【摘要】：隨著生物數(shù)學(xué)理論的不斷發(fā)展,中立型泛函微分方程已經(jīng)被越來(lái)越廣泛地用于描述生物種群模型的演化規(guī)律。中立型泛函微分方程一般被用來(lái)描繪當(dāng)前時(shí)刻狀態(tài)變化率依賴于歷史時(shí)刻狀態(tài)變化率的發(fā)展系統(tǒng),在種群模型中這意味著當(dāng)前時(shí)刻的種群數(shù)量增長(zhǎng)率依賴于歷史某時(shí)刻的增長(zhǎng)率。首先,對(duì)具年齡結(jié)構(gòu)的雙曲模型進(jìn)行約化,得到了一類具有年齡結(jié)構(gòu)的種群增長(zhǎng)的中立型方程。通過(guò)選擇不同的出生函數(shù),得到了兩類要研究的模型,第一類模型的增長(zhǎng)率按照l(shuí)ogistic形式進(jìn)行,第二類模型的增長(zhǎng)率按照指數(shù)形式進(jìn)行。其次,針對(duì)第一類模型,討論了模型平衡解的穩(wěn)定性,通過(guò)分析特征方程,得到了關(guān)于零解和正平衡解的穩(wěn)定性結(jié)果。接下來(lái)分成兩種情形研究了方程的Hopf分支性質(zhì)。第一種是幼年個(gè)體死亡率被忽略的情形;第二種是幼年個(gè)體死亡率沒(méi)有被忽略的情形。應(yīng)用中心流形定理與規(guī)范型理論,研究了正平衡解處的Hopf分支方向與分支周期解的穩(wěn)定性。針對(duì)第二類模型,研究了按照指數(shù)形式增長(zhǎng)的微分方程平衡解的穩(wěn)定性,并分兩種情形研究了方程的Hopf分支性質(zhì)。此外,得到了正平衡解處的Hopf分支方向與分支周期解的穩(wěn)定性。最后,以第二類模型為例研究了方程的全局Hopf分支。利用全局Hopf分支定理給出了方程周期解的大范圍存在性條件。同時(shí),對(duì)理論分析結(jié)果給予了數(shù)值算例支撐。
[Abstract]:With the continuous development of biological mathematics theory, neutral functional differential equations have been more and more widely used to describe the evolution of biological population models. Neutral functional differential equations are generally used to describe the development system in which the state change rate of the current time depends on the state change rate of the historical moment. In the population model, this means that the population growth rate at the current time depends on the growth rate at a certain time in history. Firstly, the hyperbolic model with age structure is reduced, and a class of neutral equation of population growth with age structure is obtained. By selecting different birth functions, two kinds of models to be studied are obtained. the growth rate of the first model is carried out in the form of logistic, and the growth rate of the second model is carried out in the exponential form. Secondly, for the first kind of model, the stability of the equilibrium solution of the model is discussed, and the stability results of the zero solution and the positive equilibrium solution are obtained by analyzing the characteristic equation. Next, the Hopf bifurcation properties of the equation are studied in two cases. The first is that the infant mortality rate is ignored; the second is the case where the juvenile individual mortality rate is not ignored. In this paper, the Hopf bifurcation direction and the stability of bifurcation periodic solutions at the positive equilibrium solution are studied by using the central manifolds theorem and the canonical form theory. For the second kind of model, the stability of equilibrium solutions of differential equations growing in exponential form is studied, and the Hopf bifurcation properties of the equations are studied in two cases. In addition, the stability of the Hopf bifurcation direction and the bifurcation periodic solution at the positive equilibrium solution is obtained. Finally, the global Hopf bifurcation of the equation is studied by taking the second kind of model as an example. By using the global Hopf bifurcation theorem, the existence conditions of periodic solutions for the equation are given. At the same time, numerical examples are given to support the theoretical analysis results.
【學(xué)位授予單位】：哈爾濱工業(yè)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1條

1 魏俊杰,黃啟昌;泛函微分方程分支理論發(fā)展概況[J];科學(xué)通報(bào);1997年24期

相關(guān)博士學(xué)位論文 前1條

1 蘇穎;單種群模型的分支問(wèn)題[D];哈爾濱工業(yè)大學(xué);2011年

，

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