Hopf bifurcation of a delayed single population model with patch structure

TL;DR

This paper investigates how dispersal rates influence the Hopf bifurcation in a delayed single population model with multiple patches, revealing limits as dispersal tends to zero or infinity.

Contribution

It demonstrates the effects of dispersal rate variations on Hopf bifurcation points in a structured population model, including limiting behaviors.

Findings

Hopf bifurcation exists in the model with delay and patch structure.

As dispersal rate approaches zero, bifurcation value approaches the minimum local value.

As dispersal rate approaches infinity, bifurcation value approaches the average model's value.

Abstract

In this paper, we show the existence of Hopf bifurcation of a delayed single population model with patch structure. The effect of the dispersal rate on the Hopf bifurcation is considered. Especially, if each patch is favorable for the species, we show that when the dispersal rate tends to zero, the limit of the Hopf bifurcation value is the minimum of the "local" Hopf bifurcation values over all patches. On the other hand, when the dispersal rate tends to infinity, the Hopf bifurcation value tends to that of the "average" model.