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Hou, Zhanyuan (2023) On global dynamics of type-k competitive Kolmogorov differential systems. Nonlinearity, 36 (7). p. 3796. ISSN 0951-7715
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 A 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 SH and SV, 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 SH or SV: each trajectory in the interior of the first orthant is asymptotic to one in SH and one in SV. Geometric and asymptotic features of the global attractor A are investigated. It is proved that the partition A is divided into AH, A0 and AV such that AH, A0 are on SH and AV, A0 are on SV. Thus, A0 contains all the omega-limit sets for all interior trajectories of any type-K subsystems and the closure of the union of AH, AV as a subset of A is invariant and the upper boundary of the basin of repulsion of the origin. This A has the same asymptotic feature as the modified carrying simplex for a competitive system: every nontrivial trajectory below A is asymptotic to one in A and the omega-limit set is in A for every other nontrivial trajectory.
Available under License Creative Commons Attribution Non-commercial No Derivatives 4.0.
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