Internal phosphorus recycling in lakes is an important nutrient source that promotes algal growth. Its persistence impedes the effort to improve water quality and thus poses a challenge to the management of eutrophication in lakes, especially in shallow lakes where the occurrence of internal phosphorus recycling is reportedly more common. This paper aims to provide crucial insights on the effects of internal phosphorus recycling on eutrophication dynamics for effective management of lake eutrophication. For this purpose, a mathematical model for lake eutrophication, comprising two compartments of algae and phosphorus, is first formulated for application to a eutrophic tropical lake named Tasik Harapan in Universiti Sains Malaysia. Numerical bifurcation analysis of the model is then performed to assess the combined influences of internal phosphorus recycling, algal mortality and external phosphorus loading on Tasik Harapan eutrophication dynamics. Specifically, co-dimension one bifurcation analysis of algal mortality rate is carried out by means of XPPAUT for various external phosphorus loading rates. The emergence of limit cycle for a certain range of algal mortality rate could be related to the hydra effect (i.e., algal concentration increases in response to greater algal mortality) and the paradox of enrichment (i.e., destabilization of algae in nutrient rich environment). To trace the locus of co-dimension one bifurcation, co-dimension two bifurcation analysis is performed by means of MatCont. The analysis demonstrated that the inclusion of the internal phosphorus recycling term induces rich and complex dynamics of the model. These dynamics include saddle-node bifurcation, cusp, Bogdanov-Takens bifurcation, Generalized Hopf bifurcation and limit point bifurcation of cycles. The results suggest that high internal phosphorus recycling rate promotes bistability and catastrophic shift in a shallow and tropical lake like Tasik Harapan. Hence, the key to effective management of eutrophication in shallow and tropical lakes is the control of internal phosphorus recycling.

湖泊内部磷循环是促进藻类生长的重要营养来源。它的持续存在阻碍了改善水质的努力，因此对湖泊富营养化的管理构成了挑战，特别是在据报道内部磷循环更常见的浅湖中。本文旨在提供关于内部磷循环对富营养化动态影响的重要见解，以有效管理湖泊富营养化。为此，首先制定了湖泊富营养化的数学模型，包括藻类和磷的两个部分，以应用于马来西亚理科大学名为 Tasik Harapan 的富营养化热带湖泊。然后对该模型进行数值分岔分析，以评估内部磷循环、藻类死亡率和外部磷负荷对 Tasik Harapan 富营养化动态的综合影响。具体而言，通过XPPAUT对各种外部磷负荷率进行藻类死亡率的一维一分叉分析。一定范围内藻类死亡率极限循环的出现可能与水螅效应（即藻类浓度增加以响应更高的藻类死亡率）和富集悖论（即藻类在营养丰富的环境中不稳定）有关。为了追踪一维一分岔的轨迹，二维二分岔分析是通过 MatCont 进行的。分析表明，包含内部磷循环项会引起模型的丰富而复杂的动态。这些动力学包括鞍节点分岔、尖点、Bogdanov-Takens 分岔、广义 Hopf 分岔和循环的极限点分岔。结果表明，高内部磷循环率促进了像 Tasik Harapan 这样的热带浅湖的双稳态和灾难性转变。因此，有效管理浅水和热带湖泊富营养化的关键是控制内部磷循环。结果表明，高内部磷循环率促进了像 Tasik Harapan 这样的热带浅湖的双稳态和灾难性转变。因此，有效管理浅水和热带湖泊富营养化的关键是控制内部磷循环。结果表明，高内部磷循环率促进了像 Tasik Harapan 这样的热带浅湖的双稳态和灾难性转变。因此，有效管理浅水和热带湖泊富营养化的关键是控制内部磷循环。