2025/04/23 by Hou, Zhanyuan
#37B25 #37B35 #37C70 #37D10 #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2504.16860
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.