Slowly oscillating periodic solutions in a nonlinear Volterra equation with non-symmetric feedback

TL;DR

This paper investigates the existence of slowly oscillating periodic solutions in a nonlinear Volterra equation with non-symmetric feedback, relevant to population dynamics, and demonstrates convergence to a discrete difference equation in a singular limit.

Contribution

It introduces a novel approach to prove the existence of slowly oscillating solutions without symmetry assumptions, using a homeomorphism and fixed-point theory.

Findings

Existence of slowly oscillating periodic solutions established.

Periodic solutions converge to a discrete difference equation in a singular limit.

Numerical simulations support theoretical results.

Abstract

In this work we study a nonlinear Volterra equation with non-symmetric feedback that arises as a particular case of the Gurtin-MacCamy model in population dynamics. We are particularly interested in the existence of slowly oscillating periodic solutions when the trivial stationary state is unstable. Here the absence of symmetry of the nonlinearity prevents the use of many traditional strategies to obtain a priori estimates on the solution. Without a precise knowledge of the period of the solution, we manage to prove the forward invariance of a carefully constructed set of initial data whose properties imply the slowly oscillating character of all continuations. We prove the existence of periodic solutions by constructing a homeomorphism between our set and a convex subset of a different Banach space, thereby showing that it possesses the fixed-point property. Finally, in a singular limit of a parameter, we show that this periodic solution converges to the solution of a wellknown discrete difference equation. We conclude the paper with some numerical simulations to illustrate the existence of the periodic orbit as well as the singular limit behavior.