The periodically forced light-limited Droop model represents microalgae growth under co-limitation by light and a single substrate, accounting for periodic fluctuations of factors such as light and temperature. In this paper, we describe the global dynamics of this model, considering general monotone growth and uptake rate functions. Our main result gives necessary and sufficient conditions for the existence of a positive periodic solution i.e. a periodic solution characterized by the presence of microalgae) which is globally attractive. In our approach, we reduce the model to a cooperative planar periodic system. Using results on periodic Kolmogorov equations and on monotone sub-homogeneous dynamical systems, we describe the global dynamics of the reduced system. Then, using the theory of asymptotically periodic semiflows, we extend the results on the reduced system to the original model. To illustrate the applicability of the main result, we include an example considering a standard microalgae population model.

中文翻译：

---

周期性强制光限制下垂模型的动力学

周期性强制光限制下垂模型表示微藻在光和单一基质共同限制下的生长，考虑了光和温度等因素的周期性波动。在本文中，我们描述了该模型的全局动态，考虑了一般的单调增长和吸收率函数。我们的主要结果给出了存在正周期解的充分必要条件，即以微藻存在为特征的周期解，它具有全局吸引力。在我们的方法中，我们将模型简化为协作平面周期系统。使用周期性 Kolmogorov 方程和单调次齐次动力系统的结果，我们描述了简化系统的全局动力学。然后，利用渐近周期半流理论，我们将缩减系统的结果扩展到原始模型。为了说明主要结果的适用性，我们包括一个考虑标准微藻种群模型的例子。