The extremal-point conjecture for the support of the invariant measure

Let $F_0$ and $F_1$ be the two Lotka–Volterra vector fields, and let $G$ be the polynomial defining the non-axis collinearity curve

$$\tilde C=\left\{(x,y)\in\mathbb{R}_+^2:G(x,y)=0\right\}.$$

Define

$$T=\left\{(x,y)\in\mathbb{R}_+^2:G(x,y)=0\text{ and }(F_0\cdot\nabla G)(x,y)=0\right\}.$$

For $z\in T$, let $C(z)$ be the bounded region enclosed by the trajectories of $F_0$ and $F_1$ starting at $z$, together with the segment $[1/a_1,1/a_0]\times\{0\}$.

Extremal-point conjecture. The set $T$ is a singleton $\{z_0\}$, and the support of the invariant measure that is not supported by one of the two axes is

$$C(z_0)\times\{0,1\}.$$

Numerical experiments suggest that there is a unique extremal point on the relevant part of $\tilde C$, whose two flow trajectories form the boundary of the support. The conjecture identifies this point and the support of the non-axis-supported invariant measure, while the preceding discussion only establishes inclusion of the relevant stable-equilibrium trajectories in the invariant support.