几类具时滞的浮游生物扩散系统的动力学性质

发布时间：2018-06-26 14:50

本文选题：反应扩散方程 + 浮游生物模型　；参考：《哈尔滨工业大学》2016年博士论文

【摘要】：浮游生物是水生态系统的重要组成部分。浮游植物是水生态中的初级生产者,其通过光合作用消耗全球一半的二氧化碳并释放出全球一半的氧气。显而易见,研究浮游生物生态系统的动力学行为是非常重要的。本文主要分析了几类具有时滞和扩散的浮游生物模型的动力学性质,发现了浮游生物系统诸如平衡解的稳定性、Hopf分支、图灵不稳定和持久性等动力学性质,并在生物学上给出了适当的解释。主要工作如下:(一)研究了一类具有毒素影响的浮游生物模型的动力学性质。当毒素即时发作时,给出了系统全局正解的存在性条件和该解先验界估计。对毒素项具有离散时滞的情形,给出了保障边界平衡解全局稳定的充分条件;通过分析特征方程根的分布给出了保证正平衡解的稳定性和Hopf分支的存在性的条件,并利用中心流形理论和规范型方法讨论了Hopf分支的性质。对毒素项具有非局部时滞的情形,得到了保证正平衡解渐近稳定的充分条件。最后,给出了数值算例来说明理论分析的结果。研究结果表明毒素的发作时间会影响系统的动力学模式。(二)讨论了一类考虑浮游生物空间扩散和浮游动物成熟期时滞的最小浮游生态模型。在数学上,证明了边界平衡解的全局稳定性。这表明当营养水平充分低时,浮游动物会灭绝而浮游植物种群数量会达到环境的最大承载量。在富营养条件下,证明了系统存在唯一的空间齐次共存平衡解,且随着鱼类对浮游动物捕食率的上升该平衡解中浮游植物的种群密度增大而浮游动物种群密度减小。详细的给出了Hopf分支的存在性和分支性质的分析,发现成熟期时滞和鱼类对浮游动物捕食率之间的一条Hopf分支曲线。这个结果揭示了鱼类对浮游动物的捕食会抑制系统振荡的发生。最后给出了和理论分析结果相符合的数值结果。(三)讨论一类具有时滞和二次封闭项的浮游生物模型。研究结果表明扩散、时滞和封闭项的不同会影响模型的动力学行为模式。通过研究得到了一个保障系统持久性且和时滞以及扩散无关的充分条件,而具有线性封闭项的系统不会具有持久性。研究还发现具有二次封闭项的系统的边界平衡解总是不稳定的且其正平衡解总是存在的,但是具有线性封闭项的系统的边界平衡解在一定条件下是稳定的且这时其正平衡解不存在。研究结果表明扩散会带来图灵不稳定性,时滞会影响正平衡解的稳定性并引起Hopf分支产生。通过中心流形理论和规范型方法给出了判断分支周期解性质的计算公式,并给出了一些数值算例来说明理论分析的结果。
[Abstract]:Plankton is an important part of water ecosystem. Phytoplankton is the primary producer of aquatic ecology, which consumes half of the world's carbon dioxide and emits half of the world's oxygen through photosynthesis. Obviously, it is very important to study the dynamics of plankton ecosystem. In this paper, the dynamical properties of several kinds of plankton models with time delay and diffusion are analyzed. The dynamical properties of plankton systems such as the stability of equilibrium solutions, the stability of Hopf bifurcation, Turing instability and persistence are found. An appropriate biological explanation is given. The main work is as follows: (1) the dynamic properties of a class of plankton models with toxic effects are studied. When the toxin attacks in real time, the existence conditions and the priori bounds of the global positive solution of the system are given. In the case of the toxin term with discrete delay, the sufficient conditions for the global stability of the boundary equilibrium solution are given, and the conditions for the stability of the positive equilibrium solution and the existence of Hopf bifurcation are given by analyzing the distribution of the root of the characteristic equation. The properties of Hopf bifurcation are discussed by using center manifold theory and normal form method. A sufficient condition for the asymptotic stability of the positive equilibrium solution is obtained for the toxin term with nonlocal delay. Finally, numerical examples are given to illustrate the results of the theoretical analysis. The results show that the onset time of the toxin affects the dynamic model of the system. (2) A class of minimum zooplankton ecological models considering plankton spatial diffusion and zooplankton maturity delay is discussed. In mathematics, the global stability of the boundary equilibrium solution is proved. This indicates that when the nutrient level is low enough, zooplankton will become extinct and phytoplankton population will reach the maximum carrying capacity of the environment. Under the condition of eutrophication, it is proved that there exists a unique spatially homogeneous equilibrium solution, and that the population density of phytoplankton increases and the population density of zooplankton decreases with the increase of predation rate of fish to zooplankton. The existence and bifurcation properties of Hopf bifurcation are analyzed in detail. It is found that a Hopf bifurcation curve between the ripening time delay and the prey rate of fish to zooplankton is found. This result shows that the predation of fish to zooplankton can inhibit the occurrence of systematic oscillation. Finally, the numerical results which are in agreement with the theoretical analysis results are given. (3) A class of plankton models with time delay and quadratic closed term are discussed. The results show that the diffusion, delay and closed term will affect the dynamic behavior of the model. In this paper, we obtain a sufficient condition that the system is persistent and independent of time delay and diffusion, but the system with linear closed term does not have permanence. It is also found that the boundary equilibrium solutions of systems with quadratic closed terms are always unstable and their positive equilibrium solutions always exist. But the boundary equilibrium solution of the system with linear closed term is stable under certain conditions and the positive equilibrium solution does not exist. The results show that diffusion leads to Turing instability and time delay affects the stability of positive equilibrium solutions and results in Hopf bifurcation. By means of the center manifold theory and the normal form method, the calculation formulas for judging the properties of the bifurcation periodic solutions are given, and some numerical examples are given to illustrate the results of the theoretical analysis.
【学位授予单位】：哈尔滨工业大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175

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