Global dynamics of planar discrete type-K competitive systems
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910917334138880 |
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| author | Hou, Zhanyuan |
| author_facet | Hou, Zhanyuan |
| contents | For a continuously differentiable Kolmogorov map defined from the nonnegative orthant to itself, a type-K competitive system is defined. Under the assumptions that the system is dissipative and the origin is a repeller, the global dynamics of such systems is investigated. A (weakly) type-K retrotone map is defined on a bounded set, which is backward monotone in some order. Under certain conditions, the dynamics of a type-K competitive system is the dynamics of a type-K retrotone map. Under these conditions, there exists a compact invariant set A that is the global attractor of the system on the nonnegative orthant exluding the origin. Some basic properties of A are established and remaining problems are listed for further investigation for general N-dimensional systems. These problems are completely solved for planar type-K competitive systems: every forward orbit is eventually monotone and converges to a fixed point; the global attractor A consists of two monotone curves each of which is a one-dimensional compact invariant manifold. A concrete example is provided to demonstrate the results for planar systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_16860 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global dynamics of planar discrete type-K competitive systems Hou, Zhanyuan Dynamical Systems 37B35, 37B25, 37C70, 37D10 For a continuously differentiable Kolmogorov map defined from the nonnegative orthant to itself, a type-K competitive system is defined. Under the assumptions that the system is dissipative and the origin is a repeller, the global dynamics of such systems is investigated. A (weakly) type-K retrotone map is defined on a bounded set, which is backward monotone in some order. Under certain conditions, the dynamics of a type-K competitive system is the dynamics of a type-K retrotone map. Under these conditions, there exists a compact invariant set A that is the global attractor of the system on the nonnegative orthant exluding the origin. Some basic properties of A are established and remaining problems are listed for further investigation for general N-dimensional systems. These problems are completely solved for planar type-K competitive systems: every forward orbit is eventually monotone and converges to a fixed point; the global attractor A consists of two monotone curves each of which is a one-dimensional compact invariant manifold. A concrete example is provided to demonstrate the results for planar systems. |
| title | Global dynamics of planar discrete type-K competitive systems |
| topic | Dynamical Systems 37B35, 37B25, 37C70, 37D10 |
| url | https://arxiv.org/abs/2504.16860 |