{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,1]],"date-time":"2026-06-01T21:44:03Z","timestamp":1780350243170,"version":"3.54.1"},"reference-count":31,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2024,6,26]],"date-time":"2024-06-26T00:00:00Z","timestamp":1719360000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Philosophy and Social Sciences Research Project for Higher Education Institutions in Shanxi Province","award":["2023W061"],"award-info":[{"award-number":["2023W061"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper analyzes the dynamic behavior of a fishery model described by differential algebraic equations. Two patches, namely free fishing area and protected area, are included in the model. The migration of fish is symmetrical, i.e., the fish can migrate between the two patches. It is observed that a singularity-induced bifurcation occurs when the economic benefit of harvesting changes. When the economic benefit is positive, a state feedback controller is added to stabilize the system. Some examples and numerical simulations are presented to verify the theoretical results. In addition, harvesting of prey populations is used as a control measure to obtain the maximum economic benefits and ecological sustainability. The optimal solution is derived by using Pontryagin\u2019s maximum principle. Through extensive numerical simulations, it is shown that the optimal solution is capable of achieving ecosystem sustainability.<\/jats:p>","DOI":"10.3390\/sym16070800","type":"journal-article","created":{"date-parts":[[2024,6,27]],"date-time":"2024-06-27T13:26:43Z","timestamp":1719494803000},"page":"800","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Dynamics and Optimal Harvesting for Fishery Models with Reserved Areas"],"prefix":"10.3390","volume":"16","author":[{"given":"Wenjun","family":"Gao","sequence":"first","affiliation":[{"name":"School of Economics and Management, Shanxi Normal University, Taiyuan 030031, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiu","family":"Jia","sequence":"additional","affiliation":[{"name":"School of Mathematics and Computer Science, Shanxi Normal University, Taiyuan 030031, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ruiqing","family":"Shi","sequence":"additional","affiliation":[{"name":"School of Mathematics and Computer Science, Shanxi Normal University, Taiyuan 030031, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2024,6,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1725","DOI":"10.1016\/j.na.2005.11.049","article-title":"Influence of prey reserve in a prey-predator fishery","volume":"65","author":"Kar","year":"2006","journal-title":"Nonlinear Anal."},{"key":"ref_2","first-page":"195","article-title":"A model for prey-predator fishery with marine reserve","volume":"2","author":"Kar","year":"2007","journal-title":"J. Fish. Aquat. Sci."},{"key":"ref_3","first-page":"2481","article-title":"Analysis of a prey-predator fishery model with prey reserve","volume":"1","author":"Zhang","year":"2007","journal-title":"Appl. Math. Sci."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1142\/S1793524510000982","article-title":"Modeling prey-predator type fishery with reserve area","volume":"3","author":"Ghosh","year":"2010","journal-title":"Int. J. Biomath."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1776","DOI":"10.1080\/00207160.2010.527001","article-title":"Dynamics of fisheries with prey reserve and harvesting","volume":"88","author":"Khamis","year":"2011","journal-title":"Int. J. Comput. Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1016\/j.crvi.2004.10.018","article-title":"Can marine protected areas enhance both economic and biological situations?","volume":"328","author":"Ami","year":"2005","journal-title":"Comptes Rendus Biol."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"111754","DOI":"10.1016\/j.jenvman.2020.111754","article-title":"Fishery management in a marine protected area with compliance gaps: Socio-economic and biological insights as a first step on the path of sustainability","volume":"280","author":"Coppa","year":"2021","journal-title":"J. Environ. Manag."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1016\/j.jenvman.2006.12.001","article-title":"A bioeconomic model of a single-species fishery with a marine reserve","volume":"86","author":"Kar","year":"2008","journal-title":"J. Environ. Manag."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1016\/j.proeng.2017.08.154","article-title":"Mathematical modeling applied to sustainable management of marine resources","volume":"194","author":"Biswas","year":"2017","journal-title":"Procedia Eng."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"212","DOI":"10.1016\/j.mbs.2012.07.008","article-title":"Economic perspective of marine reserves in fisheries: A bioeconomic model","volume":"240","author":"Chakraborty","year":"2012","journal-title":"Math. Biosci."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1007\/s10441-014-9227-7","article-title":"A mathematical model of a fishery with variable market price: Sustainable fishery\/over-exploitation","volume":"62","author":"Balde","year":"2014","journal-title":"Acta Biotheor."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"3048","DOI":"10.1016\/j.apm.2012.07.030","article-title":"A prey-predator model with harvesting for fishery resource with reserve area","volume":"37","author":"Lv","year":"2013","journal-title":"Appl. Math. Model."