發(fā)布時(shí)間：2018-02-25 05:01

  本文關(guān)鍵詞： 捕食系統(tǒng) 傳染病模型 隨機(jī)模型 庇護(hù)效應(yīng) 捕獲效應(yīng) 時(shí)滯效應(yīng) 傳染率 功能反應(yīng) 穩(wěn)定性　出處：《蘭州大學(xué)》2008年碩士論文　論文類型：學(xué)位論文

【摘要】： 近年來，以Lotka和Volterra為代表的種群動力學(xué)和以Kermack及McKendrick為代表的流行病動力學(xué)，已經(jīng)有了相當(dāng)?shù)陌l(fā)展，它們分別在開發(fā)利用資源和預(yù)防治療疾病方面都起到了不同程度的指導(dǎo)作用。由于流行病必然在物種之間傳播，所以為了更符合實(shí)際情況，應(yīng)該把種群動力學(xué)和流行病動力學(xué)結(jié)合起來考慮，但是這方面的工作至今還寥寥無幾�；诖�，本文中我們研究了疾病在捕食系統(tǒng)中的傳播情況，建立模型，通過數(shù)學(xué)分析和數(shù)值模擬，主要得到以下結(jié)論： 1．當(dāng)食餌具有傳染病時(shí)，我們建立了食餌有病的生態(tài)-流行病模型，討論了解的有界性，應(yīng)用特征根法得到了平衡點(diǎn)局部漸近穩(wěn)定的充分條件。進(jìn)一步，，分析了平衡點(diǎn)的全局漸近穩(wěn)定性，得到了邊界平衡點(diǎn)和正平衡點(diǎn)全局穩(wěn)定性的充分條件，并且我們還找到了“基本再生數(shù)”R_0，為疾病的控制提供了理論基礎(chǔ)。 2．當(dāng)捕食者有病時(shí)，我們建立了捕食者有病的生態(tài)-流行病模型，討論了解的有界性以及平衡點(diǎn)的穩(wěn)定性。并且在環(huán)境擾動為白噪聲的情況下，我們建立了隨機(jī)微分方程模型，并找到了一個(gè)Lyapunov函數(shù)，結(jié)果表明在正平衡點(diǎn)附近的一個(gè)隨機(jī)擾動并不改變此平衡點(diǎn)的穩(wěn)定性。 3．對于這樣一個(gè)簡單的生態(tài)一流行病模型蘊(yùn)含著復(fù)雜的動力學(xué)行為，甚至出現(xiàn)混沌現(xiàn)象。 4．對于上述系統(tǒng)中出現(xiàn)的混沌現(xiàn)象，我們提出了三種混沌控制方法：加入庇護(hù)效應(yīng)；捕獲效應(yīng)；時(shí)滯效應(yīng)。通過建立數(shù)學(xué)模型，并對其進(jìn)行數(shù)學(xué)分析和數(shù)值模擬，我們發(fā)現(xiàn)這三種效應(yīng)都能阻止種群震蕩的發(fā)生，增強(qiáng)系統(tǒng)的穩(wěn)定性。 最后，我們詳細(xì)的討論了傳染率，功能反應(yīng)和空間因素對系統(tǒng)動力學(xué)行為的影響，結(jié)合前人的一些工作，提出了今后努力的方向。
[Abstract]:In recent years, population dynamics, represented by Lotka and Volterra, and epidemiological dynamics, represented by Kermack and McKendrick, have developed considerably. They have played a different role in the development and utilization of resources and in the prevention and treatment of diseases. Since epidemics must spread between species, in order to be more realistic, The combination of population dynamics and epidemic dynamics should be considered, but little has been done so far. Based on this, we have studied the spread of disease in predation system and established a model. Through mathematical analysis and numerical simulation, the main conclusions are as follows:. 1. When the prey has infectious disease, we establish the eco-epidemic model of the prey disease, discuss the boundedness of the solution, and obtain the sufficient condition of the local asymptotic stability of the equilibrium point by using the characteristic root method. In this paper, the global asymptotic stability of the equilibrium point is analyzed, and the sufficient conditions for the global stability of the boundary equilibrium point and the positive equilibrium point are obtained. Furthermore, we find the "basic reproducing number" R0, which provides a theoretical basis for disease control. 2. When the predator is ill, we establish the ecological epidemic model of predator disease, discuss the boundedness of solution and the stability of equilibrium point, and establish the stochastic differential equation model when the environmental disturbance is white noise. A Lyapunov function is found. The results show that a random perturbation near the positive equilibrium does not change the stability of the equilibrium. 3. For such a simple ecological epidemic model, there are complex dynamic behaviors and even chaotic phenomena. 4. For the chaotic phenomena in the system mentioned above, we propose three chaos control methods: adding sheltering effect, capturing effect, time-delay effect, mathematical model, mathematical analysis and numerical simulation. We find that these three effects can prevent the occurrence of population oscillation and enhance the stability of the system. Finally, we discuss in detail the effects of infection rate, functional response and spatial factors on the dynamic behavior of the system. Combined with some previous work, we put forward the direction of future efforts.
【學(xué)位授予單位】：蘭州大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2008
【分類號】：R311

【參考文獻(xiàn)】

相關(guān)期刊論文 前2條

1 孫樹林,原存德;捕食者有病的生態(tài)-流行病SIS模型的分析[J];工程數(shù)學(xué)學(xué)報(bào);2005年01期

2 孫樹林;原存德;;捕食者具有流行病的捕食-被捕食(SI)模型的分析[J];生物數(shù)學(xué)學(xué)報(bào);2006年01期

本文編號：1533073