Considering \begin{equation} \begin{cases} \dot{m}=pa(1-m)h-\mu m \\ \dot{h}=qad(1-h)m-\alpha h \end{cases} \end{equation} where $m$ is the proportion of infected mosquitoes and $h$ is the proportion of infected humans and $d$ represents the density of mosquitoes per human ($d=\frac{M}{H}).$ I'm interested in analysing if $(0,0)$ is a unstable equilibria given that $pqa^2d=\alpha\mu$, whereas this condition gives us that the stability theorem is inconclusive $(|J(F)_{(0,0)}|=0;tr(J(F)_{(0,0)})<0)$. I also know that there is no other equilibrium point as $pqa^2d \leq \alpha \mu$.
As $Q=[0,1]^2$ is simply connected, by Bendixson's Criterion $(tr(J(F)_{(0,0)})<0)$, there is no closed orbit that is totally contained in $Q$; so by Poincaré-Bendixson's Theorem, if we get rid of an homoclynic orbit, then we could say that $(0,0)$ is assymptotically stable. But looking at the initial monotonicity of the solutions beginning at $\partial Q$
Any suggestions?
