On the Stability of Generalized Volterra Equations with M-Function

Abstract

This paper discusses the stability of a positive equilibrium point with respect to a strictly positive orthant of solutions of the generalized Volterra equation
dx_i/dt=-x_if_i(x) i=1, …, n
which has been used to describe physical, biophysical, chemical or biochemical systems. The population dynamics among interacting biological species and the reaction dynamics in some chemical or biochemical processes are two specific examples.
When f(x)=b+Ax, a sufficient condition for a positive equilibrium point x^*=-A^-1b to be stable is that A is an M-matrix. The authors show that this sufficient condition can be extended for a nonlinear f(x). It is proved that f(x) is an M-function when f(x) is nonlinear with respect to x.