IDEAS home Printed from https://ideas.repec.org/a/spr/pardea/v3y2022i6d10.1007_s42985-022-00216-w.html
   My bibliography  Save this article

A time-periodic reaction-diffusion population model of a species subjected to harvesting on the boundary of a protected habitat

Author

Listed:
  • K. D. Searle

    (Stellenbosch University
    University of Edinburgh)

  • J. H. Vuuren

    (Stellenbosch University)

Abstract

The objective in this paper is to determine analytically a maximally sustainable pro-rata density-dependent harvest rate of a hypothetical biological species on the spatial boundary of a protected habitat (i.e. no harvesting of the species is allowed in the interior of the spatial domain). This is achieved by analysing an initial-boundary value model involving a reaction-diffusion equation in which the reaction term is a concave function of the population density, depends periodically on time, and varies spatially. The model is equipped with Robin boundary conditions modelling a continuum of potential pro-rata density-dependent harvest rates on the spatial boundary. A set of necessary and sufficient criteria for the existence of a unique positive periodic attractor of solutions to the model is established by employing the theory of comparison principles and monotone iteration schemes. Thereafter, a long-time asymptotic analysis of the population density is undertaken by invoking classical results from the theory of eigenproblems. In this way, the aforementioned necessary and sufficient conditions are rendered more practical in terms of an upper bound on the maximal allowable pro-rata density-dependent harvest rate. Moreover, important properties of the time-periodic attractor of solutions are established to guarantee the existence of a density pro-rata harvest rate which maximises the total harvest per unit time at equilibrium.

