Stability and bifurcation in a diffusive Lotka–Volterra system with delay

In this paper, we investigate the dynamics of a class of diffusive Lotka–Volterra equation with time delay subject to the homogeneous Dirichlet boundary condition in a bounded domain. The existence of spatially nonhomogeneous steady state solution is investigated by applying Lyapunov–Schmidt reduction. The stability and nonexistence of Hopf bifurcation at the spatially nonhomogeneous steady-state solution with the changes of a specific parameter are obtained by analyzing the distribution of the eigenvalues. Moreover, we illustrate our general results by applications to models with a single delay and one-dimensional spatial domain.