【摘要】：動(dòng)力系統(tǒng)是非線性學(xué)科的重要組成部分,非線性問題存在于諸多學(xué)科和生活的各個(gè)領(lǐng)域中,比如,數(shù)學(xué)、物理學(xué)、生物學(xué)、醫(yī)學(xué)、工程、力學(xué)以及經(jīng)濟(jì)學(xué)等都可以用非線性動(dòng)力系統(tǒng)來解釋。特別的,生物數(shù)學(xué)作為生物和數(shù)學(xué)的交叉學(xué)科,近幾年得到了快速的發(fā)展。為了建立切合實(shí)際的數(shù)學(xué)模型,需要考慮很多因素,比如,時(shí)間、空間、時(shí)滯、隨機(jī)、脈沖,階段等。對(duì)此,本文主要研究離散的傳染病模型和帶有時(shí)滯的食餌-捕食者模型以及隨機(jī)離散的捕食與被捕食者模型的動(dòng)力學(xué)行為,主要內(nèi)容分為四章如下:1.首先敘述了生物動(dòng)力系統(tǒng)國內(nèi)外發(fā)展、目的和意義,其次簡單敘述了本論文需要用到的一些基本定義和定理,最后介紹了本論文所做的工作。2.主要分析一類離散傳染病模型SI系統(tǒng)的動(dòng)力學(xué)行為。首先,根據(jù)特征方程的特征根的情況得到了不動(dòng)點(diǎn)穩(wěn)定性的條件;其次,根據(jù)中心流行定理和分岔理論得到了系統(tǒng)在不動(dòng)點(diǎn)發(fā)生倍周期分岔和Neimark-Sacker分岔的條件;最后,數(shù)值模擬驗(yàn)證該結(jié)論的正確性。3.主要分析具有雙時(shí)滯的食餌-捕食者模型的動(dòng)力學(xué)行為。首先根據(jù)特征根分布情況判斷在平衡點(diǎn)處系統(tǒng)的Hopf分岔的存在性;其次,通過運(yùn)用泛函微分方程的規(guī)范性理論以及中心流行定理對(duì)系統(tǒng)的Hopf分岔的方向、周期解進(jìn)行分析;最后,數(shù)值模擬驗(yàn)證理論的正確性。4.主要分析對(duì)帶有隨機(jī)滯后食餌-捕食者模型離散系統(tǒng)的漸近穩(wěn)定性和Hopf分岔問題進(jìn)行分析。首先,用正交多項(xiàng)式逼近的方法將隨機(jī)離散系統(tǒng)轉(zhuǎn)化為確定性系統(tǒng);其次,根據(jù)Hopf分岔理論得到了隨機(jī)系統(tǒng)發(fā)生Hopf分岔的臨界值并結(jié)合中心流行定理分析;最后,數(shù)值模擬說明正確性。
[Abstract]:Dynamic systems are an important part of nonlinear disciplines. Nonlinear problems exist in many disciplines and fields of life, such as mathematics, physics, biology, medicine, engineering. Mechanics and economics can be explained by nonlinear dynamic systems. In particular, biological mathematics as an interdisciplinary subject of biology and mathematics, has been rapid development in recent years. In order to establish a practical mathematical model, many factors need to be considered, such as time, space, time delay, randomness, pulse, stage and so on. In this paper, the dynamic behaviors of discrete infectious disease model, prey predator model with time delay and stochastic discrete predator model are studied. The main contents are as follows: 1. This paper first describes the development, purpose and significance of biodynamic system at home and abroad, then briefly describes some basic definitions and theorems that need to be used in this paper, and finally introduces the work done in this paper. 2. The dynamic behavior of a class of discrete infectious disease model (SI) systems is analyzed. Firstly, according to the characteristic root of the characteristic equation, the stability condition of the fixed point is obtained. Secondly, according to the central popular theorem and the bifurcation theory, the condition of the double periodic bifurcation and the Neimark-Sacker bifurcation of the system at the fixed point is obtained. Finally, the numerical simulation verifies the correctness of the conclusion. 3. The dynamic behavior of predator-prey model with two delays is analyzed. Firstly, the existence of Hopf bifurcation of the system at the equilibrium point is judged according to the distribution of the characteristic root, secondly, the direction of the Hopf bifurcation and the periodic solution of the system are analyzed by using the normative theory of functional differential equation and the central popular theorem. Finally, numerical simulation verifies the correctness of the theory. 4. The asymptotic stability and Hopf bifurcation of discrete systems with stochastic lag predator-prey model are analyzed. Firstly, the stochastic discrete system is transformed into a deterministic system by orthogonal polynomial approximation. Secondly, according to the Hopf bifurcation theory, the critical value of Hopf bifurcation for the stochastic system is obtained and the central popular theorem is analyzed. Finally, the numerical simulation shows the correctness.
【學(xué)位授予單位】：蘭州交通大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O19

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