games Article Cooperative Game for Fish Harvesting and Pollution Control Mouhamadou Samsidy Goudiaby 1, * , Ben Mansour Dia 2 , Mamadou L. Diagne 3 and Hamidou Tembine 4   Citation: Goudiaby, M.S.; Dia, B.M.; Diagne, M.L.; Tembine, H. Cooperative Game for Fish Harvesting and Pollution Control. Games 2021, 12, 65. https://doi.org/ 10.3390/g12030065 Academic Editors: Ulrich Berger and Richard McLean Received: 5 July 2021 Accepted: 16 August 2021 Published: 19 August 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 Département de Mathématiques, UFR des Sciences et Technologies, Université Assane Seck de Ziguinchor, Ziguinchor BP 523, Senegal 2 College of Petroleum Engineering and Geosciences (CPG), King Fahd University of Petroleum and Minerals (KFUPM), Dhahran 31261, Saudi Arabia; ben.dia@kfupm.edu.sa 3 Département de Mathématiques, UFR des Sciences et Technologies, Université de Thiés, Thiés BP 967, Senegal; mlamine.diagne@univ-thies.sn 4 Learning & Game Theory Laboratory (L&G-Lab), Division of Engineering, Saadiyat Campus, New York University Abu Dhabi (NYUAD), Abu Dhabi P.O. Box 129188, United Arab Emirates; tembine@nyu.edu * Correspondence: msgoudiaby@univ-zig.sn; Tel.: +221-76-902-81-60 Abstract: This paper studies fishery strategies in lakes, seas, and shallow rivers subject to agricultural and industrial pollution. The flowing pollutants are modeled by a nonlinear differential equation in a general manner. The logistic growth model for the fish population is modified to cover the pollution impact on the fish growth rate. We start by presenting the stability analysis of the dynamical system to discern the different types of the evolution of the fish population according to human actions. A cooperative game is formulated to design strategies for preserving the fish population by controlling the pollution as well as the fish stock for harvesting. The sufficient conditions for implementing the cooperative strategy are investigated through an incentive design approach with an adaptive taxation policy for the players. Numerical results are presented to illustrate the benefit of the cooperative for fish population preservation but also for the players’ rewards. Keywords: biodiversity preservation; stability analysis; optimal control; cooperative game 1. Introduction Prior to the 2019 coronavirus disease (COVID-19) pandemic, towns and cities had grown and attracted more than fifty percent of the world population. Their sustainability and competitive efficiency have become increasingly important issues to decision-makers. A city, more simply, is a geographic area where live people, with hard infrastructures and changing systems that interconnect the city with itself. Because of pandemics, economic drivers, energy, climate change, and resource supply, the concept of biodiversity preser- vation has gained more attention for the next generation of cities, called smart cities. The “smart city” can include a large and extensive range of systems, networks, and infras- tructure elements. However, in this paper we use the term “smart city” to refer to more intelligent, more efficient, and more sustainable cities, and we focus on pollution control and economic supply. In support of the environment, risk management and sustainable de- velopment; natural resource management including systems for the reduction of pollutants; increasing energy efficiency; managing the human response to environmental stresses, and sustaining biodiversity are key elements of smart cities. Seas and lakes provide a variety of economic uses: they are sources of drinking water, fishing, and recreational sports, and they provide pleasant locations for smart homes and drainage for agriculture. Runoff from fields flows into the streams and rivers that feed the lake. Much of the agricultural runoff includes phosphorus from fertilizers and animal wastes. Phosphorus is the primary nutrient of algae and weeds in the water. When it grows excessively from an infusion of phosphorus, algae blooms reduce the oxygen content of the lake and release toxins into the water. In addition, large amounts of waste produced by some industries and municipal sewage enter natural water bodies, either without treatment Games 2021, 12, 65. https://doi.org/10.3390/g12030065 https://www.mdpi.com/journal/games