Hopf bifurcation for a delayed diffusive logistic population model in the advective heterogeneous environment

TL;DR

This paper analyzes a delayed reaction-diffusion-advection model for population dynamics in heterogeneous environments, revealing how spatial scale and advection influence Hopf bifurcation and population stability.

Contribution

It introduces a weighted inner product to compute normal forms, highlighting the impact of advection and spatial scale on bifurcation in the model.

Findings

Existence of nonconstant positive steady states.

Hopf bifurcation is influenced by advection and spatial scale.

Weighted inner product method for normal form calculation.

Abstract

In this paper, we investigate a delayed reaction-diffusion-advection equation, which models the population dynamics in the advective heterogeneous environment. The existence of the nonconstant positive steady state and associated Hopf bifurcation are obtained. A weighted inner product associated with the advection rate is introduced to compute the normal forms, which is the main difference between Hopf bifurcation for delayed reaction-diffusion-advection model and that for delayed reaction-diffusion model. Moreover, we find that the spatial scale and advection can affect Hopf bifurcation in the heterogenous environment.