Understanding the impact of eutrophication on the dynamics of aquatic food webs, remains a long-term challenge in ecal models generally predict the destabilisation of such webs, under increasing eutrophication levels, with large oscillations of species densities that can result in their extinction. This is at odds with a number of ecological observations that show stable dynamics even for high nutrient loads. The apparent discrepancy between theory and observations is known as the Rosenzweig's 'paradox of enrichment' and various solutions to the problem have been proposed over the years. In this study, we explore stabilisation of dynamics of a tri-trophic plankton model in eutrophic environment which occurs as a result of interplay of space heterogeneity, ecological stoichiometry, and food taxis of predators. We build a variety of models of increasing complexity, to explore various scenarios of phytoplankton growth, zooplankton food-dependent vertical movement, and different stoichiometric limitations of zooplankton. We show that the synergy among the vertical gradient in phytoplankton growth, phytoplankton structuring in terms of their stoichiometric ratio, and food-dependent vertical movement of zooplankton, would result in a postponing of destabilisation of eutrophic systems as compared to a well-mixed system. Our approach reveals a high complexity of the bifurcation structure of the system when key model parameters, such as the degree of eutrophication and light shading, are varied. We find coexistence of limit cycles and stable equilibria and that the possibility of multiple attractors in the system can result in hysteresis phenomena when the nutrient load is manipulated. These results are relevant and should be taken into account in lake restoration programs.

中文翻译：

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探索空间和化学计量异质性在富营养化浮游生态系统自上而下控制中的作用。

理解富营养化对水生食物网动态的影响仍然是长期的挑战，ecal模型通常预测随着富营养化水平的提高，这种网的不稳定，物种密度的剧烈波动可能导致其灭绝。这与许多生态观测结果不一致，即使在高营养负荷下，这些生态观测结果也显示出稳定的动态。理论与观察之间的明显差异被称为罗森茨威格的“致富悖论”，多年来，人们提出了各种解决方案。在这项研究中，我们探索了富营养环境中三营养浮游生物模型的动力学稳定性，这是由于空间异质性，生态化学计量和捕食者的食物出租车相互作用而产生的。我们建立了越来越复杂的各种模型，以探索浮游植物生长，浮游动物依赖食物的垂直运动以及浮游动物不同化学计量限制的各种情况。我们显示出浮游植物生长的垂直梯度，按其化学计量比进行浮游植物结构化以及浮游植物的食物依赖性垂直运动之间的协同作用，将导致富营养化系统与稳定混合系统相比不稳定的推迟。当关键模型参数（例如富营养化程度和遮光度）发生变化时，我们的方法表明系统的分叉结构非常复杂。我们发现极限循环和稳定平衡并存，并且在操纵营养负荷时，系统中多个吸引子的可能性会导致磁滞现象。这些结果是相关的，在湖泊恢复计划中应予以考虑。