Three-dimensional population systems

In this paper we consider the interaction matrix B   of a three-dimensional population system, with positive determinant and we prove that the invariant Σ(B)=(traceofB)×(sumofprincipalminorsofB)-(determinantofB), characterizes the sign of the real parts of the eigenvalues of B. Secondly, we write this invariant in a convenient form for us, in order to find some classes of matrices B   such that sign(Σ(DB))=sign(Σ(B))Σ(DB))=sign(Σ(B)) for any diagonal matrix D with positive diagonal elements. Finally, we give three applications of this result concerning three-dimensional population systems.