Stability and positivity of equilibria for subhomogeneous cooperative systems (Q640171)

The following result is well-known: Let the origin of the positive linear system \(\dot x= Ax\) be globally asymptotically stable. Then so does the system \(\dot x= DAx\) for all diagonal \(D\) with positive diagonal entries (\(D\)-stability). The focus of the paper is on extending the notion of \(D\)-stability to nonlinear positive systems of the type \[ \dot x= \text{diag}(d(x))f(x), \] where \(d:\mathbb{R}^n\to \mathbb{R}^n\) is a \(C^1\) mapping satisfying \(d(x_1,\dots, x_n)= (d(x_1),\dots, d(x_n))\) with \(d_i: \mathbb{R}\to\mathbb{R}\), \(d_i(x_i)> 0\) for \(x_i> 0\). The main result requires that \(f\) is a subhomogeneous of degree \(\tau> 0\), that is, \(f(\lambda v)\leq \lambda^\tau f(v)\) for all \(v\in \mathbb{R}^n_+\), \(\lambda\in\mathbb{R}\) with \(\lambda\geq 1\).