基于离散分数阶微积分的蜘蛛网经济学模型平衡点稳定性研究及应用

蜘蛛网模型本质上是解释某些生产周期较长的商品在失去均衡状态下后发生的不同波动情况的一种动态分析理论。本项目针对现有蜘蛛网模型大多数是连续型的，需求函数和供应函数关于时间和价格的假设都过于粗放，且缺少数值模拟对比实验等缺陷，拟将离散分数阶微积分理论拓展应用到经典蜘蛛网模型中。首先考虑带有离散分数阶导数的蜘蛛网模型，在获得模型的解后，进一步考虑平衡点的稳定性并给出充分条件。另外，考虑到由于现有技术问题而产生的延迟影响，本项目还研究带有时滞的离散分数阶导数的蜘蛛网模型。最后以实际生活中猪肉价格的数据为基础，设计与现有蜘蛛网模型对比实验，借助Matlab等数学软件进行数值仿真模拟，旨通过数值模拟实验验证以上创新成果的有效性，期待对所得到的结果有一个比较直观的认识，直接反应出不同蜘蛛网模型之间的优劣性，从而为经济学上出现的蜘蛛网模型现象解释提供理论依据，同时也将进一步完善蜘蛛网模型领域的研究体系。

The cobweb models are essentially a dynamic analysis theory that explain the different changes that occur after some commodities with reduced production cycles are out of equilibrium. In view of the fact that most of existing cobweb models are continuous, the demand function and supply function's assumptions about time and price are too extensive, the lack of numerical simulation comparison experiments and other defects, this project aims at investigating classical cobweb models with discrete fractional calculus. We first consider a cobweb model with discrete fractional order derivative. After obtaining its solution, we then consider the stability of the equilibrium value as well as give the sufficient condition. In addition, considering the delay effect due to the existing technical problems, we also study a discrete fractional cobweb model with a time delay. Finally, based on the data of pork prices in real life and with the help of mathematical softwares such as Matlab, we design comparison tests with that for the current cobweb models and demonstrate the effectiveness of the above innovation achievements. It is hoped that the results obtained will be more intuitive which directly reflect the pros and cons of different cobweb models. In conclusion, our research works of the proposed project will provide strong theoretical and technical supports for the cobweb phenomenon existing on economics.