We analyze the impacts of pollution on fishery sector using a dynamical system approach. The proposed model presupposes that the economic development causes emissions that either remediate or accumulate in the oceans. The model possesses a block structure where the solutions of the rate equations for the pollutant and the economic activity act as an input for the biomass and effort equation. We also account for distributed delay effects in both the pollution level and the economic activity level in our modeling framework. The weight functions in the delay terms are expressed in terms of exponentially decaying functions, which in turn enable us to convert the modeling framework to a higher-order autonomous dynamical system by means of a linear chain trick. When both the typical delay time for the economic activity and the typical delay time for the pollution level are much smaller than the biomass time scale, the governing system is analyzed by means of the theory for singularly perturbed dynamical systems. Contrary to what is found for population dynamical systems with absolute delays, we readily find that the impact of the distributed time lags is negligible in the long-run dynamics in this time-scale separation regime.

中文翻译：

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开放式渔业中的时间延迟和污染

我们使用动力系统方法分析了污染对渔业部门的影响。所提出的模型假设经济发展导致排放在海洋中得到修复或积累。该模型具有块结构，其中污染物和经济活动的速率方程的解充当生物量和努力方程的输入。我们还在我们的建模框架中考虑了污染水平和经济活动水平的分布式延迟效应。延迟项中的权重函数以指数衰减函数的形式表示，这反过来又使我们能够通过线性链技巧将建模框架转换为高阶自主动力系统。当经济活动的典型延迟时间和污染水平的典型延迟时间都远小于生物量时间尺度时，利用奇摄动动力系统理论对控制系统进行分析。与对具有绝对延迟的人口动力系统的发现相反，我们很容易发现，在这种时间尺度分离机制的长期动力学中，分布式时间滞后的影响可以忽略不计。