A short proof for Hopf bifurcation in Gurtin-MacCamy's population dynamics model

TL;DR

This paper offers a concise proof of the Hopf bifurcation theorem in Gurtin-MacCamy's population model using the implicit function theorem, simplifying previous approaches and requiring only mild smoothness conditions.

Contribution

It introduces a shorter, more efficient proof for Hopf bifurcation in the model, leveraging the Crandall-Rabinowitz approach with minimal smoothness assumptions.

Findings

Shorter proof of Hopf bifurcation theorem

Requires only bounded variation smoothness

Significant reduction in proof length

Abstract

In this paper, we provide a short proof for the Hopf bifurcation theorem in the Gurtin-MacCamy's population dynamics model. Here we use the Crandall and Rabinowitz's approach, based on the implicit function theorem. Compared with previous methods, here we require the age-specific birth rate to be slightly smoother (roughly of bounded variation), but we have a huge gain for the length of the proof.