基于微分代数方程的随机生物经济系统建模与复杂性分析

The existing regulation and control of biological resource based on singular system models mostly assumes that the system operation environment is deterministic. The real problem is not the case, the random factors are everywhere. For example, climate, temperature, sudden disease has great influence on the development of biological system, regulation and control will. At the same time, stochastic differential equations have been widely applied in biological field described by the normal system model. However, this kind of model considered less on between some physical quantity constraint. This project attempts to use stochastic differential equations and stochastic algebraic equations combined with describing the problem of regulation and control of biological system. The utility models have the advantages of both considers random factors and consideration between some physical quantity constraint. In the aspect of system analysis, the stochastic stability, stochastic bifurcation, impulse on system are researched. The key and difficulty of the project is the analysis of stochastic bifurcation and impulse phenomena in singular stochastic biological economy systems. The successful implementation of the project will provide new ways and methods for the prediction of the trend of the change of biological populations and the early prediction of emergencies, to ensure the sustainable survival of biological populations and the steady development of economic benefits. It will also have a positive impact on the improvement of the modeling method on exploitation of biological resources, stochastic stability, stochastic bifurcation and impulse theory of singular stochastic systems.

现有基于广义系统模型的生物资源的开发调节问题大多假定系统运行的环境是确定性的。实际问题并非如此，随机因素无处不在，例如环境、气候、突发疾病等对生物系统的开发、调节会有非常大的影响。与此同时,随机微分方程在正常系统模型描述的生物领域得到了大量应用。然而,这类模型对某些物理量之间的约束考虑较少。本项目试图利用随机微分方程和随机代数方程联合描述生物资源的开发调节问题，建立广义随机生物经济模型，其优点是既考虑随机因素又考虑某些物理量之间的约束。在系统分析方面，研究系统的随机稳定性、随机分岔、脉冲等问题。本项目的难点和关键问题是广义随机生物经济系统中随机分岔和脉冲现象的研究。该项目的成功实施，将为生物种群的变化趋势的预测、突发事件的早期预测提供新的途径和方法，确保生物种群持续生存和人们经济效益稳定发展。同时也将对生物资源开发的建模方法、广义随机系统稳定性、分岔及脉冲理论的完善产生积极影响。

已有基于广义系统模型的生物资源的开发调节问题大多假定系统运行的环境是确定性的。实际问题并非如此，随机因素无处不在，例如环境、气候、突发疾病等对生物系统的开发、调节会有非常大的影响。与此同时,随机微分方程在正常系统模型描述的生物领域得到了大量应用。然而,这类模型对某些物理量之间的约束考虑较少。本项目利用随机微分方程和随机代数方程联合描述生物资源的开发调节问题。建立广义随机捕食-被捕食生物经济模型、广义随机竞争生物经济模型、具有时滞的随机竞争生物经济模型、具有阶段结构的随机竞争生物经济模型。其优点是既考虑随机因素又考虑某些物理量之间的约束，而且考虑种群成长时滞对种群发展的影响。在系统分析方面，研究系统的随机稳定性、随机分岔、种群的持久生存性和灭绝性等问题。利用最优控制理论，研究了最优价格控制策略及对种群的最优捕获策略。研究成果为生物种群的变化趋势的预测、突发事件的早期预测提供新的途径和方法，为生物种群的合理开发提供理论支撑，确保生物种群持续生存和人们经济效益稳定发展。同时也将对生物资源开发的建模方法、广义随机系统稳定性、分岔理论的完善产生积极影响。此外本项目利用随机微分方程理论，建立两类随机传染病模型，给出疾病灭绝和持久存在的条件。研究成果为传染病的预防和控制提供理论方法。