simplex is C1.