摘 要
種群動力學(xué)已發(fā)展為生物數(shù)學(xué)的一個非常重要的分支學(xué)科，它在生態(tài)學(xué)理論中，特別是動植物保護和生態(tài)環(huán)境的治理與開發(fā)等領(lǐng)域都有著非常重要的作用。生物學(xué)家針對種群的相互作用關(guān)系進行大量的實驗，并對實驗數(shù)據(jù)進行統(tǒng)計分析以及合理細致的機理分析，建立微分、積分或者差分方程形式的數(shù)學(xué)模型，以用來描述、預(yù)測、調(diào)節(jié)和控制物種的發(fā)展過程和發(fā)展趨勢.數(shù)學(xué)家用數(shù)學(xué)的理論和方法針對所建立的數(shù)學(xué)模型，研究生態(tài)系統(tǒng)中種群的長期變化規(guī)律，得到一些理論上的結(jié)果,再用來解釋和解決具體的生物學(xué)問題。這對于研究生態(tài)學(xué)有重要意義，可以使人們更好的認識自然。
自然界中的生態(tài)系統(tǒng)是多種多樣的，常見的有單食物鏈模型，競爭模型。數(shù)學(xué)上常用常微分方程組，偏微分方程組，時滯微分方程組等描述.對模型主要考慮的問題有:系統(tǒng)正平衡解(周期解)的存在性，平衡解的局部和全局穩(wěn)定性、系統(tǒng)的一致持續(xù)生存等性質(zhì)。常用的理論有:微分方程(組)的定性理論,分歧理論,不動點指數(shù)和拓撲度理論。本文通過對DeAngelis模型解的研究來表現(xiàn)種群之間的關(guān)系。
文中主要研究該模型的平衡點與極限環(huán)，結(jié)合生態(tài)規(guī)律對捕食者-食餌模型進行分析。

關(guān)鍵詞 平衡點 捕食者-食餌模型 局部漸近穩(wěn)定性

Abstract
Population dynamics has been developed as a bio-mathematical a very important branch of science, theory in ecology, especially the flora and fauna protection and the ecological environment in areas such as governance and development are a very important role. Biologist for the populations a great deal of interaction between the experimental and statistical analysis of experimental data as well as a reasonable mechanism of detailed analysis, differential, integral or differential equation forms of the mathematical model to describe, predict, regulate and control the　development of species and the development of trend. mathematician with mathematical theories and methods for the mathematical model created to study the ecosystem of the long-term population changes of some theoretical results, and then used to explain and address specific biological issues. This is the study of ecology are important, you can make people a better understanding of the natural. At present, the population has become a mathematical model for the development of resources, a more rational use of resources and environmental protection an important tool.
Natural ecosystems in a variety of common single-food chain model, competition model. Mathematical common ordinary differential equations, partial differential equations, delay differential equations to describe groups. To model the main issues to consider : system is a balanced solution (of periodic solutions) the existence of equilibrium solution of the local and global stability, the continued survival of the system, such as the nature of consensus. the theory of commonly used are: differential equations (Unit) of the qualitative theory, differences between the theory of fixed point index and the topological degree theory.
In this paper,wo know the model’s Equilibrium point and Limited annulus,combine the rule of zoology to study the Predator - Prey Model.

Key words Equilibrium point 　Predator - Prey Model　 Local asymptotic stabilit


目 錄
摘 要 I
ABSTRACT II

第1章 緒 論 1
1.1 引 言 1
1.2 課題研究背景與目的 4
1.2.1 研究背景 4
1.2.2 研究目的 5
第2章 預(yù)備知識 8
2.1平衡點的定義 8
2.2極限環(huán)的定義 9
第3章 平衡點與極限環(huán)的討論 10
3.1正平衡點的討論 10
3.2平面線性系統(tǒng)平衡點分類 12
3.3極限環(huán)的討論 16
第4章 數(shù)值模擬及主要結(jié)果的生態(tài)學(xué)意義 18
4.1數(shù)值模擬 18
4.2主要結(jié)果的生態(tài)學(xué)意義 20
總 結(jié) 21
致 謝 22
參 考 文 獻 23
附 錄1 24
附 錄2 28
附 錄3 33