Nonexistence of periodic solutions in delayed Lotka–Volterra systems

Abstract

In this paper we derive su6cient conditions for the nonexistence of nonconstant periodic solutions of Volterra di8erential equations with distributed delays where the delay kernels are chosen among -functions or their suitable convex normalized combinations. The reason of this choice for the kernels is that the Volterra delay di8erential equations can thus be transformed in an expanded system of ordinary di8erential equations by the standard “linear chain trick” method [7]. To this expanded o.d.e. Volterra system we can apply the conditions, encoded by the logarithmic norm of some Jacobian related matrix, that Li and Muldowney [5] have obtained for the nonexistence of (nontrivial) periodic solutions for autonomous ordinary di8erential equations in R , conditions that generalize to the case N ? 2 the Bendixon and Dulac critera. The general structure of the o.d.e. systems obtained from Volterra di8erential delay systems (when the delay kernels are convex normalized combinations of -functions) has been studied, mainly in relation to boundedness and existence of an equilibrium and