Laboratory of Nonlinear Analysis and Applied Mathematics, Department of Mathematics, University of Tlemcen, Algeria.
Normandie Univ, UNIHAVRE, LMAH, FR-CNRS-3335, ISCN, 76600 Le Havre, France.
Math Biosci. 2023 Aug;362:109022. doi: 10.1016/j.mbs.2023.109022. Epub 2023 May 29.
We propose a bio-economic model of a fishery describing the variations of the fish stock, the fishing effort and the price of the resource on the market supposed to depend on supply and demand. The originality of this model comes from taking into account the storage of part of the resource for a certain time before being put up for sale on the market. Taking into account the supposedly fast price dynamics compared to the other mechanisms involved and after integration of the stock equation, the system is reduced to a system of two delayed differential equations. The qualitative analysis of the model is carried out with the search for equilibrium points and the study of their stability. The study shows the existence of a catastrophic equilibrium corresponding to the extinction of the resource and one or two sustainable fishery equilibrium points that can coexist under certain conditions. The model shows that storing part of the resource makes it possible to avoid a catastrophic situation with the extinction of the fish stock and to stabilize the fishery in the long term. The study also shows that the price variation of the resource has a stabilizing effect by avoiding the appearance of periodic solutions associated with a stable limit cycle surrounding a sustainable fishery equilibrium point resulting from a Hopf bifurcation, which is contrary to the case without price variation where this is possible.
我们提出了一个渔业的生物经济模型，描述了鱼群、捕捞努力和市场上资源价格的变化，这些变化预计取决于供求关系。该模型的新颖之处在于考虑了部分资源在投放市场前储存一段时间。考虑到与其他机制相比，价格的动态变化应该相对较快，并且在整合了库存方程之后，该系统被简化为一个具有两个时滞微分方程的系统。通过寻找平衡点并研究其稳定性，对模型进行了定性分析。研究表明，存在一个对应于资源灭绝的灾难性平衡点，以及在某些条件下可以共存的一个或两个可持续渔业平衡点。该模型表明，储存部分资源可以避免鱼群灭绝的灾难性情况，并在长期内稳定渔业。研究还表明，资源价格的变化通过避免与稳定的可持续渔业平衡点相关的周期性解的出现来产生稳定作用，这是由于Hopf 分岔导致稳定的极限环围绕可持续渔业平衡点出现，而在没有价格变化的情况下，这种情况是可能的。
