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.

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