几类生态系统的非线性分析及研究

【作者】 宋涛；

【导师】 黄东卫；

【作者基本信息】 天津工业大学 ， 应用数学， 2007， 硕士

【摘要】 生态系统的非线性分析是数学生态理论的一个重要组成部分。随着非线性科学的发展，人们发现它在实际应用方面的前景越来越广阔。利用非线性科学能描绘，预测以至调节和控制物种的发展过程与发展趋势，是人类合理开发资源，使用资源和保护资源的有效理论依据之一。第一部分，文章主要运用微分方程稳定性理论和定性理论的方法，全面细致的分析和讨论了一类食饵种群具有常数收获率的Kolmogorov系统的性质，分析了系统的平衡点的性态，在给定参数满足一定条件下，得到了该系统极限环存在与不存在的充分条件，并对系统进行了数值模拟，最后讨论了此系统的生态意义。从以上定性分析所得出的结论可知，此类系统在经过充分长的时间后，只要在一定条件下，捕食者与被捕食者的密度或趋于稳定在某定值附近，或者产生周期性的变化。对此类方程研究的结果，能有效的掌握生态系统的发展规律，更好的实现对此类系统的预测和控制。在第二部分，文章针对海洋赤潮食物链，根据赤潮生物的数量变化及各种影响因素随时间的变化，建立了营养物质浓度变化与藻类相互制约的营养动力学模型，并将其它要素尽量反映在模型的各种参数之中。利用非线性动力学理论，对模型的稳定性及分岔行为进行了研究，得到了发生Hopf分岔时的参数值，并通过计算机数值模拟，对模型进行理论分析和讨论。根据理论分析结果，能够较为明确的求出发生赤潮的理论参数，从而为更深入具体的研究赤潮产生的机理提供参考，为预测和控制赤潮发生提供了依据。更多还原

【Abstract】 Nonlinear analysis of ecological system is an important part of Mathematical ecology. With the development of nonlinear science, the study on it has more and more wide prospects. The nonlinear science which applied can describe,calculate in order to regulate and control the developmental processes and the developmental trend of species, which is one of theoretical basis for people exploiting resources, making use of resources and protecting resources.In the first section, by using the stability theories and the qualitative theories of ordinary differential equation, we study a Kolmogorov system with constant harvesting rate for preys group and analyze the nature of equilibrium points. Then we obtain the sufficient condition of nonexistence and existence to limit cycle under the given parameter satisfied certain conditions, and approved the theoretic analyse by the numerical simulation. At last, the appropriate biological significance of the system is discussed.In the second section, a model of differential equation on nutrient—diatom— N.Scintillans is constructed, based on structural analysis of the essential factors associated with the red tide of N.Scintillans,. Using the methods of the modern nonlinear dynamics, stability and bifurcation behaviors of the model equations are discussed. The value of bifurcation parameter is calculated when Hopf bifurcation was happening and the Mathematical simulation on numerical calculation was also presenented. The results provide scientific basis for predicting and controlling red tide of N.Scintillans to a certain extent.更多还原