When dealing with positive nonlinear systems, conventional theory requires too much for the stability of equilibrium points located on the boundary of the positive orthant, which encourages the consideration of stability with respect to the positive orthant. Generalizing this concept to the stability of a family of equilibrium points with Non-vanishing Basin of Attraction (NvBA)-stability has been introduced, which often becomes of interest when positive systems undergo bifurcations. In this technical note, we present a simple condition that can be employed for determining the NvBA-stability with respect to the positive orthant. The proposed condition is applicable to a class of nonlinear systems of certain Jacobian structure and is tailormade for this system. An illustrative example is included in order to help the exposition of the technical note.