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.

本文的目的是通过分析确定在其栖息地的空间边界上假设生物物种的最大可持续按比例分布密度变化的收获率，否则 该物种将成为保护区（即，不允许在该物种的区域中进行该物种的收获）。栖息地的内部）。这是通过分析一个抽象数学模型来实现的，如果该模型受到栖息地潜在比例的连续性影响，那么该物种在其栖息地上的时空演化在空间边界上依赖密度的收割率。该模型采用包含反应扩散方程的初始边界值问题的形式，其中反应项是总体密度的凹函数，并规定了Robin边界条件。通过调用本征问题理论的经典结果，进行了人口密度的长期渐近分析。通过这种方式，为模型解存在严格的正平衡吸引子，建立了与比例密度相关的收获率的必要条件和充分条件。此外，建立了该平衡吸引子的重要必要特性，以保证存在密度比例 平衡时可使单位时间的总收获最大化的收获率。