本文選題：捕食-食餌模型 + 反應(yīng)擴(kuò)散　； 參考：《安徽師范大學(xué)》2017年碩士論文

【摘要】：生物斑圖動(dòng)力學(xué)作為非線性科學(xué)的主要分支之一,它的研究非常廣泛和豐富.產(chǎn)生生物斑圖的機(jī)理有很多,最簡(jiǎn)單的一種是反應(yīng)擴(kuò)散系統(tǒng),最先由Turing在1952年發(fā)表的《形態(tài)形成的化學(xué)基礎(chǔ)》中提出,因此通常被稱為Turing失穩(wěn)或由擴(kuò)散引起的失穩(wěn).簡(jiǎn)單的說(shuō),就是由于擴(kuò)散使得穩(wěn)定的平衡點(diǎn)變得不穩(wěn)定.通過(guò)構(gòu)造具有種群動(dòng)力學(xué)特征的數(shù)學(xué)模型,進(jìn)行動(dòng)力學(xué)形態(tài)分析,可以用來(lái)解釋種群之間相互作用而形成的空間斑圖,同時(shí)結(jié)合數(shù)值模擬的結(jié)果,說(shuō)明種群向時(shí)空混沌的轉(zhuǎn)變可以解釋種群在空間中的持續(xù)、滅絕、進(jìn)化等問(wèn)題.本文將利用線性化分析理論、Lyapunov函數(shù)方法、Routh-Hurwitz準(zhǔn)則以及多重尺度分析方法,研究三類帶反應(yīng)擴(kuò)散項(xiàng)的捕食-食餌模型,以下是論文的主要研究?jī)?nèi)容:1.研究了一類具比率依賴功能反應(yīng)的捕食-食餌模型的Turing斑圖生成與選擇問(wèn)題.通過(guò)線性化分析,得到Turing空間,利用多重尺度分析方法推導(dǎo)系統(tǒng)的振幅方程并進(jìn)行斑圖選擇,在Turing空間中選擇合適的參數(shù),得到包括點(diǎn)狀、條狀以及二者共存的Turing斑圖.2.基于自然界中猛獸群體的追捕現(xiàn)象,研究了一類帶有負(fù)交叉擴(kuò)散項(xiàng)的一般二維模型,并對(duì)一類具比率依賴的捕食-食餌模型進(jìn)行理論和數(shù)值研究,所得結(jié)果表明:負(fù)交叉擴(kuò)散項(xiàng)(-d21)影響Turing斑圖生成及選擇.在其它參數(shù)固定情況下d21必須小于某個(gè)臨界值時(shí),系統(tǒng)才會(huì)出現(xiàn)Turing不穩(wěn)定現(xiàn)象.3.研究一類三種群食物鏈模型的強(qiáng)耦合交叉擴(kuò)散系統(tǒng).首先通過(guò)構(gòu)造Lyapunov函數(shù)證明唯一的正平衡點(diǎn)在ODE系統(tǒng)下是全局漸近穩(wěn)定的,當(dāng)交叉擴(kuò)散系數(shù)均為零時(shí),唯一的正平衡點(diǎn)仍是全局漸近穩(wěn)定的,但是,當(dāng)引入交叉擴(kuò)散時(shí),正平衡點(diǎn)則變得不穩(wěn)定.利用Routh-Hurwitz準(zhǔn)則和Descartes符號(hào)法則證明了大的交叉擴(kuò)散系數(shù)(k21或k32足夠大時(shí))可以導(dǎo)致平衡點(diǎn)由原來(lái)的穩(wěn)定變得不穩(wěn)定.最后利用數(shù)學(xué)軟件Matlab對(duì)我們的結(jié)果進(jìn)行數(shù)值模擬,得到了不同類型的Turing斑圖,包括六邊形、條狀以及二者共存的斑圖.
[Abstract]:As one of the main branches of nonlinear science, biographic dynamics has been widely studied. There are many mechanisms for producing biological patterns, the simplest of which is the reaction-diffusion system, which was first proposed by Turing in the Chemical basis of Morphology published in 1952. Therefore, it is usually referred to as the instability of Turing or the instability caused by diffusion. Simply put, it is because of diffusion that the stable equilibrium becomes unstable. By constructing a mathematical model with the characteristics of population dynamics, the dynamic morphological analysis can be used to explain the spatial pattern formed by the interaction between populations, and at the same time, the results of the numerical simulation can be combined with the results of the numerical simulation. The transformation of population to spatiotemporal chaos can explain the persistence, extinction and evolution of population in space. In this paper, three kinds of predator-prey models with reaction diffusion term are studied by using the Lyapunov function method, Routh-Hurwitz criterion and multi-scale analysis method. The following is the main content of this paper: 1. The problem of Turing pattern generation and selection for a predator-prey model with ratio dependent functional response is studied. Through the linearization analysis, the Turing space is obtained. The amplitude equation of the system is deduced and the pattern selection is carried out by using the multi-scale analysis method. The suitable parameters are selected in the Turing space, and the Turing pattern. 2, which includes the dot shape, the bar shape and the coexistence of the two, is obtained. Based on the hunting phenomenon of predator population in nature, a general two-dimensional model with negative cross-diffusion term is studied, and a kind of predator-prey model with ratio dependence is studied theoretically and numerically. The results show that the negative cross-diffusion term (-d21) affects the generation and selection of Turing patterns. In the case of other fixed parameters D21 must be less than a certain critical value before the system will appear Turing instability. 3. A strong coupling cross diffusion system for a class of three species food chain model is studied. It is proved that the unique positive equilibrium is globally asymptotically stable under the ODE system by constructing the Lyapunov function. When the cross-diffusion coefficients are 00:00, the unique positive equilibrium is globally asymptotically stable. However, when the cross-diffusion is introduced, the unique positive equilibrium is globally asymptotically stable. The positive equilibrium becomes unstable. Using the Routh-Hurwitz criterion and the Descartes sign rule, it is proved that when the cross diffusion coefficient is large enough, the equilibrium point can change from the original stability to the instability. Finally, the numerical simulation of our results is carried out by using the mathematical software Matlab, and different types of Turing patterns are obtained, including hexagonal, bar and coexistence patterns.
【學(xué)位授予單位】：安徽師范大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O175

【參考文獻(xiàn)】

相關(guān)期刊論文 前7條

1 張建強(qiáng);張e，

本文編號(hào)：1880543