Permanence of Weakly Coupled Vector Fields

Abstract

Let F1 , . . . , Fk be k dissipative vector fields on finite dimensional Euclidean spaces that preserve the skeleton of the positive orthant. Permanence of all sufficiently weak couplings of these vector fields corresponds to robust permanence of the uncoupled vector field F1 × ... × Fk. A sufficient condition for robust permanence of F1 × ... × Fk involving unsaturated Morse decompositions is provided. In the case of coupled food chain vector fields and coupled two-dimensional vector fields, this sufficient condition is shown to be necessary. As an illustration, these results are applied to weakly coupled logistic-Holling predator-prey systems.