Monotonicity properties arising in a simple model of Wolbachia invasion for wild mosquito populations

The paper rigorously establishes: (i) positive invariance and a compact absorbing rectangle X = [0,N♯]×[0,W♯] (hence precompactness) , (ii) competitiveness and strong order-preservation in Int R^2_+ via an order-isomorphism and an irreducible Metzler Jacobian , (iii) the two boundary equilibria EN, EW are locally asymptotically stable while the interior Ec is a saddle via det J(Ec) < 0 , and (iv) an unordered separatrix passing through Ec that separates the basins and equals the stable manifold of Ec, tangent to its stable eigenvector . The candidate solution proves the same end results with a slightly different toolkit (Hirsch planar theory for competitive systems and a shorter det J(Ec) calculation), and is essentially correct. The only minor gap is the informal linearization near E0; the paper handles E0 carefully via invariance of the axes and logistic dynamics .