Conjecture on stability and convergence of the coordinate-wise positive equilibrium

Let $p$, $s$, $\sigma$, and $\kappa$ be the model parameters, and let $(\tilde x_1,\tilde x_{2a})$ denote the coordinate-wise positive equilibrium of the system. Let $\tilde{\kappa}$ and $\kappa'$ be the parameter thresholds appearing in the statement. For a coordinate-wise positive initial condition $(x_1(0),x_{2a}(0))$ with $x_1(0)+x_{2a}(0)<1$, consider the trajectory of the system.

Stability and convergence conjecture. (A) If

$$\sigma > \frac{2s^2}{p(1-s))},$$

and $\kappa\in(\tilde{\kappa},\kappa')$, then $(\tilde x_1,\tilde x_{2a})$ is locally asymptotically stable, and the trajectory converges to $(\widetilde x_1,\widetilde x_{2a})$. (B) If

$$\sigma < \frac{2s^2}{p(1-s))},$$

and $\kappa\in(\kappa',\tilde{\kappa})$, then $(\tilde x_1,\tilde x_{2a})$ is unstable and both eigenvalues of the corresponding Jacobian matrix have strictly positive real part.

The conjecture addresses the parameter regimes not fully treated in the paper and describes the stability of the interior equilibrium and the resulting long-term behaviour of the two-dimensional Moran dynamical system.