Global phase portrait conjecture for the predator-prey system in region II-b

Let the predator-prey system be considered on the closed positive quadrant of the Poincaré disc, with parameters $b,c,\b7\delta$ satisfying the indicated inequalities. The parameter space contains regions II-a and II-b and surfaces $S_2$ and $S_3$; in region II-a, the phase portrait is the one shown in Figure (C).

Global phase portrait conjecture. If

$$0<b\delta<c-\delta,$$ $$\delta(\delta(b+1)+c(b-1))^2-4c(c-\delta)^2(c-\delta(b-1))<0,$$

and

$$1+c-\delta-b-b\delta>0,$$

then the global phase portrait is topologically equivalent to the phase portrait shown in Figure (C). Equivalently, the phase portrait is conjectured to be the same in region II-b and over the surfaces $S_2$ and $S_3$.

The theorem preceding this conjecture establishes the phase portraits in the other parameter regimes, and the phase portrait in region II-a is proved. The claim concerns the unresolved region II-b and the indicated boundary surfaces.