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

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.