A Class of Matrices Connected with Volterra Prey-Predator Equations

Abstract

If ($p_{ij} $) is a real $n \times n$ matrix, conditions are given which ensure $ ( a_i p_{ij} )\leqq 0$ for some set of positive constants $a_i $, where $( a_i p_{ij} )\leqq 0$ means that the associated quadratic form is nonpositive. Within this class, a companion paper gives an effectively complete solution to the problem of asymptotic stability for n-variable Volterra prey-predator equations; that is why the class is important. Both the algebraic problem considered here and the analytic problem of stability involve a novel blend of topological and combinatorial considerations.