On the global flow of a 3-dimensional Lotka–Volterra system

In the study of the black holes with a Higgs field appears in a natural way the Lotka–Volterra differential system ẋ=x(y−1),ẏ=y(1+y−2x2−z2),ż=zy, in R3R3. Here we provide the qualitative analysis of the flow of this system describing the αα-limit set and the ωω-limit set of all orbits of this system in the whole Poincaré ball, i.e. we identify R3R3 with the interior of the unit ball of R3R3 centered at the origin and we extend analytically this flow to its boundary, i.e. to the infinity.