In this paper, we study a mathematical bio-economic model of a fishery with varying price. The three dimensional model describes the time evolution of the resource, the fishing effort and the price. The model is original because it considers a nonlinear harvesting function assumed to depend upon stock size and fishing effort with a saturation effect with respect to the resource as well as a price equation depending on demand and supply which is in addition proportional to price. Assuming that the price varies at a fast time scale, we are able to use ”aggregation of variables methods” in order to reduce the model in a two dimensional model at a slow time scale. This aggregated (reduced) model is analyzed. Several numerical simulations of the model are performed to substantiate our analytical findings. The existence of nonlinear harvesting makes the dynamics of the model more complicated, including multiple equilibria, bi-stability and limit cycle. Such large amplitude cycle variations are not desirable because they generate periods of overfishing at periods of very low activity. We then study the effects of marine reserves on the dynamics of the fishery, showing that for an adequate number of small reserves, limit cycle oscillations are switched off.

在本文中，我们研究了价格变化的渔业的数学生物经济模型。三维模型描述了资源的时间演变，捕捞努力和价格。该模型是原始模型，因为它考虑了一种非线性捕捞函数，该函数假定取决于种群数量和捕捞努力，并且对资源具有饱和效应，并且取决于依赖于供需的价格方程，该方程与价格成正比。假设价格在快速的时间尺度上变化，我们能够使用“变量汇总法”，以便在较慢的时间尺度上将模型缩小为二维模型。分析该汇总（简化）模型。对模型进行了几次数值模拟，以证实我们的分析结果。非线性收获的存在使模型的动力学更加复杂，包括多重平衡，双稳定性和极限环。如此大的振幅周期变化是不希望的，因为它们会在活动非常低的时间段产生过度捕捞的时间段。然后，我们研究了海洋保护区对渔业动态的影响，表明对于足够数量的小型保护区，极限循环振荡被关闭。