一类污染环境中的广义生物经济系统建模与控制

发布时间：2018-05-22 21:08

本文选题：广义生物经济系统 + 奇异诱导分岔　；参考：《沈阳工业大学》2015年硕士论文

【摘要】：环境污染对生物系统的影响越来越严重，为了更好地研究物种的发展和变化规律，建立生物数学模型时考虑污染因素十分必要。同时，时滞也在实际模型中普遍存在，一个很小的时滞可能会造成系统崩溃，，在考虑受时滞影响的生物系统时，则需要进一步借助时滞微分方程来描述系统。本文主要以广义系统理论、经济学原理、奇异诱导分岔理论、Hopf分岔理论、时滞系统理论为基础，对一类污染中的广义生物经济系统进行了研究。本文首先介绍了环境污染给生物种群带来的影响，广义生物系统和时滞生物系统的研究意义及研究现状，并由此提出本文所研究问题，给出本文所用到的基础知识。其次，建立了受毒素影响的广义生物经济模型，利用系统的稳定性理论和Routh-hurwitz定理对系统在平衡点处的稳定性进行分析，针对系统的分岔现象设计状态反馈控制器消除分岔，并使用实际例子参数，通过仿真得到施加控制器后的状态响应图，说明所得控制器可以对系统进行有效控制。最后，研究了一类受时滞因素和污染因素影响的广义生物经济模型，通过对时滞广义系统做适当的线性变换，并使用参数化等方法将时滞广义系统转换成一般的时滞微分系统，对所得微分时滞系统在平衡点处的稳定性进行分析，得出系统在正平衡点处出现Hopf分岔，推导出相关参数，说明Hopf分岔的稳定性和方向性，并通过数值仿真验证本文所得结论的合理性。
[Abstract]:The influence of environmental pollution on biological systems is becoming more and more serious. In order to better study the development and variation of species, it is necessary to consider the pollution factors in the establishment of biological mathematical models. At the same time, time delay also exists in the real model. A small delay may cause the system collapse. When considering the biological system affected by the delay, it is necessary to describe the system with delay differential equation. On the basis of generalized system theory, economic principle, singular induced bifurcation theory, Hopf bifurcation theory and time-delay system theory, a class of generalized bio-economic systems in pollution is studied in this paper. In this paper, the effects of environmental pollution on biological population, the significance and current research status of generalized biological systems and time-delay biological systems are introduced, and the problems and basic knowledge used in this paper are presented. Secondly, the generalized bio-economic model affected by toxin is established. The stability of the system at the equilibrium point is analyzed by using the stability theory of the system and Routh-hurwitz theorem, and the state feedback controller is designed to eliminate the bifurcation for the bifurcation phenomenon of the system. Using the actual example parameters, the state response diagram after applying the controller is obtained by simulation, which shows that the controller can effectively control the system. Finally, a class of generalized bio-economic models affected by time-delay factors and pollution factors are studied. By making appropriate linear transformation to time-delay singular systems and using parameterized methods, the time-delay singular systems are transformed into general time-delay differential systems. The stability of the differential delay system at the equilibrium point is analyzed, and the Hopf bifurcation at the positive equilibrium point is obtained. The related parameters are deduced, and the stability and directivity of the Hopf bifurcation are explained. The rationality of the conclusion is verified by numerical simulation.
【学位授予单位】：沈阳工业大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O175;O231

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