THE DYNAMICAL BEHAVIOR ON THE CARRYING SIMPLEX OF A THREE-DIMENSIONAL COMPETITIVE SYSTEM: II. HYPERBOLIC STRUCTURE SATURATION

A competitive system on the n-rectangle: {x is an element of R-n: 0 <= x(i) <= l(i), i - 1,..., n} was considered, each species of which, in isolation, admits logistic growth with the hyperbolic structure saturation. It has an (n - 1)-dimensional invariant surface called carrying simplex Sigma as a globe attractor, hence the long term dynamics of the system is completely determined by the dynamics on Sigma. For the three-dimensional system, the whole dynamical behavior was presented. It has a unique positive equilibrium point and any limit set is either an equilibrium point or a limit cycle. The system is permanent and it is proved that the number of periodic orbits is finite and non-periodic oscillation the May-Leonard phenomenon does not exist. A criterion for the positive equilibrium to be globally asymptotically stable is provided. Whether there exist limit cycles or not remains open.