In this paper, we focus on a network system which describes spatiotemporal dynamics of single species population at different patches since species can have different features in various life stages and different behaviors in various spatial environments. With the effect of time delay and spatial dispersion, homogenous, periodic and spatiotemporally nonhomogeneous distributions are identified. The stability analysis is carried out for the discrete-space and continuous-time network on single species with time delay and the Hopf bifurcation of the single species population model in a network is explored. Formulas for determining the direction of Hopf bifurcation are derived by using the center manifold method and the normal form theorem. It is found that the network can generate spatial patterns only when time delay is present. Finally, numerical simulations are performed which agree well with our theoretical result, i.e. this discrete-space and continuous-time model admits regular temporal patterns since the delay induces Hopf bifurcations with network structure.

中文翻译：

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延迟单物种网络系统中的 Hopf 分岔

在本文中，我们专注于描述单个物种种群在不同斑块的时空动态的网络系统，因为物种在不同的生命阶段具有不同的特征，在不同的空间环境中具有不同的行为。在时间延迟和空间分散的影响下，识别出均匀的、周期性的和时空非​​均匀的分布。对具有时滞的单个物种的离散空间和连续时间网络进行稳定性分析，探索网络中单个物种种群模型的Hopf分岔。利用中心流形法和范式定理推导了确定Hopf分岔方向的公式。发现只有在存在时间延迟的情况下，网络才能生成空间模式。最后，