Stability and Hopf bifurcation analysis in a Lotka–Volterra competition–diffusion–advection model with time delay effect *

Abstract In this paper, we consider a classical two-species Lotka–Volterra competition–diffusion–advection model with time delay effect. By utilizing the implicit function theorem, we obtain the existence of at least one spatially nonhomogeneous positive steady state under some conditions on parameters. By analyzing the corresponding characteristic equation, we show the local stability of this spatially nonhomogeneous positive steady state and the occurrence of Hopf bifurcation from it. When there is no time delay, we also study the global stability of the positive steady state. Based on the idea of Chen et al (2018 J. Differ. Equ. 264 5333–5359), the stability and direction of Hopf bifurcation are derived by introducing a weighted inner product associated with the advection rate. Finally, numerical simulations are carried out to verify the theoretical analysis results.