發(fā)布時(shí)間：2018-07-06 07:34

  本文選題：脈沖微分方程 + 持久性�。� 參考：《吉林大學(xué)》2015年博士論文

【摘要】：脈沖微分方程的基本理論與方法在近幾十年里已經(jīng)得到了充分的發(fā)展和完善.由于一些自然現(xiàn)象會(huì)在特定的時(shí)刻發(fā)生“跳躍”,譬如動(dòng)物季節(jié)性的繁殖或者遷移,所以它們不能用連續(xù)或離散的微分方程來建模,但是脈沖方程能夠很好的來描述這些生物現(xiàn)象.首次使用脈沖方法是Beverton和Holt(1957)在一個(gè)半離散模型的基礎(chǔ)上想要用離散模型去逼近一個(gè)連續(xù)時(shí)間的Logistic模型.從此以后,幾乎在生命科學(xué)的每一個(gè)領(lǐng)域都出現(xiàn)了脈沖模型.本文主要研究了脈沖控制在生物數(shù)學(xué)中的兩個(gè)應(yīng)用.第一個(gè)應(yīng)用是通過研究系統(tǒng)的持久性和穩(wěn)定的非平凡周期解分支,我們?cè)趲в蠦eddington-DeAngelis功能關(guān)系的脈沖控制捕食模型中給出了食餌種群密度的完整的控制策略；第二個(gè)應(yīng)用是研究了如何利用不育昆蟲繁殖技術(shù)(SIT)在連續(xù)和脈沖兩種類型的捕食模型中成功地殺滅自然昆蟲.本論文共分為四章.第一章主要介紹了脈沖微分方程的一些基本概念和基礎(chǔ)理論,包括脈沖微分方程的定義,常見分類以及穩(wěn)定性等概念.我們特別給出了研究脈沖微分方程的兩個(gè)重要定理：比較定理和線性周期方程的Floquet定理.之后我們?cè)敿?xì)地介紹了脈沖微分方程的應(yīng)用.到目前為止,脈沖微分方程已經(jīng)廣泛地應(yīng)用到傳染病學(xué),醫(yī)藥學(xué)以及人口模型等各個(gè)方面.第二章主要研究了一個(gè)帶有Beddington-DeAngelis功能關(guān)系的脈沖控制捕食模型.在帶有Beddington-DeAngelis功能關(guān)系的捕食模型中,B-D關(guān)系影響了捕食者的捕食能力,從而在營養(yǎng)關(guān)系的表達(dá)式上表現(xiàn)出來S.Nundlol-la, L.Mailleretb和F.Grognarda(2010)首先研究了帶有一種特殊的脈沖控制的Beddington-DeAngelis功能關(guān)系的捕食模型,建立了周期性釋放捕食者產(chǎn)生的平凡周期解,并得到了平凡周期解的全局穩(wěn)定性條件.我們著重研究了該系統(tǒng)的持久性并尋找到了非平凡周期解.結(jié)果表明,在此類脈沖控制中,當(dāng)脈沖控制率大于某個(gè)特定的臨界值或者脈沖釋放周期小于對(duì)應(yīng)的臨界值時(shí),系統(tǒng)的對(duì)應(yīng)于食餌滅絕的周期解是局部漸近穩(wěn)定的.我們給出了系統(tǒng)的持久存在性條件,并且得到隨著參數(shù)變化當(dāng)食餌滅絕的周期解失去它的穩(wěn)定性時(shí)系統(tǒng)出現(xiàn)了穩(wěn)定的非平凡的周期解.最終我們根據(jù)平凡周期解和非平凡周期解的穩(wěn)定性給出了食餌種群密度的完整控制策略.第三章我們研究了不育昆蟲繁殖技術(shù)(SIT)在一般的捕食模型中如何成功的控制或殺滅自然昆蟲.我們?cè)谶B續(xù)型模型中討論了不育昆蟲技術(shù)在捕食模型中應(yīng)用的可行性.這個(gè)模型是以Murray(2002)的模型為基礎(chǔ).在Murray的著作中,不育昆蟲的數(shù)量總被保持成一個(gè)常數(shù).我們將Murray的模型擴(kuò)展成為了一個(gè)一般的捕食模型并進(jìn)行了理論分析,討論了模型的動(dòng)力學(xué)行為并計(jì)算出了使自然昆蟲滅絕的臨界值.最終我們知道,帶有不育昆蟲繁殖技術(shù)(SIT)的捕食模型具有非常豐富的,有趣的以及復(fù)雜的動(dòng)力學(xué)性質(zhì),其有可能出現(xiàn)多個(gè)穩(wěn)定平衡點(diǎn),鞍結(jié)點(diǎn)分支,Hopf分支等.它具有兩個(gè)重要的臨界值：固定的SIT臨界值和捕食臨界值,它們的大小比較決定了捕食者在不育昆蟲技術(shù)中的作用.第四章我們研究了脈沖控制在不育昆蟲繁殖技術(shù)(SIT)中的應(yīng)用,為了計(jì)算不育昆蟲釋放的數(shù)量,我們引入了這個(gè)模型,一個(gè)周期或脈沖釋放的模型,這個(gè)模型是由一個(gè)帶有脈沖部分的常微分方程組組成.跟連續(xù)型系統(tǒng)相比,脈沖的SIT在實(shí)際生產(chǎn)中更容易被操作.我們計(jì)算出了自然昆蟲滅絕的臨界值并得到了脈沖SIT模型平凡周期解的全局穩(wěn)定性條件.
