Abstract In view of time delay in the transport of nutrients, a delayed reaction-diffusion system with homogeneous Neumann boundary conditions is presented to understand the formation of the heterogeneous distribution of bacteria and nutrients in the sediment. With the effects of time delay and diffusion, the system will experience various dynamical behaviors, such as stability, the Turing instability, successive switches of stability of equilibria, the Hopf and the Hopf-Hopf bifurcations. To further understand the dynamics of the Hopf-Hopf bifurcation, the multiple time scale (MTS) technique is employed to derive the amplitude equations at this co-dimensional bifurcation point, and the dynamical classification near such bifurcation point is also identified by analyzing the obtained amplitude equations. Some numerical simulations are carried out to demonstrate the validity of the theoretical analysis.

中文翻译：

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延迟养分-微生物模型中的 Hopf-Hopf 分岔

摘要 针对养分传输的时间延迟问题，提出了一种具有均质Neumann边界条件的延迟反应-扩散系统，以了解沉积物中细菌和养分异质分布的形成。在时间延迟和扩散的影响下，系统将经历各种动力学行为，如稳定性、图灵不稳定性、平衡稳定性的连续切换、Hopf 和 Hopf-Hopf 分岔。为进一步了解Hopf-Hopf分岔的动力学，采用多时间尺度（MTS）技术推导出该共维分岔点处的振幅方程，并通过分析得到的该分岔点附近的动力学分类进行识别幅度方程。