The delay and network are incorporated to describe the spatiotemporal behavior of a food-limited population dynamical system. By using the standard approach of upper and lower solutions, we have shown the global existence and uniqueness of solutions to the system. By analyzing eigenvalue spectrum, we show that the delay can cause the long-term behavior of the system from stability to instability, that is, the positive equilibrium is asymptotically stable in the absence of delay, but loses its stability such that the Hopf bifurcation occurs when the time delay increases beyond a threshold. By the norm form and the center manifold theory, we study the stability and direction of the Hopf bifurcation. We propose some formulas to control the stability and period of the bifurcating periodic solutions. Moreover, numerical simulations reveal that the network structure can switch the type of spatiotemporal patterns.

结合了时延和网络来描述食物受限的人口动态系统的时空行为。通过使用上下解决方案的标准方法，我们已经展示了系统解决方案的全局性和唯一性。通过分析特征值谱，我们发现时延会引起系统从稳定性到不稳定的长期行为，即，在没有时延的情况下，正平衡是渐近稳定的，但会失去稳定性，从而发生Hopf分叉当时间延迟超过阈值时。通过范式和中心流形理论，我们研究了Hopf分支的稳定性和方向。我们提出一些公式来控制分支周期解的稳定性和周期。此外，