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 Sigma(B) = (trace of B) x (sum of principal minors of B) - (determinant of B), 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(Sigma(DB)) = sign(Sigma(B)) tor any diagonal matrix D with positive diagonal elements. Finally, we give three applications of this result concerning three-dimensional Population systems. (C) 2007 Elsevier Ltd. All rights reserved.