本文選題：有毒性效應(yīng)的營養(yǎng)-浮游植物反應(yīng)擴(kuò)散模型　切入點：穩(wěn)定性　出處：《江蘇師范大學(xué)》2017年碩士論文

【摘要】：近年來反應(yīng)擴(kuò)散方程的研究日益受到重視,反應(yīng)擴(kuò)散方程涉及的大量問題來自物理學(xué),化學(xué)和生物學(xué)中眾多的數(shù)學(xué)模型,從而有強(qiáng)烈的實際背景.對生物學(xué)中的營養(yǎng)-浮游植物模型模型進(jìn)行研究,不僅具有重大的理論意義,也具有很大的實用價值.Chakraborty等人提出了一個復(fù)雜的營養(yǎng)-浮游植物模型,研究結(jié)果表明系統(tǒng)的漸近穩(wěn)定性和圖靈不穩(wěn)定性依賴于毒性效應(yīng)θ[9].θ提高時,則會出現(xiàn)圖靈不穩(wěn)定.本文主要研究了一個簡單的具有毒性效應(yīng)的營養(yǎng)-浮游植物模型,通過運(yùn)用比較原理,偏微分定理中的能量估計,隱函數(shù)定理等方法更好地研究解的定性性質(zhì).主要研究內(nèi)容如下:第一章對營養(yǎng)-浮游植物模型的研究背景和意義做了介紹,并簡單的介紹了本文的主要工作.第二章研究了其反應(yīng)擴(kuò)散方程的解的先驗估計和長時間漸近行為,并討論了拋物系統(tǒng)的吸引子的存在性第三章關(guān)心的是對應(yīng)的橢圓型方程的常數(shù)解的穩(wěn)定性,特別是圖靈不穩(wěn)定性.第四章致力于對橢圓型方程的解進(jìn)行先驗估計,將是討論非常數(shù)穩(wěn)態(tài)解的不存在性和存在性的基礎(chǔ).在第五章和第六章中,主要分析了當(dāng)擴(kuò)散系數(shù)在一定范圍內(nèi)變化時,橢圓方程的非常數(shù)解的存在性與不存在性,分別運(yùn)用了能量估計,隱函數(shù)定理和Leray-Schauder拓?fù)涠葘ζ溥M(jìn)行討論.第七章總結(jié)了本論文的研究結(jié)果,并與提出的具有Holling-Ⅲ型功能反應(yīng)的毒性效應(yīng)的模型進(jìn)行比較[9].
[Abstract]:In recent years, more and more attention has been paid to the study of the reaction diffusion equation. A large number of problems related to the reaction diffusion equation come from many mathematical models in physics, chemistry and biology, so it has a strong practical background.The study of nutrition-phytoplankton model in biology is not only of great theoretical significance, but also of great practical value. Chakraborty et al put forward a complex nutrition-phytoplankton model.The results show that the asymptotic stability and Turing instability of the system depend on the toxic effect 胃 [9]. When 胃 increases, Turing instability will occur.In this paper, a simple nutrition-phytoplankton model with toxic effect is studied. By using the comparison principle, the energy estimation in the partial differential theorem and the implicit function theorem, the qualitative properties of the solution are better studied.The main contents are as follows: the first chapter introduces the background and significance of nutrition-phytoplankton model, and briefly introduces the main work of this paper.In chapter 2, we study the priori estimation and long-time asymptotic behavior of the solution of the reaction-diffusion equation, and discuss the existence of the attractor of the parabolic system. In Chapter 3, we focus on the stability of the constant solution of the corresponding elliptic equation.Especially Turing instability.In chapter 4, a priori estimate of the solutions of elliptic equations is proposed, which will be the basis for discussing the nonexistence and existence of the steady-state solutions of non-constant numbers.In the fifth and sixth chapters, the existence and non-existence of the nonconstant solutions of the elliptic equation are analyzed when the diffusion coefficient varies within a certain range. The energy estimation, implicit function theorem and Leray-Schauder topological degree are used to discuss the existence and non-existence of the solutions respectively.In chapter 7, the results of this paper are summarized and compared with the proposed model with the toxic effect of Holling- 鈪，

本文編號：1719261