【摘要】：研究生物種群的數(shù)量增長(zhǎng)模型對(duì)于人類(lèi)社會(huì)的發(fā)展有著重要的意義,其在控制人口、調(diào)配社會(huì)資源、監(jiān)控和改善生態(tài)環(huán)境、保護(hù)物種和開(kāi)發(fā)養(yǎng)殖業(yè)等方面都有重要應(yīng)用。近年來(lái),生物數(shù)學(xué)得到不斷發(fā)展,生物數(shù)學(xué)模型也得到了很好的運(yùn)用。人們?cè)谘芯糠N群時(shí),最為關(guān)心問(wèn)題是種群是否具有一個(gè)正的平衡態(tài),以及這個(gè)平衡態(tài)是否能夠保持穩(wěn)定。而在數(shù)學(xué)上,種群平衡態(tài)就是關(guān)于種群競(jìng)爭(zhēng)模型解的穩(wěn)定性問(wèn)題。本文對(duì)Lotka-Volterra競(jìng)爭(zhēng)模型、連續(xù)的種群競(jìng)爭(zhēng)模型和復(fù)數(shù)域上的種群競(jìng)爭(zhēng)模型分別進(jìn)行了分析。其中,Lotka-Volterra種群競(jìng)爭(zhēng)模型奠定了競(jìng)爭(zhēng)模型的基礎(chǔ)。本文首先把分形幾何中Julia集的思想和方法應(yīng)用于Lotka-Volterra種群競(jìng)爭(zhēng)模型,并建立了該競(jìng)爭(zhēng)模型的Julia集,采用反饋控制方法對(duì)其加以控制,并且考慮了不同參數(shù)下競(jìng)爭(zhēng)模型的同步,使得其中一個(gè)Julia集同步到另一個(gè)Julia集。其次,對(duì)連續(xù)的種群競(jìng)爭(zhēng)模型進(jìn)行了分析,離散化并建立了該競(jìng)爭(zhēng)模型的Julia集,采用反饋控制法以及最優(yōu)控制法對(duì)其加以控制,同時(shí)計(jì)算每一Julia集所對(duì)應(yīng)的分形盒維數(shù),利用分形盒維數(shù)的值刻劃Julia集和吸引域的復(fù)雜性。以上研究?jī)?nèi)容是把Julia集的思想和方法放在實(shí)數(shù)系統(tǒng)范圍內(nèi)考慮的,但Julia集自身是定義在復(fù)數(shù)域上的,因此文章最后把Julia集推廣到復(fù)系統(tǒng)種群競(jìng)爭(zhēng)模型中。將種群競(jìng)爭(zhēng)模型拓展到復(fù)數(shù)域,并把分形Julia集思想應(yīng)用于離散化的種群競(jìng)爭(zhēng)模型,利用Jury準(zhǔn)則判斷系統(tǒng)的穩(wěn)定性,并建立復(fù)種群競(jìng)爭(zhēng)模型的Julia集,從種群初始數(shù)量考慮對(duì)模型的影響。并通過(guò)反饋控制和追蹤控制對(duì)模型Julia集加以合理的控制;在實(shí)際生態(tài)系統(tǒng)中,就是對(duì)種群數(shù)量加入人為的干擾,從而達(dá)到保護(hù)生物種群,保持生態(tài)平衡的目的。為方便,本文將討論的范圍僅限于兩種群之間的競(jìng)爭(zhēng)關(guān)系上。
[Abstract]:The model of population growth of graduate students is of great significance to the development of human society. It has important applications in controlling population, allocating social resources, monitoring and improving ecological environment, protecting species and developing breeding industry. In recent years, the biological mathematics has been continuously developed, and the biological mathematical model has been well used. In the study of population, the most important question is whether the population has a positive equilibrium state and whether the equilibrium state can be kept stable. In mathematics, population equilibrium is about the stability of the solution of population competition model. In this paper, the Lotka-Volterra competition model, the continuous population competition model and the population competition model in the complex number domain are analyzed respectively. Among them, Lotka-Volterra population competition model laid the foundation of competition model. In this paper, the idea and method of Julia set in fractal geometry are first applied to the Lotka-Volterra population competition model, and the Julia set of the competition model is established, and the feedback control method is used to control it, and the synchronization of the competition model under different parameters is considered. Synchronizes one of the Julia sets to the other Julia set. Secondly, the continuous population competition model is analyzed, the Julia set of the competition model is discretized, the feedback control method and the optimal control method are used to control it, and the fractal box dimension corresponding to each Julia set is calculated. The complexity of the Julia set and the domain of attraction is characterized by the value of fractal box dimension. The ideas and methods of Julia set are considered in the real system, but the Julia set itself is defined in the complex number field, so the Julia set is extended to the complex system population competition model at the end of this paper. The population competition model is extended to the complex number domain, and the fractal Julia set is applied to the discrete population competition model. The stability of the system is judged by using the Jury criterion, and the Julia set of the complex population competition model is established. The effect of the initial population number on the model is considered. The model Julia set is controlled reasonably by feedback control and tracking control, and in the actual ecosystem, the artificial disturbance is added to the population quantity to protect the biological population and maintain the ecological balance. For convenience, the scope of this paper is limited to the competitive relationship between two species.
【學(xué)位授予單位】：山東大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類(lèi)號(hào)】：O175;O189

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