[Abstract]:The basic theories and methods of impulsive differential equations have been fully developed and perfected in recent decades. Because some natural phenomena will "jump" at a specific time, such as the seasonal reproduction or migration of animals, they can not be modeled by continuous or discrete micro equations, but the pulse equation can be very good. To describe these biological phenomena, the first use of pulse method is that Beverton and Holt (1957) want to use discrete models to approach a continuous time Logistic model on the basis of the 1.5 discrete model. From then on, the pulse model has appeared in almost every field of life science. This paper mainly studies the pulse control in the birth. Two applications in physical mathematics. The first application is to study the complete control strategy of the prey population density in the impulsive predator-prey model with the Beddington-DeAngelis function relationship by studying the persistence of the system and the stable non trivial periodic solution. The second application is to study how to make use of the sterile insect propagation. SIT has successfully killed natural insects in two types of predator-prey models in continuous and pulse types. This paper is divided into four chapters. The first chapter introduces some basic concepts and basic theories of impulsive differential equations, including the definition of impulsive differential equations, common classification and the concepts of stability. Two important theorems of differential equations: the comparison theorem and the Floquet theorem of linear periodic equations. After that, we introduce the application of impulsive differential equations in detail. Up to now, the impulsive differential equations have been widely applied to infectious diseases, medicine and population models. The second chapter mainly studies one with Beddi The impulse control model of the ngton-DeAngelis function relationship. In a predator model with Beddington-DeAngelis function, the B-D relationship affects the predator's predatory ability, thus showing S.Nundlol-la in the expression of the nutrition relation, L.Mailleretb and F.Grognarda (2010) first studies with a special pulse control. The predator-prey model of the Beddington-DeAngelis function relationship is made, the periodic solution generated by the periodic release predator is established, and the global stability condition of the ordinary periodic solution is obtained. We focus on the persistence of the system and find the nontrivial periodic solution. The periodic solution corresponding to the extinction of the prey is locally asymptotically stable when a particular critical value or the pulse release period is less than the corresponding critical value. We give the persistent existence condition of the system and obtain the stability of the system as the periodic solution of the extinction of the prey loses its stability. In the end, we give a complete control strategy for the population density of prey on the basis of the stability of the ordinary periodic solution and the nontrivial periodic solution. In the third chapter, we have studied how the sterile insect propagation technique (SIT) successfully controlled or killed the natural insects in the general predator model. This model is based on the model of Murray (2002). In the work of Murray, the number of sterile insects is always kept as a constant. We extend the model of Murray into a general predation model and carry out a theoretical analysis, and discuss the dynamic behavior of the model and discuss the dynamic behavior of the model and discuss the dynamic behavior of the model. We have calculated the critical value of the extinction of natural insects. Finally, we know that the predation model with the sterile insect propagation technique (SIT) is very rich, interesting and complex dynamics, and it may have many stable equilibrium points, the saddle node branch, the Hopf branch, etc. it has two important critical values: fixed SIT presence. In the fourth chapter, we studied the application of pulse control in the sterile insect propagation technology (SIT). In order to calculate the number of sterile insects released, we introduced the model type, a cycle or a pulse release model, which is the model. It is composed of a group of ordinary differential equations with a pulse part. Compared with the continuous system, the SIT of the pulse is easier to be operated in the actual production. We calculate the critical value of the extinction of natural insects and obtain the global stability conditions of the ordinary periodic solution of the pulse SIT model.
【學(xué)位授予單位】：吉林大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2015
【分類號(hào)】：O175
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本文編號(hào)：2101987