Abstract In this paper, we have proposed a reaction–diffusion system of partial differential equations which model the plankton-nutrient interaction mediated by a toxin-determined functional response. It has been established that microalgae, a clean and green source of energy, can be potentially used for carbon capture and sequestration. The common biofuels (bio-diesel and ethanol) are efficiently extracted from microalgae of different shapes and sizes. A spatio-temporal model has been presented to guide exploration and harvesting of microalgae (e.g., dinoflagellates, cilliates, chlorella, etc.). The spatial distribution of the phytoplankton (microalgae) is determined by growth pattern of the biotic subsystem (phytoplankton and zooplankton); e.g., whether it is oscillatory or aperiodic. The model incorporates a toxin-determined functional response of the zooplankton, which can be parametrized for specific phytoplankton–zooplankton combinations in different aquatic bodies such as ponds, seas, and oceans. The present model does not take into account higher zooplankton’s role in maintaining the core subsystem. The temporal model is analytically investigated in terms of the existence criteria and stability analysis (both linear and nonlinear) of the possible equilibria and the spatio-temporal model is studied in terms of global stability, Turing instability and existence of Hopf-bifurcation which help us to explore the dynamical behavior of the spatial model system. Numerical simulations are carried out to support the obtained theoretical results. Simulation experiments and computed densities thereof (equal densities are codes by same color) suggest that the spatial distribution of microalgae is complex; e.g., spatial density of microalgae varies chaotically for certain parameter sets. Harvesting schedule can be designed based on information thus derived. It should be implemented carefully in case the spatial density distribution is chaotic. The sustainability of the marine system for future use has been the prime concern. Parameters of harvesting strategy (time, intensity and technology) are determined in such a way that exploitation causes minimal damage to the environment and the yield of the harvest is maximal. Future studies would consider larger carnivorous fishes (e.g., Squids, Dolphins) on system’s dynamics. The effect of oceanic noise and colloidal swarming of zooplankton in the presence of bacteria will also be incorporated.

中文翻译：

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海洋系统中微藻的空间分布：反应扩散模型

摘要 在本文中，我们提出了一种偏微分方程的反应-扩散系统，该系统模拟了由毒素决定的功能反应介导的浮游生物-养分相互作用。已经确定，微藻是一种清洁的绿色能源，可用于碳捕获和封存。常见的生物燃料（生物柴油和乙醇）是从不同形状和大小的微藻中有效提取的。已经提出了一种时空模型来指导微藻（例如，甲藻、纤毛虫、小球藻等）的探索和收获。浮游植物（微藻）的空间分布由生物子系统（浮游植物和浮游动物）的生长模式决定；例如，它是振荡的还是非周期性的。该模型结合了浮游动物的毒素确定的功能响应，可以对不同水生体（如池塘、海洋和海洋）中的特定浮游植物-浮游动物组合进行参数化。目前的模型没有考虑到高级浮游动物在维持核心子系统方面的作用。从可能平衡的存在标准和稳定性分析（线性和非线性）方面对时间模型进行了分析研究，并从全局稳定性、图灵不稳定性和 Hopf 分岔的存在性方面研究了时空模型，这有助于我们探索空间模型系统的动力学行为。进行数值模拟以支持获得的理论结果。模拟实验及其计算密度（等密度为同色编码）表明微藻空间分布复杂；例如，对于某些参数集，微藻的空间密度是无序变化的。可以根据由此获得的信息设计收获计划。如果空间密度分布混乱，则应谨慎实施。供未来使用的海洋系统的可持续性一直是首要关注的问题。采伐策略的参数（时间、强度和技术）的确定方式是，开采对环境造成的破坏最小，收获的产量最大。未来的研究将考虑更大的食肉鱼类（例如，鱿鱼、海豚）对系统动力学的影响。