基于比率的三种群捕食者—食饵系统的稳定性及最优捕获问题的研究

发布时间：2018-04-04 13:07

本文选题：基于比率　切入点：捕食者-食饵系统　出处：《北方民族大学》2017年硕士论文

【摘要】：近年来,生物数学的迅速发展促进了渔业、医学、农业、林业、生物信息学和细胞生物学等诸多领域的发展.而种群动力学则是生物数学中最经典、最广泛的研究领域之一.它为发展生物资源优化管理和有害生物综合治理等应用领域奠定了坚实的基础,为可再生资源的优化管理提供了一定的理论指导意义.探究基于比率的三种群捕食者-食饵系统并分析其稳定性对于保护种群资源具有深刻的实际意义.由于许多人为因素影响物种的捕获,因此合理开发和保护种群资源显得尤为重要.为此,本论文主要致力于种群动力学行为和更新资源优化管理的研究,主要内容有:(1)对基于比率的三种群捕食者-食饵模型进行研究,运用微分方程理论,RouthHurwitz判别法和Lyapunov稳定性定理探讨正平衡点的稳定性并给出其充分条件.(2)利用微分方程稳定性理论,Routh-Hurwitz判别法,Lyapunov稳定性定理,Pontryagin最大值原理,探讨了在双密度制约的情形下,具有B-D功能反应的捕食者-食饵系统的稳定性和最优捕获策略,为生物种群的持久生存和开发管理提供了理论依据和指导意义.(3)研究了具有Michaelis-Menten功能反应函数的捕食者-食饵系统,根据微分方程稳定性理论,Routh-Hurwitz判别法和Lyapunov稳定性定理得到正平衡点存在及系统渐近稳定的充分条件,最后采用Pontryagin最大值原理推导并获得系统的最优税收策略.并利用MATLAB进行数值模拟,检验结果的正确性.
[Abstract]:In recent years, the rapid development of biological mathematics has promoted the development of fishery, medicine, agriculture, forestry, bioinformatics and cell biology.Population dynamics is one of the most classical and extensive research fields in biological mathematics.It lays a solid foundation for the development of optimal management of biological resources and integrated pest management and provides certain theoretical guidance for the optimal management of renewable resources.It is of great practical significance to explore the predator-prey system based on ratio and analyze its stability for the protection of population resources.Because many artificial factors affect species capture, it is very important to develop and protect population resources reasonably.Therefore, this thesis is mainly devoted to the research of population dynamics behavior and renewable resource optimization management. The main content of this paper is to study the predator-prey model of three species based on ratio.This paper discusses the stability of positive equilibrium point by using the theory of differential equation and Lyapunov stability theorem, and gives its sufficient condition. 2. Using the stability theory of differential equation, we use the theory of Routh-Hurwitz discriminant to discuss the Lyapunov stability theorem and the maximum value principle of Pontryagin.The stability and optimal capture strategy of a predator-prey system with B-D functional response under double density constraints are discussed.In this paper, the predator-prey system with Michaelis-Menten functional response function is studied, which provides a theoretical basis and guidance for the sustainable survival and development management of biological populations.According to the Routh-Hurwitz criterion and Lyapunov stability theorem of differential equations, sufficient conditions for the existence of positive equilibrium point and asymptotic stability of the system are obtained. Finally, the optimal tax policy of the system is derived and obtained by using the maximum Pontryagin principle.MATLAB is used to carry out numerical simulation to verify the correctness of the results.
【学位授予单位】：北方民族大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

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