Department of Mathematics, University of Louisiana at Lafayette, 1401 Johnston Street, Lafayette, LA, 70504, USA.
Faculty of Life Sciences, School of Zoology, Tel Aviv University, Tel Aviv-Yafo, Israel.
J Math Biol. 2023 Aug 17;87(3):45. doi: 10.1007/s00285-023-01969-7.
Nutrient-Phytoplankton-Zooplankton-Detritus (NPZD) models, describing the interactions between phytoplankton, zooplankton systems, and their ecosystem, are used to predict their ecological and evolutionary population dynamics. These organisms form the base two trophic levels of aquatic ecosystems. Hence understanding their population dynamics and how disturbances can affect these systems is crucial. Here, starting from a base NPZ modeling framework, we incorporate the harmful effects of phytoplankton overpopulation on zooplankton-representing a crucial next step in harmful algal bloom (HAB) modeling-and split the nutrient compartment to formulate an NPZD model. We then mathematically analyze the NPZ system upon which this new model is based, including local and global stability of equilibria, Hopf bifurcation condition, and forward hysteresis, where the bi-stability occurs with multiple attractors. Finally, we extend the threshold analysis to the NPZD model, which displays both forward hysteresis with bi-stability and Hopf bifurcation under different parameter regimes. We also examine ecological implications after incorporating seasonality and ecological disturbances. Ultimately, we quantify ecosystem health in terms of the relative values of the robust persistence thresholds for phytoplankton and zooplankton and find (i) ecosystems sufficiently favoring phytoplankton, as quantified by the relative values of the plankton persistence numbers, are vulnerable to both HABs and (local) zooplankton extinction (ii) even healthy ecosystems are extremely sensitive to nutrient depletion over relatively short time scales.
营养-浮游植物-浮游动物-碎屑（NPZD）模型描述了浮游植物、浮游动物系统及其生态系统之间的相互作用，用于预测它们的生态和进化种群动态。这些生物构成了水生生态系统的两个基本营养级。因此，了解它们的种群动态以及干扰如何影响这些系统是至关重要的。在这里，我们从一个基本的 NPZ 建模框架出发，将浮游植物过度繁殖对浮游动物的有害影响纳入其中，这代表了有害藻类水华（HAB）建模的重要下一步，并将营养物质隔室分开，以制定 NPZD 模型。然后，我们对基于该新模型的 NPZ 系统进行了数学分析，包括平衡点的局部和全局稳定性、Hopf 分岔条件和前向滞后，其中双稳定性与多个吸引子同时存在。最后，我们将阈值分析扩展到 NPZD 模型，该模型在不同参数条件下表现出前向滞后和双稳定性以及 Hopf 分岔。我们还研究了季节性和生态干扰纳入后的生态影响。最终，我们根据浮游植物和浮游动物的稳健持久性阈值的相对值来量化生态系统的健康状况，并发现（i）生态系统对浮游植物的偏好程度足够高，即浮游生物持久性数量的相对值较高，容易受到 HAB 和（局部）浮游动物灭绝的影响（ii）即使是健康的生态系统对相对较短时间尺度的营养物质耗尽也非常敏感。
