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

Author

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  • 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
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    References listed on IDEAS

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    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. 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).
    4. 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.
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