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1016\/j.fishres.2015.07.020","article-title":"Assessment of catches, landings and fishing effort as useful tools for MPA management","volume":"172","author":"Batista","year":"2015","journal-title":"Fish Res."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"100906","DOI":"10.1016\/j.ecocom.2020.100906","article-title":"A bioeconomic model of a fishery with saturated catch and variable price: Stabilizing effect of marine reserves on fishery dynamics","volume":"45","author":"Moussaoui","year":"2021","journal-title":"Ecol. Complex."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1016\/j.cnsns.2017.05.013","article-title":"Impact of marine reserve on maximum sustainable yield in a traditional prey-predator system","volume":"54","author":"Paul","year":"2018","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1016\/j.marpol.2019.04.011","article-title":"Lobster fishery and marine reserve interactions in central New Zealand","volume":"105","author":"Eddy","year":"2019","journal-title":"Mar. Policy"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1016\/j.matcom.2018.08.009","article-title":"Model-based on fishery management systems with selective harvest policies","volume":"156","author":"Pei","year":"2019","journal-title":"Math. Comput. Simul."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"110200","DOI":"10.1016\/j.chaos.2020.110200","article-title":"Analysis of fractional fishery model with reserve area in the context of time-fractional order derivative","volume":"140","author":"Mansal","year":"2020","journal-title":"Chaos Solitons Fractals"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"8946","DOI":"10.1016\/j.apm.2013.04.024","article-title":"Dynamic analysis of a fractional order prey-predator interaction with harvesting","volume":"37","author":"Javidi","year":"2013","journal-title":"Appl. Math. Model."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"102232","DOI":"10.1016\/j.jocs.2024.102232","article-title":"On the dynamics of a nutrient-plankton system with Caputo and Caputo-Fabrizio fractional operators","volume":"76","author":"Dehingia","year":"2024","journal-title":"J. Comput. Sci."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"100441","DOI":"10.1016\/j.ijoes.2023.100441","article-title":"A comparative study of fractional-order models for supercapacitors in electric vehicles","volume":"19","author":"Zhang","year":"2024","journal-title":"Int. J. Electrochem. Sci."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Ang, T.K., and Safuan, H.M. (2021). Dynamical behaviors and optimal harvesting of an intraguild prey-predator fishery model with Michaelis-Menten type predator harvesting. Biosystems, 202.","DOI":"10.1016\/j.biosystems.2021.104357"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"107992","DOI":"10.1016\/j.cnsns.2024.107992","article-title":"Optimal harvest for predator-prey fishery models with variable price and marine protected area","volume":"134","author":"Chu","year":"2024","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"114902","DOI":"10.1016\/j.chaos.2024.114902","article-title":"Dynamic analysis and optimal control strategies of a predator-prey mathematical model for the pest eradication in oil palm plantation","volume":"183","author":"Zevika","year":"2024","journal-title":"Chaos Solitons Fractals"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.apm.2024.05.007","article-title":"Parameter identification of a reaction-diffusion predator-prey system based on optimal control theory","volume":"133","author":"Miao","year":"2024","journal-title":"Appl. Math. Model."},{"key":"ref_26","first-page":"8778","article-title":"Optimal control of harvest and bifurcation of a prey-predator model with stage structure","volume":"217","author":"Chakraborty","year":"2011","journal-title":"Appl. Math. Comput."},{"key":"ref_27","first-page":"613","article-title":"Bifurcation and control of a bioeconomic model of a prey-predator system with a time delay","volume":"5","author":"Chakraborty","year":"2011","journal-title":"Nonlinear Anal."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1016\/j.mbs.2017.02.004","article-title":"Biological conservation through marine protected areas in the presence of alternative stable states","volume":"286","author":"Ghosh","year":"2017","journal-title":"Math. Biosci."},{"key":"ref_29","unstructured":"Clark, W. (1990). Mathematical Bioeconomics: The Optimal Management of Renewable Resources, Wiley."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1992","DOI":"10.1109\/9.478226","article-title":"Local bifurcations and feasibility regions in differential-algebraic systems","volume":"40","author":"Venkatasubramanian","year":"1995","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_31","unstructured":"Kamien, M.I., and Schwartz, N.L. (1991). Dynamic Optimization: The Calculus of Variations and Optimal Control in Economics and Management, Elsevier."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/7\/800\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:04:44Z","timestamp":1760108684000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/7\/800"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,6,26]]},"references-count":31,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2024,7]]}},"alternative-id":["sym16070800"],"URL":"https:\/\/doi.org\/10.3390\/sym16070800","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,6,26]]}}}