Time delays and pollution in an open‐access fishery

Abstract

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.