Global climate change affects marine species including phytoplankton, which constitute the base of the marine food web, by changing the primary productivity. Global warming affects the ocean surface temperature, in turn leading to a change in the oxygen production of phytoplankton. In this work, the fractional oxygen–phytoplankton–zooplankton mathematical model is considered by the Caputo fractional operator. The production rate of photosynthesis is determined by a temperature function. The model is, therefore, based on the idea that the rate of photosynthesis changes due to the impact of global warming, while phytoplankton oxygen production increases and decreases. We analyze the model with the Caputo fractional derivative differently from the classical case of the model and we compare the results with the integer order derivative when $\alpha$ tends to 1. Existence and uniqueness properties of the oxygen–plankton model have been proved by means of a local Lipschitz condition. It was shown that the species are more sustainable than its corresponding classical case in the Caputo model. Our results show that the effect of global warming on the oxygen production rate has been observed to be quite severe, resulting in oxygen depletion and plankton extinction.

中文翻译：

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气候变化下的分数阶氧-浮游生物系统

全球气候变化通过改变初级生产力来影响包括浮游植物在内的海洋物种，而浮游植物是构成海洋食物网的基础。全球变暖影响海洋表面温度，进而导致浮游植物的产氧量发生变化。在这项工作中，Caputo分数运算符考虑了分数氧气-浮游植物-浮游动物的数学模型。光合作用的生产率取决于温度函数。因此，该模型基于这样的思想：光合作用的速率由于全球变暖的影响而改变，而浮游植物的氧气产量则增加和减少。我们使用Caputo分数导数分析模型的方法与模型的经典情况不同，并且将结果与整数阶导数进行比较$\alpha$趋向于1.浮游生物模型的存在性和唯一性已通过局部Lipschitz条件得到了证明。结果表明，该物种比Caputo模型中相应的经典案例更具可持续性。我们的结果表明，已经观察到全球变暖对氧气产生速率的影响非常严重，导致氧气消耗和浮游生物灭绝。