本文選題：反應(yīng)擴(kuò)散方程 + 浮游生物模型　； 參考：《哈爾濱工業(yè)大學(xué)》2016年博士論文

【摘要】：浮游生物是水生態(tài)系統(tǒng)的重要組成部分。浮游植物是水生態(tài)中的初級(jí)生產(chǎn)者,其通過(guò)光合作用消耗全球一半的二氧化碳并釋放出全球一半的氧氣。顯而易見,研究浮游生物生態(tài)系統(tǒng)的動(dòng)力學(xué)行為是非常重要的。本文主要分析了幾類具有時(shí)滯和擴(kuò)散的浮游生物模型的動(dòng)力學(xué)性質(zhì),發(fā)現(xiàn)了浮游生物系統(tǒng)諸如平衡解的穩(wěn)定性、Hopf分支、圖靈不穩(wěn)定和持久性等動(dòng)力學(xué)性質(zhì),并在生物學(xué)上給出了適當(dāng)?shù)慕忉尅Ｖ饕ぷ魅缦?(一)研究了一類具有毒素影響的浮游生物模型的動(dòng)力學(xué)性質(zhì)。當(dāng)毒素即時(shí)發(fā)作時(shí),給出了系統(tǒng)全局正解的存在性條件和該解先驗(yàn)界估計(jì)。對(duì)毒素項(xiàng)具有離散時(shí)滯的情形,給出了保障邊界平衡解全局穩(wěn)定的充分條件;通過(guò)分析特征方程根的分布給出了保證正平衡解的穩(wěn)定性和Hopf分支的存在性的條件,并利用中心流形理論和規(guī)范型方法討論了Hopf分支的性質(zhì)。對(duì)毒素項(xiàng)具有非局部時(shí)滯的情形,得到了保證正平衡解漸近穩(wěn)定的充分條件。最后,給出了數(shù)值算例來(lái)說(shuō)明理論分析的結(jié)果。研究結(jié)果表明毒素的發(fā)作時(shí)間會(huì)影響系統(tǒng)的動(dòng)力學(xué)模式。(二)討論了一類考慮浮游生物空間擴(kuò)散和浮游動(dòng)物成熟期時(shí)滯的最小浮游生態(tài)模型。在數(shù)學(xué)上,證明了邊界平衡解的全局穩(wěn)定性。這表明當(dāng)營(yíng)養(yǎng)水平充分低時(shí),浮游動(dòng)物會(huì)滅絕而浮游植物種群數(shù)量會(huì)達(dá)到環(huán)境的最大承載量。在富營(yíng)養(yǎng)條件下,證明了系統(tǒng)存在唯一的空間齊次共存平衡解,且隨著魚類對(duì)浮游動(dòng)物捕食率的上升該平衡解中浮游植物的種群密度增大而浮游動(dòng)物種群密度減小。詳細(xì)的給出了Hopf分支的存在性和分支性質(zhì)的分析,發(fā)現(xiàn)成熟期時(shí)滯和魚類對(duì)浮游動(dòng)物捕食率之間的一條Hopf分支曲線。這個(gè)結(jié)果揭示了魚類對(duì)浮游動(dòng)物的捕食會(huì)抑制系統(tǒng)振蕩的發(fā)生。最后給出了和理論分析結(jié)果相符合的數(shù)值結(jié)果。(三)討論一類具有時(shí)滯和二次封閉項(xiàng)的浮游生物模型。研究結(jié)果表明擴(kuò)散、時(shí)滯和封閉項(xiàng)的不同會(huì)影響模型的動(dòng)力學(xué)行為模式。通過(guò)研究得到了一個(gè)保障系統(tǒng)持久性且和時(shí)滯以及擴(kuò)散無(wú)關(guān)的充分條件,而具有線性封閉項(xiàng)的系統(tǒng)不會(huì)具有持久性。研究還發(fā)現(xiàn)具有二次封閉項(xiàng)的系統(tǒng)的邊界平衡解總是不穩(wěn)定的且其正平衡解總是存在的,但是具有線性封閉項(xiàng)的系統(tǒng)的邊界平衡解在一定條件下是穩(wěn)定的且這時(shí)其正平衡解不存在。研究結(jié)果表明擴(kuò)散會(huì)帶來(lái)圖靈不穩(wěn)定性,時(shí)滯會(huì)影響正平衡解的穩(wěn)定性并引起Hopf分支產(chǎn)生。通過(guò)中心流形理論和規(guī)范型方法給出了判斷分支周期解性質(zhì)的計(jì)算公式,并給出了一些數(shù)值算例來(lái)說(shuō)明理論分析的結(jié)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.
【學(xué)位授予單位】：哈爾濱工業(yè)大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類號(hào)】：O175

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