本文選題：兩種群競爭 + 正平衡點(diǎn)　； 參考：《華中師范大學(xué)》2017年碩士論文

【摘要】：現(xiàn)在數(shù)學(xué)與生態(tài)學(xué)的交叉學(xué)科發(fā)展越來越來成熟,通過建立數(shù)學(xué)模型來研究生態(tài)問題、揭示生態(tài)現(xiàn)象的規(guī)律已經(jīng)成為一種重要的方法,并被眾多學(xué)者所采用.從馬爾薩斯研究人口理論,用單種群人口數(shù)量為模型開始,一直到Lotka-Volterra模型,前人這些基本模型都為后人的研究奠定了基礎(chǔ),并形成了一種模式:建立數(shù)學(xué)模型,運(yùn)用數(shù)學(xué)的方法進(jìn)行計(jì)算和數(shù)值模擬,最后根據(jù)計(jì)算的結(jié)果來分析生物意義.競爭模型是學(xué)者大量研究的模型,本文主要探討了四類不同的兩種群競爭模型的平衡點(diǎn)以及全局穩(wěn)定性,并剖析了它們的生物學(xué)意義:第二章是具有常數(shù)輸入率的生物競爭模型,經(jīng)過計(jì)算知此模型存在唯一的正平衡點(diǎn),通過構(gòu)造環(huán)域證明了正平衡點(diǎn)的全局漸近穩(wěn)定性;第三章是一種群受到密度制約,一種群具有常數(shù)輸入率的生物競爭模型,經(jīng)過計(jì)算知此模型存在唯一的邊界平衡點(diǎn),計(jì)算和討論了可能存在的正平衡點(diǎn),當(dāng)滿足不同的條件時(shí),此模型可以存在兩個(gè)正平衡點(diǎn)、也可以存在一個(gè)正平衡點(diǎn)、甚至不存在正平衡點(diǎn),然后通過構(gòu)造環(huán)域證明了平衡點(diǎn)的全局漸近穩(wěn)定性;第四章是兩種群都受到密度制約的生物競爭模型,也即是Lotka-Volterra競爭模型;計(jì)算可知存在三個(gè)邊界平衡點(diǎn)、存在一個(gè)正平衡點(diǎn),通過構(gòu)造環(huán)域證明了平衡點(diǎn)的全局漸近穩(wěn)定性;第五章是一種群受到Allee效應(yīng),一種群受到密度制約的生物競爭模型,經(jīng)過計(jì)算知此模型存在唯一的邊界平衡點(diǎn),計(jì)算和討論了可能存在的正平衡點(diǎn),當(dāng)?shù)珴M足不同的條件時(shí),此模型可以存在兩個(gè)正平衡點(diǎn)、也可以存在一個(gè)正平衡點(diǎn)、甚至不存在正平衡點(diǎn),然后通過構(gòu)造環(huán)域證明了平衡點(diǎn)的全局漸近穩(wěn)定性.
[Abstract]:Nowadays, the interdisciplinary discipline of mathematics and ecology is becoming more and more mature. It has become an important method to establish mathematical models to study ecological problems and reveal the laws of ecological phenomena, which has been adopted by many scholars. From Malthus' study of population theory, using single population as the model and all the way to the Lotka-Volterra model, the previous basic models have laid the foundation for future generations to study, and formed a kind of model: to establish a mathematical model, The mathematical method is used to calculate and simulate, and the biological significance is analyzed according to the result of the calculation. Competition model is a model studied by many scholars. This paper mainly discusses the equilibrium point and global stability of four different two species competition models. In chapter 2, the biological competition model with constant input rate is presented. The unique positive equilibrium point is found by calculation. The global asymptotic stability of the positive equilibrium point is proved by constructing the annular domain. The third chapter is a biological competition model with constant input rate, which is a species restricted by density. It is known that there is a unique boundary equilibrium point in the model, and the possible positive equilibrium point is calculated and discussed when different conditions are satisfied. The model can have two positive equilibrium points or one positive equilibrium point or even no positive equilibrium point. Then the global asymptotic stability of the equilibrium point is proved by constructing the ring domain. The fourth chapter is the biological competition model, that is, Lotka-Volterra competition model, in which both species are restricted by density, and it is found that there are three boundary equilibrium points and a positive equilibrium point, and the global asymptotic stability of the equilibrium point is proved by constructing the annular domain. Chapter 5 is a biological competition model in which a population is subjected to the Allee effect and a population is restricted by density. It is known that there is a unique boundary equilibrium point in the model, and the possible positive equilibrium point is calculated and discussed when different conditions are satisfied. The model can have two positive equilibrium points or one positive equilibrium point or even no positive equilibrium point. Then the global asymptotic stability of the equilibrium point is proved by constructing a ring domain.
【學(xué)位授予單位】：華中師范大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O175

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