Stability and bifurcation analysis of a reaction–diffusion equation with distributed delay

Dynamics of a general reaction–diffusion equation with distributed delay are considered. The effects of the weak kernel and the strong kernel on the dynamics of the system are both investigated. By analyzing the characteristic equations in detail and taking the average delay as a bifurcation parameter, the stability of the constant equilibrium and the existence of Hopf bifurcations are obtained. The absolute stability and the conditional stability can be explicitly determined by the coefficients of the linearized system. For the case of the strong kernel, the average delay may induce the stability switches, but it is not able to occur for the case of the weak kernel. The algorithm for determining the direction and stability of the bifurcating periodic solutions is derived. Finally, the obtained theoretical results are applied to several single-species models, and the numerical simulations are illustrated to verify the theoretical results.

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