Global stability of the natural equilibrium in the persistence component of the SIT-free model

Consider the SIT-free model obtained from the full model by setting $S(t)\equiv 0$, with $\zeta>0$ and $\delta_L>0$. Suppose that $\mathcal R_0^{\mathrm{q}}>1$ and $K_E>K_E^{\ast}$, and let $X_-^{**}$ and $X_+^{**}$ denote, respectively, the Allee equilibrium and the natural equilibrium of the SIT-free model. Define the positive wild-population invariant subspace

$$\Omega_{\mathrm{sf}}=\bigl\{(E,L,P,F_u,F_{mw},F_{ms},M_w,M_s)\in\Omega:E,L,P,F_u,F_{mw},M_w>0,\ F_{ms}=M_s=0\bigr\}.$$

Let

$$\mathcal B_0=\bigl\{X_0\in\Omega_{\mathrm{sf}}:\lim_{t\to\infty}X(t;X_0)=\mathcal E_0\bigr\}$$

denote the set of positive wild-population initial states whose solutions converge to the mosquito-free equilibrium, and let

$$\mathcal W^s_+(X_-^{**})=\bigl\{X_0\in\Omega_{\mathrm{sf}}:\lim_{t\to\infty}X(t;X_0)=X_-^{**}\bigr\}$$

denote the stable set of the Allee equilibrium in $\Omega_{\mathrm{sf}}$. Finally, let $\mathcal P_+$ be the connected component containing $X_+^{**}$ of the forward-invariant set

$$\Omega_{\mathrm{sf}}\setminus\bigl(\mathcal B_0\cup\mathcal W^s_+(X_-^{**})\bigr).$$

Global-stability conjecture. The natural equilibrium $X_+^{**}$ is globally asymptotically stable relative to $\mathcal P_+$: $\mathcal P_+$ is positively invariant, $X_+^{**}$ is Lyapunov stable relative to $\mathcal P_+$, and every solution with initial condition in $\mathcal P_+$ satisfies

$$\lim_{t\to\infty}X(t)=X_+^{**}.$$

Numerical simulations suggest persistence-side stability for the SIT-free model with the mate-search delay and larval density dependence retained. The mosquito-free equilibrium and the Allee equilibrium create excluded extinction and threshold basins, respectively, so the conjecture concerns the component of positive wild-population states outside those sets. A rigorous proof of global asymptotic stability on this component remains open.