本文選題：Gilpin-Ayala種群增長模型 + 周期解　； 參考：《陜西師范大學》2015年碩士論文

【摘要】：可再生生物資源(如森林資源、牧業(yè)資源、漁業(yè)資源等)能夠根據(jù)自身的特點,借助于自然循環(huán)和生物自身生長、發(fā)育或者繁殖而不斷更新,并保持一定的儲量.現(xiàn)代社會由于人類以高投入、高消耗、高污染的粗放型方式謀求經(jīng)濟的快速增長,社會生產(chǎn)對生物資源的攝取消耗能力己遠遠超過自身其更新循環(huán)能力,造成資源枯竭.因此,可再生生物資源優(yōu)化管理問題的研究和資源的可持續(xù)發(fā)展密切相關(guān).近年來關(guān)于如何利用有限的可再生資源,實現(xiàn)其可持續(xù)發(fā)展的相關(guān)問題,引起了許多學者的關(guān)注.某些情形下,人們可能希望在保證物種和生態(tài)環(huán)境可持續(xù)發(fā)展的前提下,追求經(jīng)濟凈收益的最大化,而在有些情況下,可能需要研究某時間段內(nèi)在收獲量一定的前提下,使得種群在周期末獲得最大的剩余量等問題.常見的收獲策略包括連續(xù)收獲、脈沖收獲、時間以及時間依賴收獲等.為了定量地研究各種關(guān)鍵因素對種群系統(tǒng)可持續(xù)發(fā)展的影響,確切地刻畫各種控制策略并評估其有效性,需要以人類對生物資源的消耗為背景,建立數(shù)學模型對種群系統(tǒng)的發(fā)展進行描述,通過對數(shù)學模型進行系統(tǒng)的理論分析,獲得資源自身生長規(guī)律與人類收獲或者放養(yǎng)等開發(fā)行為之間的關(guān)系.評估、分析和預(yù)測在不同的參數(shù)條件下種群系統(tǒng)的變化趨勢,為資源管理者能夠合理管理可再生資源提供理論指導(dǎo)以及決策依據(jù).本文主要應(yīng)用脈沖及連續(xù)微分系統(tǒng)的極值原理以及種群動力學的基本理論討論具有脈沖收獲或連續(xù)收獲的種群系統(tǒng)的優(yōu)化控制問題.研究結(jié)果不僅豐富了種群動力學的理論研究,而且為實際生態(tài)問題的解決提供決策依據(jù),有一定的實際意義.本文的主要研究內(nèi)容和成果包括以下幾個方面：(1)研究在周期環(huán)境下的Gilpin-Ayala種群系統(tǒng)的連續(xù)收獲問題.對Gilpin-Ayala種群系統(tǒng)進行比例收獲,以一個周期內(nèi)總收獲量最大為目標研究獲得了最優(yōu)收獲策略.首先研究系統(tǒng)周期解的存在性及穩(wěn)定性,并利用關(guān)于微分系統(tǒng)的極值原理和一些分析技巧,獲得了最優(yōu)控制策略及最優(yōu)收益的確切表達式.還研究了一類非自治Gilpin-Ayala種群系統(tǒng)在有限時間內(nèi)的最大收益問題,對種群系統(tǒng)進行連續(xù)收獲,目的是在給定的時間區(qū)間內(nèi)使得最終的總收益最大.根據(jù)優(yōu)化問題的極值原理及最速逼近原理,對于不同的初值條件,研究獲得了由分段函數(shù)表達的最優(yōu)收獲函數(shù)以及最大收獲量.(2)研究一類由周期Gilpin-Ayala模型描述的脈沖收獲系統(tǒng)的優(yōu)化控制問題.在固定時刻對種群實施比例收獲,考慮收獲成本因素,以最大經(jīng)濟凈收益為管理目標,研究不同的收獲努力量對經(jīng)濟收益的影響,并獲得最優(yōu)的收獲策略.首先研究該系統(tǒng)周期解的存在唯一性和全局漸近穩(wěn)定性.進一步利用脈沖微分方程的極值原理,得到最優(yōu)收獲策略滿足的數(shù)值方程組.(3)研究在有限時間區(qū)間內(nèi),由Gilpin-Ayala模型描述的線性脈沖收獲系統(tǒng)的優(yōu)化控制問題.收獲函數(shù)包括比例收獲和常量收獲,在收獲量一定的條件下,以種群在周期末的存儲量最大為目標函數(shù),對于任意給定的初值條件,研究不同收獲時刻對種群的影響以獲得最優(yōu)的收獲策略.首先通過脈沖微分方程的極值原理得到最優(yōu)收獲時刻應(yīng)滿足的必要條件,討論了在時間周期足夠長的條件下具有多次脈沖收獲的最優(yōu)收獲策略；進一步考慮了在給定時間范圍內(nèi)的最大收獲次數(shù)及相應(yīng)的最優(yōu)收獲策略問題.
[Abstract]:Renewable biological resources (such as forest resources, animal husbandry resources, fishery resources, etc.) can be constantly updated with the help of their own characteristics, with the help of natural circulation and biological growth, development or reproduction, and maintain certain reserves. In modern society, the rapid economic growth is sought by human beings with high input, high consumption and high pollution. The ability of social production to consume biological resources is far more than its own regeneration and recycling capacity, resulting in the exhaustion of resources. Therefore, the research on the optimization and management of renewable biological resources is closely related to the sustainable development of resources. In recent years, the problems of how to make use of limited renewable resources to achieve their sustainable development are discussed. In some cases, people may want to maximize the net income of the economy under the premise of ensuring the sustainable development of the species and the ecological environment. In some cases, it may be necessary to study the maximum surplus of the population at the end of the cycle at the end of a certain period of harvest. The common harvesting strategies include continuous harvest, pulse harvest, time and time dependence. In order to quantitatively study the impact of various key factors on the sustainable development of the population system, accurately depict various control strategies and evaluate their effectiveness, a mathematical model is needed to establish a mathematical model to the background