Svicobians of compartment models and the \(DaD\)-stability of Svicobians (Q1594552)

In various applied areas, compartmental models are used. The Jacobian of the system of model equations takes usually the form \(A= (S-Q)D\). A real-valued, indecomposable, and structurally nonsingular matrix of that form is called a Svicobian if \(S\) is a skew-symmetric matrix and if \(Q\) and \(D\) are diagonal matrices with a nonnegative and a positive principal diagonal, respectively. A real-valued stable matrix \(A\) is called \(DaD\)-stable if the matrix product \(\text{AD}_1\) is stable for any positive diagonal matrix \(D_1\) and the difference \(A- D_2\) is stable for any nonnegative diagonal matrix \(D_2\). Main results: 1. All Svicobians of sizes \(2\times 2\) and \(3\times 3\) are \(DaD\)-stable. 2. If the digraph of a nonsingular Svicobian cannot be colored according to the rules of the black-white test, then the Svicobian is \(DaD\)-stable. 3. All Svicobians with a negative main diagonal are \(DaD\)-stable.