Abstract
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 ) = ( trace of B ) × ( 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( 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.
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