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 etal (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.

在本文中，我们考虑了具有时滞效应的经典两种 Lotka-Volterra 竞争-扩散-平流模型。利用隐函数定理，我们得到了在某些参数条件下至少存在一个空间非齐次正稳态。通过分析相应的特征方程，我们展示了这种空间非均匀正稳态的局部稳定性以及由此产生的Hopf分岔。在没有时间延迟的情况下，我们还研究了正稳态的全局稳定性。基于 Chen et al (2018 J. Differ. Equ. 2645333–5359），Hopf 分岔的稳定性和方向是通过引入与平流率相关的加权内积推导出来的。最后,通过数值模拟对理论分析结果进行验证。