On global dynamics of type-K competitive Kolmogorov differential systems

Abstract

This paper deals with global asymptotic behaviour of the dynamics for N-dimensional type-K competitive Kolmogorov systems of differential equations defined in the first orthant. It is known that the backward dynamics of such systems is type-K monotone. Assuming the system is dissipative and the origin is a repeller, it is proved that there exists a compact invariant set Σ which separates the basin of repulsion of the origin and the basin of repulsion of infinity and attracts all the non-trivial orbits. There are two closed sets S H and S V , their restriction to the interior of the first orthant are ( N − 1 ) -dimensional hypersurfaces, such that the asymptotic dynamics of the type-K system in the first orthant can be described by a system on either S H or S V : each trajectory in the interior of the first orthant is asymptotic to one in S H and one in S V . Geometric and asymptotic features of the global attractor Σ are investigated. It is proved that the partition Σ = Σ H ∪ Σ 0 ∪ Σ V holds such that Σ H ∪ Σ 0 ⊂ S H and Σ V ∪ Σ 0 ⊂ S V . Thus, Σ0 contains all the ω-limit sets for all interior trajectories of any type-K subsystems and the closure Σ H ∪ Σ V ‾ as a subset of Σ is invariant and the upper boundary of the basin of repulsion of the origin. This Σ has the same asymptotic feature as the modified carrying simplex for a competitive system: every nontrivial trajectory below Σ is asymptotic to one in Σ and the ω-limit set is in Σ for every other nontrivial trajectory.