本文選題：捕食者-食餌 + 時(shí)滯��； 參考：《哈爾濱工業(yè)大學(xué)》2017年碩士論文

【摘要】：為了保護(hù)物種的多樣性,維護(hù)生態(tài)平衡,需要對(duì)種群動(dòng)力學(xué)模型進(jìn)行深入研究,揭示出種群之間的相互作用關(guān)系。在種群動(dòng)力學(xué)中,捕食者-食餌模型因其重要性一直受到各界學(xué)者的關(guān)注。在描述種群數(shù)量變化時(shí),需要考慮到物種的成熟期和能量的轉(zhuǎn)化時(shí)間,因此有必要在系統(tǒng)中引入時(shí)滯,以便更好地反應(yīng)實(shí)際情況。所以本文討論了一類具時(shí)滯的捕食者-食餌模型,并在模型中引入了食物補(bǔ)貼項(xiàng)的影響。首先,討論了系統(tǒng)正平衡點(diǎn)的存在唯一性,在此基礎(chǔ)上利用特征方程根的分布分析方法分析其穩(wěn)定性,得到了在正平衡點(diǎn)處存在局部Hopf分支的充分條件。又由中心流形定理和規(guī)范型理論,分析了正平衡點(diǎn)處Hopf分支的性質(zhì),包括分支的方向、分支周期解的穩(wěn)定性以及周期解周期的變化等。其次,在局部Hopf分支的基礎(chǔ)上進(jìn)一步研究系統(tǒng)周期解的大范圍存在性問(wèn)題。由全局Hopf分支定理可以得到每個(gè)連通分枝是無(wú)界的,接著證明了系統(tǒng)的解具有正性,又利用常微分方程高維Bendixson定理證明系統(tǒng)沒(méi)有非常值?-周期解,進(jìn)而得到了周期解的全局存在性結(jié)論。最后,本文分為兩部分進(jìn)行數(shù)值模擬。第一部分以時(shí)滯為參數(shù),觀察系統(tǒng)在不同時(shí)滯處的穩(wěn)定性和全局Hopf分支的存在性,對(duì)之前的理論結(jié)果給予了算例支撐;第二部分分別以食物補(bǔ)貼投放率、環(huán)境承載量、捕食者消耗食餌的最大速率和轉(zhuǎn)換因子為參數(shù)。通過(guò)模擬觀察其對(duì)第一個(gè)分支值的影響,從而得到各參數(shù)對(duì)系統(tǒng)穩(wěn)定區(qū)間的影響,同時(shí)解釋了各種情況下的生物學(xué)意義。
[Abstract]:In order to protect species diversity and maintain ecological balance, it is necessary to deeply study the population dynamics model and reveal the interaction between populations. In population dynamics, predator-prey model has been concerned by scholars because of its importance. In order to better reflect the actual situation, it is necessary to take into account the mature period of species and the time of energy conversion when describing the population quantity change, so it is necessary to introduce time delay in the system. In this paper, we discuss a kind of predator-prey model with time delay, and introduce the effect of food subsidy term into the model. Firstly, the existence and uniqueness of the positive equilibrium point of the system are discussed. On this basis, the stability of the system is analyzed by using the distribution analysis method of the root of the characteristic equation, and the sufficient conditions for the existence of local Hopf bifurcation at the positive equilibrium point are obtained. Based on the center manifold theorem and normal form theory, the properties of Hopf bifurcation at positive equilibrium point are analyzed, including the direction of bifurcation, the stability of bifurcation periodic solution and the variation of periodic solution. Secondly, on the basis of local Hopf bifurcation, the existence of periodic solutions in a large scale is studied. By using the global Hopf bifurcation theorem, we can obtain that every connected branch is unbounded, then we prove that the solution of the system is positive, and by using the high dimensional Bendixson theorem of ordinary differential equation, we prove that the system has no nonconstant value or periodic solution. The global existence of periodic solutions is obtained. Finally, this paper is divided into two parts for numerical simulation. In the first part, the stability of the system at different time delays and the existence of global Hopf bifurcation are observed, and the previous theoretical results are supported by examples. The maximum rate of predator consumption and the conversion factor are parameters. The influence of each parameter on the stability interval of the system is obtained by observing the influence of the first branch value by simulation, and the biological significance under various conditions is explained.
【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 宋永利,韓茂安,魏俊杰;多時(shí)滯捕食-食餌系統(tǒng)正平衡點(diǎn)的穩(wěn)定性及全局Hopf分支[J];數(shù)學(xué)年刊A輯(中文版);2004年06期

，

本文編號(hào)：1911655