Stability and bifurcation analysis of a reaction–diffusion equation with spatio-temporal delay

The spatio-temporal patterns of a general reaction–diffusion equation with non-local delay are considered. By analyzing the characteristic equations and applying the complex plane theory, the stability of the constant steady state and the possible Hopf bifurcations are obtained with the mean delay as a bifurcation parameter. The absolute stable, conditional stable and unstable region can be explicitly divided by the coefficients of the term without delay and the term with non-local delay. By investigating the effect of any weight function on the dynamics of the system, we establish the existence of spatially homogeneous and inhomogeneous bifurcating periodic solutions. And the algorithm for determining the direction and stability of Hopf bifurcation is derived by computing the normal form on the center manifold. The result shows that, for the case of strong kernel, the average delay may induce the stability switches. Finally, the theoretical results are applied to a diffusive food-limited population model. Numerical simulations show the existence of the orbit connecting the unstable spatially inhomogeneous periodic solutions to stable spatially homogeneous periodic solutions.