The dynamical behaviour of type-K competitive Kolmogorov systems and its application to three-dimensional type-K competitive Lotka–Volterra systems

This paper aims to exploit the geometrical and dynamical properties of general type-K competitive systems. We prove that there is a defined countable family of disjoint invariant n−1 cells that attract all non-convergent persistent trajectories in the type-K competitive system. For strongly type-K competitive systems, we prove that there exists an invariant n−1 manifold attracting all nontrivial orbits if the system is dissipative. We also establish the index theory of equilibria for general Kolmogorov systems. Combining such an index theory and the global attractivity, we classify the three-dimensional type-K Lotka–Volterra systems. There are four classes. We exploit the Hopf bifurcation and the global stability for positive equilibrium.