Suggested Citation

  • K. D. Searle & J. H. Vuuren, 2022. "A time-periodic reaction-diffusion population model of a species subjected to harvesting on the boundary of a protected habitat," Partial Differential Equations and Applications, Springer, vol. 3(6), pages 1-19, December.
    • Handle: RePEc:spr:pardea:v:3:y:2022:i:6:d:10.1007_s42985-022-00216-w
    DOI: 10.1007/s42985-022-00216-w
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s42985-022-00216-w
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s42985-022-00216-w?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Brown, Katrina & Adger, W. Neil & Tompkins, Emma & Bacon, Peter & Shim, David & Young, Kathy, 2001. "Trade-off analysis for marine protected area management," Ecological Economics, Elsevier, vol. 37(3), pages 417-434, June.
    2. Emamizadeh, Behrouz & Farjudian, Amin & Liu, Yichen, 2018. "Optimal harvesting strategy based on rearrangements of functions," Applied Mathematics and Computation, Elsevier, vol. 320(C), pages 677-690.
    3. Daniel Pauly & Villy Christensen & Sylvie Guénette & Tony J. Pitcher & U. Rashid Sumaila & Carl J. Walters & R. Watson & Dirk Zeller, 2002. "Towards sustainability in world fisheries," Nature, Nature, vol. 418(6898), pages 689-695, August.
    4. Searle, K.D. & van Vuuren, J.H., 2021. "An abstract mathematical model for sustainable harvesting of a biological species on the boundary of a protected habitat," Ecological Modelling, Elsevier, vol. 452(C).
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Searle, K.D. & van Vuuren, J.H., 2021. "An abstract mathematical model for sustainable harvesting of a biological species on the boundary of a protected habitat," Ecological Modelling, Elsevier, vol. 452(C).
    2. Alberto Roca Florido & Emilio Padilla Rosa, 2024. "Analysing the impacts of a reform on harmful fishery subsidies in Spain using a social accounting matrix," Journal of Economic Structures, Springer;Pan-Pacific Association of Input-Output Studies (PAPAIOS), vol. 13(1), pages 1-29, December.
    3. Flückiger, Matthias & Ludwig, Markus, 2015. "Economic shocks in the fisheries sector and maritime piracy," Journal of Development Economics, Elsevier, vol. 114(C), pages 107-125.
    4. Amrullah Rosadi & Paul Dargusch & Taryono Taryono, 2022. "Understanding How Marine Protected Areas Influence Local Prosperity—A Case Study of Gili Matra, Indonesia," IJERPH, MDPI, vol. 19(20), pages 1-19, October.
    5. Tyler D Eddy & Jonathan P A Gardner & Alejandro Pérez-Matus, 2010. "Applying Fishers' Ecological Knowledge to Construct Past and Future Lobster Stocks in the Juan Fernández Archipelago, Chile," PLOS ONE, Public Library of Science, vol. 5(11), pages 1-12, November.
    6. Li, Shangge & Jian, Jinfeng & Poopal, Rama Krishnan & Chen, Xinyu & He, Yaqi & Xu, Hongbin & Yu, Huimin & Ren, Zongming, 2022. "Mathematical modeling in behavior responses: The tendency-prediction based on a persistence model on real-time data," Ecological Modelling, Elsevier, vol. 464(C).
    7. Sara Apresentação & Mafalda Rangel & Assunção Cristas, 2024. "Towards Sustainability: A Framework for Evaluating Portuguese Small-Scale Fisheries," Sustainability, MDPI, vol. 16(8), pages 1-14, April.
    8. Grealis, Eoin & O’Donoghue, Cathal, 2015. "The Economic Impact of the Irish Bio-Economy: Development and Uses," Research Reports 210704, National University of Ireland, Galway, Socio-Economic Marine Research Unit.
    9. Heinze, Alan & Bongers, Frans & Ramírez Marcial, Neptalí & García Barrios, Luis E. & Kuyper, Thomas W., 2022. "Farm diversity and fine scales matter in the assessment of ecosystem services and land use scenarios," Agricultural Systems, Elsevier, vol. 196(C).
    10. Daniel Rainham & Rory Cantwell & Timothy Jason, 2013. "Nature Appropriation and Associations with Population Health in Canada’s Largest Cities," IJERPH, MDPI, vol. 10(4), pages 1-16, March.
    11. Gunnar Brandt & Agostino Merico & Björn Vollan & Achim Schlüter, 2012. "Human Adaptive Behavior in Common Pool Resource Systems," PLOS ONE, Public Library of Science, vol. 7(12), pages 1-9, December.
    12. Felipe Vásquez & Jeanne Simon & Ximena Paz-Lerdon, 2015. "Determining the Feasibility of Establishing New Multiple Use Marine Protected Areas in Chile," Past Working Papers 13, Universidad del Desarrollo, School of Business and Economics, revised 2015.
    13. Sumaila, Ussif R. & Walters, Carl, 2005. "Intergenerational discounting: a new intuitive approach," Ecological Economics, Elsevier, vol. 52(2), pages 135-142, January.
    14. Akpalu, Wisdom & Vondolia, Godwin K., 2012. "Bioeconomic model of spatial fishery management in developing countries," Environment and Development Economics, Cambridge University Press, vol. 17(2), pages 145-161, April.
    15. Lucrezi, Serena & Milanese, Martina & Markantonatou, Vasiliki & Cerrano, Carlo & Sarà, Antonio & Palma, Marco & Saayman, Melville, 2017. "Scuba diving tourism systems and sustainability: Perceptions by the scuba diving industry in two Marine Protected Areas," Tourism Management, Elsevier, vol. 59(C), pages 385-403.
    16. Hajkowicz, Stefan, 2007. "Allocating scarce financial resources across regions for environmental management in Queensland, Australia," Ecological Economics, Elsevier, vol. 61(2-3), pages 208-216, March.
    17. Christina M. Giovas, 2021. "Sustainable Indigenous Fishing in the Pre-Contact Caribbean: Evidence and Critical Considerations from Carriacou, Grenada," Sustainability, MDPI, vol. 13(16), pages 1-15, August.
    18. Dyhia Belhabib & U Rashid Sumaila & Vicky W Y Lam & Dirk Zeller & Philippe Le Billon & Elimane Abou Kane & Daniel Pauly, 2015. "Euros vs. Yuan: Comparing European and Chinese Fishing Access in West Africa," PLOS ONE, Public Library of Science, vol. 10(3), pages 1-22, March.
    19. Stefan Hajkowicz & Kerry Collins, 2007. "A Review of Multiple Criteria Analysis for Water Resource Planning and Management," Water Resources Management: An International Journal, Published for the European Water Resources Association (EWRA), Springer;European Water Resources Association (EWRA), vol. 21(9), pages 1553-1566, September.
    20. Suzanne van Osch & Stephen Hynes & Shirra Freeman & Tim O’Higgins, 2019. "Estimating the Public’s Preferences for Sustainable Aquaculture: A Country Comparison," Sustainability, MDPI, vol. 11(3), pages 1-24, January.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:pardea:v:3:y:2022:i:6:d:10.1007_s42985-022-00216-w. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.