Hopf bifurcations in a reaction-diffusion model with a general advection term and delay effect

TL;DR

This paper analyzes a reaction-diffusion population model with delay and advection, establishing conditions for Hopf bifurcations and stability of periodic solutions, with applications to ecological models exhibiting Allee effects.

Contribution

It introduces a general framework for analyzing Hopf bifurcations in reaction-diffusion models with delay and advection, including new theoretical results and applications.

Findings

Existence of spatially inhomogeneous steady states near eigenvalues.

Confirmation of Hopf bifurcations through characteristic equation analysis.

Stability and direction of bifurcations determined via normal form theory.

Abstract

This paper investigates a class of reaction-diffusion population models defined on a bounded domain, characterized by a general time-delayed per capita growth rate and a general advection term. Notably, the growth rate encompasses both Logistic-type and weak Allee effect-type dynamical behaviors. By applying the Lyapunov method, we establish the existence of spatially inhomogeneous steady states when a parameter approaches the principal eigenvalue of a non-self-adjoint elliptic operator. A detailed analysis of the characteristic equation further confirms the existence of Hopf bifurcations originating from these steady states. Subsequently, by applying center manifold reduction and normal form theory, we ascertain the direction of these Hopf bifurcations and the stability of the resulting periodic orbits. Finally, the proposed general theoretical results are successfully applied to a "food-limited" population model and a weak Allee effect-driven population model, each of which incorporates diffusion, time delay, and advection, thus confirming the validity of our approach.