Analysis of stability and bifurcation and design of optimal control for phytoplankton–zooplankton–fish model with additional food and fish harvesting

This paper investigates the dynamics of a four-dimensional plankton-fish system that contains three communities of plankton: non-toxic phytoplankton, toxic-producing phytoplankton, and zooplankton. The zooplankton consumes both non-toxic and toxic-producing phytoplankton. First, the positivity and boundedness of the solutions in the plankton-fish model are evaluated. Then, we derive the conditions for the existence of ecologically possible equilibria and their local and global asymptotic nature. Furthermore, we evaluate the criteria for the existence of Hopf-bifurcation and transcritical bifurcation. By applying Pontryagin’s maximum principle, we investigate the optimal control of fish harvesting, where fishing effort is taken as a control parameter. By using phase-space diagrams, time responses, bifurcation diagrams, and numerical analyses validate the theoretical results. The numerical results including maximum Lyapunov exponent verify chaotic dynamics of the system. We notice that a suitable choice of harvesting, phytoplankton growth rate, and additional food parameters can control the chaotic behavior of the presented model. The proposed novel mathematical model gives deeper insights to theoretical ecologists on how toxic-producing phytoplankton affects zooplankton and fish populations in aquatic systems. Furthermore, the proposed model is more appropriate for fish harvesting to maximize profit as well as species conversion in the presence of toxic phytoplankton.

本文针对包含三类浮游生物群落的四维浮游生物-鱼类系统展开研究：三类浮游生物分别为无毒浮游植物、产毒浮游植物以及浮游动物，其中浮游动物可同时摄食无毒与产毒浮游植物。首先，本文分析该浮游生物-鱼类模型解的正定性与有界性；随后推导生态可行平衡点的存在条件及其局部与全局渐近性质。此外，本文还探讨了霍普夫分支（Hopf-bifurcation）与跨临界分支（transcritical bifurcation）的存在判据。借助庞特里亚金极大值原理（Pontryagin’s maximum principle），本文以捕捞努力量为控制参数，研究鱼类捕捞的最优控制问题。通过相空间图、时间响应曲线、分支图与数值分析，对理论结果进行验证。包括最大李雅普诺夫指数（maximum Lyapunov exponent）在内的数值结果，证实了该系统的混沌动力学特性。研究发现，通过合理选取捕捞强度、浮游植物生长率与额外食物参数，可调控所提模型的混沌行为。本文提出的新型数学模型，可为理论生态学家深入理解产毒浮游植物如何影响水生系统中的浮游动物与鱼类种群提供参考。此外，在存在有毒浮游植物的场景下，该模型更适用于鱼类捕捞优化，以实现利润最大化与种群转化的目标。