An abstract mathematical model for sustainable harvesting of a biological species on the boundary of a protected habitat

K.D. Searle and J.H. van Vuuren

Ecological Modelling, 2021, vol. 452, issue C

Abstract: The objective in this paper is to determine analytically the maximally sustainable pro-rata density-dependent harvest rate of a hypothetical biological species on the spatial boundary of its habitat, which is otherwise a protected zone (i.e. no harvesting of the species is allowed in the interior of its habitat). This is achieved by analysing an abstract mathematical model for the spatio-temporal evolution of the species density over its habitat if it is subjected to a continuum of potential pro-rata density-dependent harvest rates on the spatial boundary. The model takes the form of an initial-boundary value problem involving a reaction-diffusion equation in which the reaction term is a concave function of the population density and Robin boundary conditions are prescribed. A long-time asymptotic analysis of the population density is undertaken by invoking classical results from the theory of eigenproblems. In this way, necessary and sufficient conditions on the pro-rata density-dependent harvest rate are established for the existence of a strictly positive equilibrium attractor of model solutions. Moreover, important necessary properties of this equilibrium attractor are established to guarantee the existence of a density pro-rata harvest rate which maximises the total harvest per unit time at equilibrium.

Keywords: Optimal yield; Population dynamics; Reaction-diffusion equation; Protection zone; Maximum sustainable yield (search for similar items in EconPapers)
Date: 2021
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Citations: View citations in EconPapers (2)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:ecomod:v:452:y:2021:i:c:s030438002100154x

DOI: 10.1016/j.ecolmodel.2021.109591

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