In this paper, we consider a three species plankton–fish system that incorporates external toxicity and nonlinear harvesting. We consider that the growth of species are affected directly or indirectly by an external toxic substance and the feeding of the predator on the affected prey is considered as Holling type II functional response. All the possible biological feasible equilibrium points are determined analytically as well as numerically and performed stability analysis around these equilibrium points. It is shown that the system undergoes for Hopf bifurcation when the growth rate of prey passes some threshold value. Furthermore, Pontryagin’s maximum principle has been applied to obtain optimal control of harvesting to maximize the benefit as well as the conservation of the ecosystem. We perform numerical simulations to justify and illustrate our analytical results. Some numerical tools such as phase portraits, time evaluation and bifurcation diagrams are presented to ensure the complex dynamics in the system. Period doubling cascade route to chaos is examined and validated through the numerical calculation of Lyapunov exponents and sensitivity analysis.

在本文中，我们考虑了具有外部毒性和非线性收获的三类浮游生物-鱼类系统。我们认为，物种的生长受到外部有毒物质的直接或间接影响，捕食被捕食的猎物被认为是Holling II型功能性反应。通过分析以及数值确定所有可能的生物学可行平衡点，并围绕这些平衡点进行稳定性分析。结果表明，当猎物的生长速度超过某个阈值时，该系统将经历霍普夫分支。此外，庞特里亚金的最大原则已被应用于获得最佳的收获控制，以最大程度地提高生态系统的效益和养护。我们进行数值模拟以证明和说明我们的分析结果。提出了一些数字工具，例如相图，时间评估和分叉图，以确保系统中的复杂动态。通过Lyapunov指数的数值计算和敏感性分析，研究并验证了倍增的到达混沌的级联路径。