We study the spread of two strains of virus competing for space in a network modeled by the classical logistic ordinary differential equations. In large-scale complex networks, the underlying nonlinear dynamical system is high-dimensional and performing qualitative analysis of the differential equation becomes prohibitive. The study of such systems is often deferred to numerical simulations or local analysis about equilibrium points of the system. In this paper, we extend the work developed in [1], to formally establish a simple sufficient condition for (exponentially fast) survival of the fittest in a bi-layer weighted digraph: the weaker strain dies out regardless of the initial conditions if its maximum in-flow rate of infection across nodes is smaller than the minimum in-flow rate of the stronger strain. We bound any solution of the logistic ODE by one- dimensional solutions over certain homogeneous networks, for which the system is well understood. Our global stability approach via bounds readily applies to the discrete-time logistic model counterpart.