of human resource consumption. The development of the group system is described. Through systematic theoretical analysis of the mathematical model, the relationship between the growth law of resources and the development behavior of human harvest or breeding is obtained. The evaluation, analysis and prediction of the change trend of the population system under different parameter conditions can be made for the resource managers to manage the renewable resources reasonably. This paper mainly applies the extremum principle of pulse and continuous differential system and the basic theory of population dynamics to discuss the optimization control problem of the population system with pulsing harvest or continuous harvest. The results not only enrich the theory of population dynamics, but also solve the problem of the actual ecological problem. The main research content and results of this paper include the following aspects: (1) study the continuous harvest problem of the Gilpin-Ayala population system under the periodic environment. The proportion of the Gilpin-Ayala population system is harvested, and the optimization of the maximum total harvest in a cycle is obtained. First, we study the existence and stability of the periodic solution of the system, and obtain the optimal control strategy and the exact expression of the optimal income by using the extreme value principle and some analytical techniques of the differential system. The maximum income problem of a class of Nonautonomous Gilpin-Ayala population systems in the finite time is also studied. The purpose of continuous harvest is to make the ultimate total profit in a given time interval. According to the extreme value principle and the maximum approximation principle of the optimization problem, the optimal harvest function and the maximum yield expressed by piecewise function are obtained for different initial value conditions. (2) a class of pulse described by periodic Gilpin-Ayala model is studied. The optimal control problem of the harvest system is obtained. In the fixed time, the proportion of the population is harvested at the fixed time, the harvest cost factor is considered, and the maximum economic net income is taken as the management goal. The effects of the different harvest efforts on the economic returns are studied and the optimal harvesting strategy is obtained. By using the extreme value principle of impulsive differential equations, we get the numerical equations which are satisfied by the optimal harvesting strategy. (3) the optimization control problem of the linear pulse harvest system, which is described by the Gilpin-Ayala model in the limited time interval, is studied. The harvest function includes the proportional harvest and the constant harvest, and the species under certain harvest conditions. The maximum storage capacity at the end of the period is the objective function. For any given initial value condition, the effect of the different harvest time on the population is studied to obtain the optimal harvest strategy. First, the necessary conditions for the optimal harvest time should be obtained by the extreme value principle of the impulsive differential equation. The optimal harvesting strategy with multiple pulses is considered, and the maximum harvesting times and corresponding optimal harvesting strategies in a given time range are further considered.

【學位授予單位】：陜西師范大學
【學位級別】：碩士
【學位授予年份】：2015
【分類號】：O175

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